Nonlinear nanoscale localization of magnetic excitations in atomic lattices

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NONLINEAR NANOSCALE LOCALIZATION OF MAGNETIC EXCITATIONS IN ATOMIC LATTICES

R. LAI, A.J. SIEVERS Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, 14853-2501, USA

AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO

Physics Reports 314 (1999) 147}236

Nonlinear nanoscale localization of magnetic excitations in atomic lattices R. Lai, A.J. Sievers* Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, 14853-2501, USA Received September 1998; editor: D.L. Mills Contents 1. Introduction 1.1. Historical background 1.2. Magnetic lattices 1.3. Overview and organization 2. Ferromagnetic chain with nearest-neighbor exchange interaction and easy-plane on-site anisotropy 2.1. The nearest-neighbor 1-D model and equations of motion 2.2. Stationary intrinsic localized spin wave modes (ILSMs) 2.3. Travelling ILSMs 2.4. Interaction of ILSMs with magnetic defects 3. Isotropic ferromagnetic chain with nearest- and next-nearest-neighbor exchange interactions 3.1. The 1-D model Hamiltonian 3.2. Stationary intrinsic localized spin wave resonances (ILSRs) 3.3. Conditions for the occurrence of ILSRs 3.4. Translating ILSRs 4. Antiferromagnetic chain with on-site easy-axis anisotropy 4.1. Equations of motion 4.2. Stationary intrinsic localized spin wave gap modes (ILSGs) 4.3. Moving gap modes 4.4. Weak nonlinearity limit 4.5. Stability of stationary ILSGs

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5. Antiferromagnetic chain with on-site easy-plane or biaxial anisotropy 5.1. The model Hamiltonian 5.2. Stationary intrinsic localized spin wave modes 5.3. Existence conditions 6. Modulational instability of an extended nonlinear spin wave in an easy-axis antiferromagnet 6.1. Travelling nonlinear extended waves 6.2. Modulational instability of extended waves 6.3. Comparison between numerical simulations and analytical results 6.4. Modulational instability recurrence 7. Production of intrinsic localized spin wave modes and the CW driving of antiferromagnetic instabilities 7.1. Creation of intrinsic localization 7.2. Uniaxial FeF  7.3. Uniaxial FeCl  7.4. Biaxial (C H NH ) CuCl     8. Conclusions 8.1. Summary 8.2. Other systems and future prospects Acknowledgements References

* Corresponding author. Tel.: #1 607 2556422; fax: #1 607 2556428; e-mail: [email protected] 0370-1573/98/$ } see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 8 ) 0 0 0 9 0 - 8

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Abstract Reviewed here is the nonlinear intrinsic localization expected for large amplitude spin waves in a variety of magnetically ordered lattices. Both static and dynamic properties of intrinsic localized spin wave gap modes and resonant modes are surveyed in detail. The modulational instability of extended nonlinear spin waves is discussed as a mechanism for dynamical localization of spin waves in homogeneous magnetic lattices. The interest in this particular nonlinear dynamics area stems from the realization that some localized vibrations in perfectly periodic but nonintegrable lattices can be stabilized by lattice discreteness. However, in this rapidly growing area in nonlinear condensed matter research the experimental identi"cation of intrinsic localized modes is yet to be demonstrated. To this end the study of spin lattice models has de"nite advantages over those previously presented for vibrational models both because of the importance of intrasite and intersite nonlinear interaction terms and because the dissipation of spin waves in magnetic materials is weak compared to that of lattice vibrations in crystals. Thus, both from the theoretical and the experimental points of view, nonlinear magnetic systems may provide more tractable candidates for the investigation of intrinsic localized modes which display nanoscale dimensions as well as for the future exploration of the quantum properties of such excitations.  1999 Elsevier Science B.V. All rights reserved. PACS: 46.10.#z; 63.20.Pw; 63.20.Ry; 75.10.Hk; 75.30.Ds; 75.30.!m Keywords: Intrinsic localized spin wave modes; Antiferromagnets; Nonlinear dynamics; Discrete lattices; Nanoscale

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R.¸. would like to dedicate this review to his wife, ¸inda Huang, for her support and A.J.S. to Paul Camenzind for showing him the way.

1. Introduction 1.1. Historical background Both nonlinearity and lattice discreteness have played important roles in many branches of condensed matter physics, including crystal [1,2] and spin dynamics [3}5]. The traditional approach in solid state physics has been to treat them separately. Lattice discreteness is usually manifested by the existence of an upper bound on the plane wave spectrum whereas nonlinear terms are typically assumed to be small and treated as a perturbation to the harmonic approximation, leading to both damping and a frequency shift of plane-wave excitations in an otherwise linear lattices [6,7]. This approach has successfully explained most of the phenomena in condensed matter physics involving weak nonlinearity. The nonlinearity cannot be treated as a perturbation in all cases, as evidenced by the appearance of domain walls, kinks and solitons [8]. An important advance in dealing with nonlinearity in condensed matter physics has been the introduction of the soliton as a new type of elementary excitation. It has been suggested [9] that solitons, which had been extensively studied [10] in #uids, plasmas and optics, may be present as thermal excitations in quasi-one-dimensional materials as well, and should be treated as a new type of elementary excitation in addition to spatially extended plane wave-like modes. Since then nonlinear excitations have attracted wide interest in many branches of condensed matter physics, for example, in lattice dynamics [11], electronic polymers [12], molecular crystals [13] and magnetic systems [14,15]. Classically, these nonlinear excitations are solutions of integrable nonlinear partial di!erential equations [16,17] which can be used to describe some realistic physical systems within the continuum approximation. The paradigm of such nonlinear excitations has provided a rather useful framework for investigating a large number of phenomena in condensed matter physics, especially the thermodynamic and transport properties of low dimensional materials [18,19]. In particular, solitary excitations in one-dimensional magnetic systems have been extensively investigated. In general, there are no exact solutions to the equations of motion derived from Heisenberg ferro- and antiferromagnetic Hamiltonians. The classical continuum limit approximation of 1-D easy-plane magnets demonstrates the existence of sine-Gordon kink excitations in addition to spin waves [20,21]. Another type of continuous nonlinear excitation that can be supported by a magnetic chain is a breather, which can be visualized as a magnon bound state [22,23]. Excellent reviews of the theoretical and experimental investigation of magnetic solitary excitations have been given [14,15]. Perhaps because these nonlinear excitations have in"nite lifetime in integrable systems but are found to be unstable in non-integrable continuous systems, historically, most attention in nonlinear dynamics was devoted to integrable continuous models. Among these integrable models are the (1#1)-dimensional sine-Gordon equation, the KdV equation, and the nonlinear SchroK dinger equation, to name a few of the best-known examples [16,17,24]. In strongly nonlinear discrete systems, the topic of this review, the spatial size of a nonlinear excitation can become comparable to the lattice spacing; hence, the discreteness of the underlying physical systems is expected to have

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a signi"cant e!ect on the properties of nonlinear excitations in condensed matter physics. However, the study of nonlinear excitations in discrete lattices has been relatively rare since, except for the Toda lattice [11] and the Ablowitz}Ladik lattice [25], few discrete lattices are integrable and even these discrete lattice models appear to be integrable by construction rather than motivated by realistic physical systems. A major advance of the theory of nonlinear excitations in discrete lattices in late 80s and early 90s was the discovery that some localized vibrations in perfectly periodic but non-integrable lattices can be stabilized by lattice discreteness [26}29]. This realization has led to extensive studies of the features associated with intrinsic localization in various nonlinear nonintegrable lattices, and it has proven to be a conceptual and practical breakthrough [30}34]. In the literature these localized excitations are called either `intrinsic localized modesa (ILMs) with the emphasis on the fact that their formation involves no disorder and that they extend over a nano-length scale, or `discrete breathersa with the emphasis on their similarity to exact breather soliton solutions in nonlinear continuum theories. Although it is well known that no bound state or localized mode exists in a continuum 3D space for a scalar "eld [35], ILMs in discrete lattices are not con"ned to certain lattice dimensions [29,36}39]. These unusual modes can occur at any site and may be stationary or move slowly through the lattice [30]. One key element for realistic lattices is the existence of gapped linear dispersion relations. Depending on the nature of the interparticle forces, a variety of interesting ILMs can exist, with spatial mode patterns ranging in type from alternating (zone center) to staggered (zone boundary). The earlier work of Sievers and Takeno [27], and Page [28] has recently been formalized in terms of a number of useful existence and stability criteria [32], and many physically exciting contexts are currently emerging } in nonlinear crystal dynamics [30,34], magnetic systems [40,41], electron}phonon systems [42], molecular biophysics [43], friction [44], etc. The potential for these self-localized oscillatory excitations in equilibrium and nonequilibrium classical and quantum discrete lattices is now extensive and this thrust is becoming a major activity in nonlinear condensed matter research. The challenge at this writing is that these excitations are yet to be unequivocally identi"ed in experiment. 1.2. Magnetic lattices In magnetic systems, both exchange interactions between spins and spin anisotropy (either single-ion or dipole}dipole) are intrinsically nonlinear. Since the strength of the internal e!ective "eld acting on a spin always decreases with increasing the spin deviation from its equilibrium direction, the nonlinearity in magnetic systems is generally soft. Fig. 1 provides a qualitative illustration of how the nonlinearity can introduce new e!ects in the spin wave excitation spectrum for a discrete lattice. The dispersion curve associated with small amplitude linear excitations for a two sublattice easy axis antiferromagnet is illustrated in the center of the "gure. The eigenvector for one of the doubly degenerate uniform precession magnetic dipole active plane wave modes is shown in the top panel. An essential point of this review is to demonstrate that if su$cient transverse amplitude can be given to a particular spin then a stationary localized spin wave excitation may appear such as is illustrated in the bottom panel of the "gure. As we shall see this excitation is stabilized by the combination of the nonlinearity and the discreteness of the lattice. Its eigenvector may appear familiar since it extends over the same nanoscale lengths as those associated with the well studied defect-induced localized spin wave modes [45}48].

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Fig. 1. Schematic picture of possible excitations in a 1-D lattice of antiferromagnetically coupled spins with easy uniaxis spin anisotropy. The center panel shows the doubly degenerate dispersion relation for small amplitude excitations of the spins. One of the magnetic dipole active eigenvectors associated with the uniform mode is shown in the top frame. The bottom panel shows a particular eigenvector that can occur when a large transverse amplitude appears at an individual spin site. The larger the amplitude of the transverse excitation of this spin the lower the mode frequency below the bottom of the plane wave spectrum and the more localized the spatial extent of the excitation. One reason for the interest in these nonlinear excitations is that the scale of the localization of such intrinsic modes can be on a scale comparable to the lattice spacing itself.

It has been known since the 1960s that the destruction of the translational symmetry of a magnetic crystal may produce both localized vibrational and localized magnetic excitations [47]. The properties of these linear localized magnetic excitations in insulating doped antiferromagnetic systems were "rst studied with a number of experimental techniques, including far infrared [45], Raman scattering [49], optical #uorescence [50], and neutron scattering [51]. Of the magnetic defect systems in which far infrared active localized modes have been observed, perhaps MnF : Fe> has been examined in the greatest detail. One reason is that both single and localized  magnetic pair modes occur at frequencies where there is almost no interference from the host phonon absorption spectrum of the tetragonal MnF host. The far infrared spectra for two  di!erent crystal geometries are shown in Fig. 2. The plane wave far IR-active modes of the host crystal, consisting of the antiferromagnetic resonance (AFMR) at the lowest frequency and the electric}dipole active two magnon absorption associated with a pair of magnons of equal wavevector excited on opposite sublattices, gives absorption peaks at 106 cm\ for E#c and at 100 cm\ for ENc. A localized phonon mode occurs at 113 cm\. The impurity-induced magnetic excitations consist of a localized magnetic mode at 94.9 cm\ and a pair excitation at 144.9 cm\ which

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Fig. 2. Spectra showing intrinsic and impurity-induced absorption lines in MnF : Fe>. The sample temperature is  1.2 K. The relative strengths of the di!erent modes are drawn qualitatively to scale for a 0.2 mol% FeF doping. The  identi"cation of the di!erent excitations is given in the "gure. Two spectra are required to characterize the far-IR absorption in this tetragonal crystal (after Ref. [52]).

involves a simultaneous excitation of the spin deviations associated with the Fe> impurity spin and a shell mode centered on the neighboring ions [52]. The result is a fairly complex array of localized modes, even for a simple point defect substitution. A fundamental di!erence between intrinsic localized spin wave modes and the defect-induced localized spin modes shown in the spectra presented in Fig. 2 is that impurity modes are trapped at the defect site whereas, according to Bloch's theorem, intrinsic localized modes must be able to move through the lattice. Another di!erence is that impurity modes can appear at frequencies above or below the linear spin wave spectrum while the intrinsic localization result presented in Fig. 1 favors low-frequency modes because of the intrinsically soft nonlinearity. This frequency red shift result is somewhat similar to that found for nonlinear vibrational excitations in crystal lattices which contain realistic intersite potentials. Note that because of the large size of the intersite cubic anharmonicity, the e!ective vibrational potential is always soft [31,53}56]. 1.3. Overview and organization In the search for ways to study intrinsic localized modes and produce nano-scale localized excitations experimentally, spin lattice models have de"nite advantages over vibrational models. Relatively simple spin lattice models can yield more complex linear dispersion curves because of the importance of both intrasite and intersite interaction terms. The combination of these dispersion

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curves with the soft nonlinearity in magnetic chains can generate both nonlinear gap modes [40] and resonant modes [57,58] in the spin wave spectrum. Another important di!erence between crystal models and spin models is that realistic two-body interactions between atoms involve cubic nonlinear potential terms [54] which produce a static distortion centered at the localized excitation whereas nonlinear interactions in simple two sublattice spin lattice models exclude odd nonlinear terms, again making the nonlinear dynamical models somewhat simpler to implement. Furthermore, the dissipation of spin waves in magnetic materials is usually weak compared to that of lattice vibrations in crystals and such weak dissipation should make it easier experimentally to drive the uniform spin wave mode into a highly nonlinear regime. Hence, there is de"nite value in "rst exploring experimentally nonlinear localized excitations in magnetic systems with small damping before turning to consider other types of crystal systems which involve large damping. Thus, both from the theoretical and experimental points of view, magnetic systems may provide more tractable candidates for the study of the formation of intrinsic localized modes which display nano-scale lengths and also for the exploration of the quantum properties of such excitations. In the "rst paper focusing explicitly on intrinsic localized spin wave modes in discrete lattices Takeno and Kawasaki [37] argued that coherent quantum states should be the appropriate starting point for large amplitude collective modes like those associated with intrinsic localization. They noted that one should employ the coherent-state ansatz for the eigenfunction W(t) of the Hamiltonian H so that (1.1) W(t)"“ exp[!()("a "#"b ")] exp(a a>#b b>)"02 , L L L L L  L L where "02 is the vacuum state of the spin boson system using a standard notation. In principle, the time dependent variational principle



d dt1W(t)"i (R/Rt)!H"W(t)2"0

(1.2)

gives the nonlinear quantum mechanical equations of motion. However, since a solution to such equations has not yet been demonstrated these authors and subsequent workers in the "eld have been forced to work with the classical spin equations of motion for discrete lattices which contain the corresponding nonlinearities. This review details such classical studies focusing on the theoretical properties of intrinsic localized spin wave modes (ILSMs) in ferro- and antiferromagnetic lattices. The existence condition for ILSMs in these non-integrable models is obtained from the nonlinear SchroK dinger-type equations that describe the spin wave dynamics in the continuum limits and from the linear stability analysis of the corresponding extended nonlinear plane waves. For many examples described here the stability and mobility of ILSMs have been investigated using both analytical techniques and molecular dynamics (MD) simulations. In general, spin lattices cannot support localized modes above the linear spin wave spectrum but there are exceptions which can be obtained with the application of a dc external "eld such as for the special con"guration of a ferromagnetic chain where spins are forced to align along the hard on-site anisotropy axis by a strong external "eld [41]. The next section explores the properties of this particular arrangement. When both the external "eld and the anisotropy are turned o! Section 3

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shows that the simplest ferromagnetic system that can support ILSMs is a onedimensional ferromagnetic chain with both nearest-neighbor (NN) and next-nearest-neighbor (NNN) exchange interactions. Long-lived in-band intrinsic localized spin wave resonances (ILSRs) may exist at frequencies near the Brillouin zone boundary when the strength of the NNN exchange interaction relative to the NN exchange interaction exceeds a speci"c threshold. However, as such ILSRs in isotropic ferromagnetic chains possess a zero net ac magnetic moment, they are not magnetic dipole active. Although a gap may appear in the linear spin wave spectrum of an isotropic ferromagnetic chain when an external dc magnetic "eld is applied, no ILSGs can exist in such a gap since the external "eld only contributes a linear term in the equations of motion, producing a frequency shift of the entire spin wave spectrum. For this reason, we consider only spin chains in the absence of external dc "elds in Section 4 where it is shown that the existence of intrinsic localized spin wave gap modes (ILSGs) in anisotropic antiferromagnets is a natural consequence of the intrinsic softness of the exchange interaction and the on-site anisotropy. Since a large number of ordered magnetic spin systems appear as anisotropic antiferromagnets and the antiferromagnetic resonance (AFMR) mode is often IR-active, these ILSGs are of particular interest. In antiferromagnetic chains with on-site easy-axis anisotropy, both single- and double-peaked ILSGs are found without invoking the usual rotating wave approximation (RWA) necessary in analytical work on vibrational problems [27]. As the maximum spin deviation at the mode center or the relative strength of the anisotropy increases, the degree of localization of the ILSGs increases and the frequency drops further into the spin wave gap. Both linear stability analysis and MD simulations demonstrate that only single-peaked ILSGs are stable against perturbations whereas a perturbed double-peaked ILSG evolves into a single-peaked one. Moving ILSGs are also discussed. It is demonstrated that the mobility of an ILSG decreases as the degree of localization increases so that strongly localized modes can become pinned at low temperatures. Section 5 examines the nonlinear properties of the more complex but more general easy-plane anisotropy and biaxial anisotropy antiferromagnets. For antiferromagnetic chains with on-site easy-plane anisotropy, the spin wave dispersion curve is separated into two distinct branches, one of which extends to zero frequency. In this hard uniaxial case, intrinsic localized spin wave resonances (ILSRs) can exist with frequencies within the linear spin wave spectrum of the lower branch when the upper branch of the dispersion curve has positive curvature at the center of Brillouin zone. The key feature in this nonlinear dynamics problem is the polarization di!erence between the two plane wave branches. The smaller the frequency of the q"0 mode in the upper branch, the less strongly coupled the resulting ILSR is to the other branch of the plane wave spectrum. Numerical simulation studies demonstrate that the ILSR excitations are long-lived with regard to a random noise perturbation. When the rotational symmetry in the easy-plane is broken and the anisotropy becomes biaxial, ILSGs, in addition to ILSRs, can also exist in the gap below the lower branch of linear spin wave spectrum. The question as to the best way of experimentally creating such an atomic scale large-amplitude excitation in a homogeneous but discrete lattice is still open although, as we shall demonstrate, the existence, localization and stability of ILMs in a variety of discrete lattices have been extensively investigated numerically. To "nd a method for generating ILSMs, the modulational instability mechanism for extended nonlinear spin waves in easy-axis antiferromagnetic chains has been reviewed in Section 6, both analytically within the framework of linear stability analysis and

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numerically by means of MD simulations. The exploration of the in#uence of damping on these modes in Section 7 is su$ciently encouraging that the remainder of the section is devoted to exploring the possible generation of ILSMs for particular sublattice antiferromagnets via MD simulations. ILSM-like excitations are generated with realistic parameters. Since ILSMs in antiferromagnets have frequencies below the extended spin waves and are magnetic dipole-active, it is anticipated that their signature can be directly probed by homodyne detection methods. Finally, the main conclusions and future prospects are given in Section 8.

2. Ferromagnetic chain with nearest-neighbor exchange interaction and easy-plane on-site anisotropy Intrinsic localized spin wave modes (ILSMs) are expected to exist in perfect but nonintegrable discrete magnetic chains because of the intrinsic nonlinearity in the exchange and anisotropy interactions. Here we consider the simplest magnetic system that can possibly support ILSMs, namely, ferromagnetic chains. Although no ILSM can occur in isotropic ferromagnetic chains with only nearest-neighbor exchange interaction because the nonlinearity in the Heisenberg exchange interaction is intrinsically soft, both even-parity and odd-parity ILSMs appear in Heisenberg ferromagnetic chains with easy plane anisotropy [41,59] when a strong magnetic "eld is applied perpendicular to this plane. Like their vibrational counterpart [30}34], these highly localized ILSMs involve only a few lattice sites and have amplitude-dependent frequencies which lie outside the harmonic plane-wave bands. The existence of such ILSMs for a ferromagnetic chain with nearest-neighbor interactions requires that the strength of the single-ion anisotropy and the external magnetic "eld exceed certain critical values so that the resulting ILSM frequencies can appear above the linear spin wave band. The production of ILSMs by the application of an external magnetic "eld makes use of an experimental parameter not available with crystal lattice systems and we review some of the "ndings below. 2.1. The nearest-neighbor 1-D model and equations of motion The Hamiltonian for the ferromagnetic chain with on-site easy-plane anisotropy [41,48,59] can be written as H"!2J S ) S #D (SX)!H SX! h (SV cos ut!SW sin ut) , (2.1) L L> L  L L L L L L L L where J'0 is the exchange interaction constant, D is the uniaxial anisotropy constant, and H is  the magnitude of the external "eld applied along the zL -axis. The last term in this equation describes a circularly polarized "eld of strength h and frequency u applied in the xy-plane. Positive values of L D correspond to the case of easy-plane anisotropy. To arrive at the interesting nonlinear dynamical regime the external "eld H is taken to be su$ciently large so that in the ground state all spins are  ordered along the external "eld direction, that is, along the hard anisotropy axis. This spin arrangement requires that H 'DS in Eq. (2.1). This one-dimensional model has been demon strated successfully to describe the spin dynamics of the linear chain compound CsFeCl [60]. 

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Some magnetic superlattice structures [61] can also be described by the energy funcitonal presented in Eq. (2.1). (Note that for CsFeCl , a rather large applied magnetic "eld, in the range of  40 T would be required to align the spins parallel to the hard axis.) To investigate spin deviations from the ground state, rotational symmetry is incorporated in the de"nition of the appropriate spin variables in the usual way so that s!"(SV$iSW)/z and sX"SX/S , (2.2) L L L L L where S is the magnitude of spin. In these circularly polarized coordinates, the equations of motion become i

ds> L "H s>#2JS[s>(sX #sX )!sX(s> #s> )]!2DSsXs>!SsXh e\ SR , L L L> L\ L L> L\ LL L L dt

(2.3)

where for classical spins sX"(1!"s>" and for convenience both and the gyromagnetic ratio L L are set to 1. 2.2. Stationary intrinsic localized spin wave modes (I¸SMs) Eq. (2.3) admits time-periodic stationary solutions of the form s>"s e\ SR with real timeL L independent coe$cients s . In these stationary modes all spins are engaging in circular precession L on a cone making an angle of h "sin\(s ) with respect to the z-axis. From Eq. (2.3) the system of L L coupled nonlinear time independent equations for amplitude s becomes L Xs "s ((1!s #(1!s )!(s #s )(1!s!2As (1!s!c (1!s , L L L> L\ L> L\ L L L L L (2.4) where the various parameters are de"ned by the following equations: X"(u!H )/2JS ,  A"D/2J

(2.5) (2.6)

and c "h /2JS . (2.7) L L Stationary ILSMs in the absence of a driving "eld (c "0) have been studied by Wallis et al. [41] L where localized modes with both even and odd parity are found. There is a critical value of A, the ratio of the anisotropy to the exchange, below which ILSMs do not exist. Fig. 3 shows the linear spin wave band for two di!erent values of A (solid curves) and the corresponding frequency positions of nonlinear plane wave modes at the zone boundary (dashed lines). For the case of the isotropic ferromagnetic chain (A"0), the e!ect of the nonlinearity in the exchange interaction is to force the frequency of the large amplitude zone boundary mode down into the linear continuum, hence no localized modes can exist. The in#uence of the nonlinearity on the frequency of large amplitude zone-boundary mode is however quite di!erent when A'2.0. In this case the nonlinear zoneboundary mode frequency increases with increasing amplitude, leaving room for a localized modes to exist below it but above the linear plane wave spectrum. For the case of A"10, Fig. 3 illustrates where a nonlinear localized mode (dot-dashed line) appears above the plane wave spectrum. The critical value of A"2.0 has a straightforward explanation which is consistent with the intrinsic nonlinear softness of the exchange and anisotropy interactions. The total "eld acting on

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Fig. 3. Frequency X versus wave-vector-lattice-constant product qa for the linear spin}wave band (solid curve) for A"0 and 10. Frequencies are shown for the nonlinear bulk mode at the zone boundary (dashed line) and the nonlinear localized mode (dot-dashed line) (after Ref. [41]).

each spin consists of three parts: the applied "eld, the exchange "eld and the anisotropy "eld } only the latter two are nonlinear. In the con"guration discussed here the anisotropy "eld is antiparallel to the applied "eld and the exchange "eld. When A'2.0, the anisotropy "eld is stronger than the exchange "eld. Thus, the resultant nonlinear "eld is antiparallel to the applied "eld, and its magnitude decreases with increasing spin wave amplitude, resulting in nonlinear spin wave modes with frequencies above the linear spin wave spectrum. The conclusion that A'2.0 is required in order for the localized mode to exist is reinforced by numerical work which is illustrated in Fig. 4. The frequency di!erence between the localized mode (solid curve) and the nonlinear zone boundary mode (dashed curve) extrapolates to zero at A"2.0, indicating that no local mode exists for A(2.0. When the transverse spin amplitude becomes su$ciently small, the intrinsic localized spin wave mode acquires a large spatial extent. In this limit the continuum approximation can be invoked. Introducing the new variable W "(!1)Ls which varies slowly in space, one obtains, for the L L continuum limit (dW/dx)!aW#bW"0 ,

(2.8)

where a"(X#2A!4)/a and b"(A!2)/a, with a the lattice constant. If b'0, that is, A'2, Eq. (2.8) has a localized solution given by W(x)"W sech K





(2A!4)W K (x!x ) ,  2a

where the frequency X"4!2A#(A!2)W /2 increases with the amplitude. K

(2.9)

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Fig. 4. Frequency relative to the top of the linear spin}wave band *X versus anisotropy parameter A for the nonlinear bulk mode at the zone boundary (dashed line) and the nonlinear localized mode (solid curve) (after Ref. [41]).

Rakhmanova and Mills [59] have investigated multi-soliton states in ferromagnetic chains described by Eq. (2.1) as well as the nature of these nonlinear excitations in an applied ac magnetic "eld. For any frequency above the maximum linear spin wave frequency, one can obtain a hierarchy of solutions with slowly varying envelopes that have the appearance of one soliton, two soliton, three soliton, 2, states. Fig. 5 shows an example of this hierarchy for zero driving "eld. Here f "(!1)Ls . Computer simulations have demonstrated that these states are quite stable L L against small amplitude perturbations. When a spatially uniform driving "eld (c "cO0) is applied with frequency u above the linear L spin wave band, Eq. (2.4) has time-periodic solutions with an ILSM-like feature near the center of the chain while a uniform background appears in both wings far from the center. Fig. 6 displays such a solution for the case of c"0.04. One can see from Fig. 6b that the application of an external driving "eld introduces an oscillatory modulation in the envelope f . These oscillations increases in L amplitude as the strength of the driving "eld increases. The presence of the external driving "eld has a signi"cant e!ect on the stability of these localized spin wave modes. Although ILSMs in zero driving "eld are stable over a long period of time, these ILSMs in the presence of a driving "eld appear to have a "nite lifetime which decreases with increasing "eld strength, as demonstrated by the computer simulations shown in Fig. 7. (The simulation time corresponds to roughly 100 precessional periods.) The solid line connects values of Re+s (t),, and the dotted lines identify Im+s (t), at t "250/JS. The localized structure L L  is quite stable in Fig. 7a for a weak driving "eld (c"0.01), while no evidence of the central feature remains at t "250/JS when the strength of the driving "eld is increased to c"0.2.  These simulations indicate that when an ac driving "eld is used to excite such ferromagnetic localized spin wave modes, its amplitude cannot be too large, though at the same time it should be large enough to drive the system into the regime where the nonlinearity is signi"cant.

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Fig. 5. For the frequency X"!3.95 and A"4, the envelope function f is shown for (a) a two soliton state, (b) a three L soliton state, and (c) a four soliton state. For (a) we have s "s "3.365;10\, for (b) s "s "2.177;10\, and for  ,  , (c) s "s "5.000;10\ (after Ref. [59]).  ,

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Fig. 6. The functions (a) s and (b) f "(!1)Ls are shown for a nonlinear spin excitation in the presence of a rf field. Here L L L the dimensionless "eld strength c"0.04 and the remaining parameters are the same as in Fig. 5 (after Ref. [59]).

2.3. Traveling ILSMs To examine traveling ILSMs, one may assume that s>(t)"s (t)e OL?\SR where s (t) is in general L L L complex since the traveling ILSMs are elliptically polarized and the ellipticity increases with increasing q. Inserting this ansatz into Eq. (2.3), gives the equation of motion [61]





Xs !s ((1!"s "#(1!"s " ) ds L L L\ L> L"i !(s !s ) sin qa(1!"s ", (2.10) L> L\ L dq #[2As #(s #s ) cos qa](1!"s " L L L> L\ where q"2JSt. To integrate Eq. (2.10) forward in time, an initial set of spin con"gurations is required. The authors assume that at q"0 the quantity in curly brackets on the right-hand side of Eq. (2.10) vanishes, that is Xs (0)"s (0)((1!"s (0)"#(1!"s (0)" ) L> L L L> (2.11) ![2As (0)#(s (0)#s (0))cos qa](1!"s (0)". L L> L\ L It should be emphasized that this approximation is good only when the envelop varies slowly in space and the wavevector q is small [58,62]. With the application of free end boundary conditions, initial spin con"gurations have been obtained from Eq. (2.11) for the investigation of single traveling ILSMs as well as their collisions [61]. The approximate initial spin con"guration for traveling ILSMs works quite well for

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Fig. 7. An illustration of the stability of the one soliton state in an external rf "eld. The structure, perturbed slightly, has been followed up to a time t "500/(2JS). For strong external "elds, the lifetime of the state shortens. Each "gure  displays the response at t . In (a), c"0.01, in (b), c"0.04, in (c), c"0.15, and in (d), c"0.20. In all cases, X"!3.95 and  A"4 (after Ref. [59]).

frequencies X close to the top of the linear spin wave band and for small wavevectors. If both X and q are large, the traveling ILSM will be scattered by the lattice and energy will shake away in the form of small amplitude extended spin waves before a stable ILSM can be achieved. This observation is similar to the results reported for easy-axis antiferromagnetic chains [62] and for isotropic ferromagnetic chains with both nearest- and next-nearest-neighbor exchange interactions [58]. Fig. 8 shows the collision between two solitons (two small amplitude ILSMs with slowly varying envelope functions). The initial spin con"guration is the two-soliton state obtained from Eq. (2.11). These two excitations have the same frequency and opposite wavevector. It is evident from the "gure that they remain unchanged in shape after the collision. Simulations with other initial spin con"gurations con"rm that traveling ILSMs with small q's are soliton-like objects since they pass through each other upon collision.

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Fig. 8. Collision of two ILSM solitons. Each has X"!0.385 and qa"0.1, due to the fact that they were obtained by taking as the initial con"guration a two-soliton solution of Eq. (2.11) (after Ref. [61]).

2.4. Interaction of ILSMs with magnetic defects Just as lattice defects can give rise to localized vibrational modes [7,63] and some spin defects can result in localized spin wave defect modes in linear theory by breaking the translational invariance of the underlying lattice [46,47] an important question is how these new ILSM's interact with localized spin wave defect modes. The 1-D ferromagnetic chain lends itself naturally to the study of the interaction between nonlinear intrinsic localized modes and magnetic defects and the results have been described in Ref. [61]. In the authors' model a defect spin is placed in the middle of a ferromagnetic chain and it distinguishes itself from other spins only by its di!erent anisotropy constant A"A!*A where A is the intrinsic anisotropy constant. In the theory of linearized excitations, there always exists a localized defect spin wave mode with frequency above the top of linear spin wave band when *A'0 since for this model with the appropriate strength of the magnetic "eld the anisotropy "eld is antiparallel to the applied dc "eld. The frequency of the linear spin wave defect mode may be written as X "X #*X ,  + where *X"2((1#(*A)!1) ,

(2.12)

(2.13)

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Fig. 9. Propagation of an ILSM soliton with X"!0.385 through the spin chain with a defect. The defect spin is at site 251, and has the values: (a) *A/A"0.05; (b) *A/A"0.07 (after Ref. [61]).

and X the frequency of the zone boundary spin wave. From simulations it is found that stationary + nonlinear localized defect modes can occur when the internal frequency X of the ILSM exceeds X , the frequency of the linear defect mode. A nonlinear localized defect mode may have a spatial  pro"le very similar to that of an ILSM in a perfect periodic chain, but it is more localized than an

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ILSM with the same amplitude. For a "xed frequency, the amplitude of the nonlinear intrinsic localized defect mode decreases with increasing *A/A. Computer simulations have demonstrated that the interaction between traveling ILSMs and magnetic defects can have a diverse and rich character, depending on the relation between the internal frequency of the ILSM and the frequency of the linear defect mode. Fig. 9 displays the propagation of an ILSM with frequency X"!3.85 through a chain of 501 spins with a defect placed at site 251. The intrinsic anisotropy parameter is A"4.0 so that X "!4.0. In Fig. 9a + and Fig. 9b, the values of *A/A are 0.05 and 0.07, respectively. Hence in both cases X'X  according to Eq. (2.11) and part of the energy in the initial ILSM is trapped at the defect site to form a nonlinear localized defect mode while another part is re#ected to form a well-de"ned traveling ILSM. The rest of the energy is radiated in the form of small amplitude extended spin waves. Note that in Fig. 9b, the re#ected ILSM is con"ned between the end of the chain and the defect, and that it bounces back and forth. This result suggested that a traveling ILSM might be trapped between two defects in a 1-D chain with proper impurity anisotropy constants which are separated in space. When the value of *A/A increases so that X exceeds the frequency of the initial ILSM, the  defect can no longer support a nonlinear defect mode of frequency X. In this regime, the incident ILSM is elastically re#ected from the defect site as illustrated by the computer simulation results shown in Fig. 10a. Since it is expected that the e!ect of defects on the propagation of ILSMs would disappear as *A/A approaches zero, one natural question is whether this limit represents a trivial case. Computer simulations (not shown here) demonstrate that the ILSM passes over the defect with its velocity decreased appreciably when *A/A is small (+0.01) but positive. Some small amplitude extended spin waves are also excited after such a collision. The decrease in velocity and the excitation of extended spin wave become less signi"cant as *A/A decreases. A rather di!erent picture occurs when *A/A approaches zero from the negative side. According to linear spin wave theory no spin wave mode localized at the defect site can exist for this parameter region. The propagating ILSM is fully re#ected by the defect even when *A/A is as small as !0.0075, as demonstrated in Fig. 10b. It should be noted that related dynamical results have been found for the case of a onedimensional nonlinear vibrational diatomic lattice which incorporates realistic nearest-neighbor Born}Mayer}Coulomb potentials. When the interaction of intrinsic gap modes with stationary anharmonic mass defect impurity modes is examined in numerical simulation studies, a variety of scattering results are found depending on the mass defect magnitude and the site in the diatomic chain. Two important features of the trajectories are that the gap mode is trapped at the mass defect when the vibrational frequencies of the moving mode and the anharmonic defect mode are near resonance while the scattering is elastic when the frequencies are far apart [64].

3. Isotropic ferromagnetic chain with nearest- and next-nearest-neighbor exchange interactions Although intrinsic localized spin wave modes can not exist in isotropic ferromagnetic chains with only nearest-neighbor (NN) exchange interactions [41], long-lived intrinsic localized spin wave resonances (ILSRs) can appear below the Brillouin zone boundary frequency when nextnearest-neighbor (NNN) ferromagnetic coupling of su$cient strength is included. The inclusion of

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Fig. 10. Propagation of an ILSM soliton with X"!0.385 through the spin chain with a defect. The defect spin is at site 251, and has the values: (a) *A/A"0.35; (b) *A/A"!0.0075 (after Ref. [61]).

NNN exchange is an important step in identifying the properties of nano-scale localization stemming from the long range interactions. The key feature of the dynamics of an in-band ILSR is the coexistence at the same frequency of the quasilocalilzed ILSR and the linear spin-wave spectrum. Recently, intrinsic localized resonant modes have also been identi"ed in lattice vibration

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simulations [65] although from analytical work, vibrational resonant modes already had been proposed to be a natural consequence of intrinsic localization [29]. 3.1. The 1-D model Hamiltonian For a one-dimensional ferromagnetic chain of N spins which are coupled through both nearest-neighbor (NN) and next-nearest-neighbor (NNN) isotropic exchange interactions in the absence of anisotropy and applied dc "eld the Hamiltonian becomes H"!2J S ) S !2J S ) S , (3.1)  L L>  L L> L L where both the NN coupling constant J and the NNN coupling constant J are positive. All spins   are taken to be aligned along the z-axis in the ground state since the chain is isotropic. For a non-dissipative chain of classical spins, the equation of motion for the spin at the nth site is then dS /dt"S ;H  , L L L where the e!ective "eld, H, acting on the spin can be obtained from L H(t)"! SLH"2J (S #S )#2J (S #S ) .  L\ L>  L\ L> L For the circular variables de"ned in Eq. (2.2), the equations of motion become

(3.2)

(3.3)

i ds> L "s>(sX #sX #osX #osX )!sX(s> #s> #os> #os> ) , (3.4) L L\ L> L\ L> L L\ L> L\ L> 2J S dt  where the dimensionless parameter o"J /J measures the strength of NNN coupling relative to   NN coupling. Once again sX"(1!"s>" so that Eq. (3.4) is intrinsically nonlinear in s>. L L L The linear spin wave dispersion curve is obtained by linearizing Eq. (3.4) (approximating sX by 1). L Introducing s>"s e OL?\SR in the usual manner the dispersion curve becomes L  u (q)"8J S[sin(qa/2)#o sin(qa)] , (3.5)   where q"(2p/Na)n (n"0,$1,2,$N/2), and a is the lattice spacing between adjacent spins. The dispersion curves for di!erent NNN coupling strengths are shown in Fig. 11. Although the frequency of the Brillouin zone boundary mode is independent of the NNN coupling because a precessing spin is in-phase with its NNNs in this mode, the NNN coupling tends to raise the dispersion curve at intermediate wave numbers so that its maximum frequency u may appear at

 a wave number q other than p/a, i.e.

 p 1 (3.6) q " ! cos\(1/4o) and u "2J S(1#4o)(1#(1/4o)) ,

 

 a a when o is greater than 1/4. In this case the dispersion curve has a local minimum frequency at the Brillouin zone boundary, which opens up a new possibility, namely, that an ILSR may drop from the band edge.

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Fig. 11. Spectra of linear spin waves in isotropic ferromagnetic chains with both nearest-neighbor (NN) and nextnearest-neighbor (NNN) exchange interactions. From top to bottom, the relative strength of the NNN coupling o"J /J are 1.0, 0.4 and 0, respectively. When o is greater than 1/4 the maximum frequency would appear at q and  

 the Brillouin zone boundary becomes a local minimum. The ILSR arrow identi"es the frequency of the intrinsic localized spin wave resonance described in the text (after Ref. [58]).

3.2. Stationary intrinsic localized spin wave resonances (I¸SRs) To "nd the eigenvector of a stationary ILSR below the Brillouin zone boundary value all nonlinear terms must be included. Inserting the ansatz sH"s , sX"(1!s) s>"s e\ SR, L L L L L L into Eq. (3.4) gives the time-independent set of nonlinear equations

(3.7)

u  s "s [(1!s #(1!s #o((1!s #(1!s )] L L\ L> L\ L> 2J S L  ! (1!s[s #s #o(s #s )] , (3.8) L L\ L> L\ L> where u is the frequency of the ILSR. The eigenvectors can be classi"ed in terms of their parities  since both an odd-parity mode and an even-parity mode can occur for a range of parameter values o. Given the maximum spin deviation at the mode center s , Eq. (3.8) can be solved numerically

 to obtain the eigenvector shape and the frequency of the ILSR [58]. 3.2.1. ILSR eigenvector shapes To illustrate the di!erent shapes that can be found, the eigenvectors of an odd-parity ILSR and an even-parity ILSR for the set of parameters A"1.0 and s "0.7 are presented in Fig. 12a and

 Fig. 12b. The lattice site with maximum spin deviation is the symmetry center of the odd-parity mode, while the symmetry center of the even-parity mode is located between two adjacent sites,

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Fig. 12. Eigenvectors of stationary ILSRs of odd and even parities. The ferromagnetic chain contains 256 spins with the parameter o"1.0. Both modes have the same maximum spin deviation s "0.7: (a) Odd-parity mode, the symmetry

 center is on the lattice site with maximum spin deviation; (b) Even-parity mode, the symmetry center (the X) is between two adjacent sites. In both cases the left side shows a factor 40 expansion of the ordinate to display the resonant mode wave character in the wings (after Ref. [58]).

each with the same maximum spin deviation. The common feature of both ILSRs is that, although the large amplitude region only extends over a few lattice sites, the spin deviations do not disappear with increasing distance from the center but instead evolve into a weak plane wave pattern with increasing distance from the center. The ILSRs in the magnetic chains reviewed here are fundamentally di!erent from the resonant breathers identi"ed in the continuous  model [66] and in discrete nonlinear lattices with substrate potentials [67]. In such resonant breathers, the localized center oscillates at a fundamental frequency outside the linear spectrum while the extended plane-wave tail oscillates at the higher harmonics of the fundamental frequency. It is these higher harmonics that are in the linear spectrum. In contrast, the fundamental frequency of the ILSRs described here in these magnetic chains is coincident with the linear spectrum itself since the ILSRs are monochromatic. By examining the Fourier transform of the eigenvector using





"s(q)"" s exp(iqna) L L



(3.9)

the wave numbers associated with the weak plane wave pattern can be identi"ed. Besides Fourier components centered at q"p/a, there is a sharp peak located at q with a strength that grows with J increasing maximum spin deviation. When this q is substituted into the linear spin dispersion J relation, Eq. (3.5), the appropriate resonant frequency u is obtained.  The strength of the coupling between localized resonant modes and extended spin waves depends on two factors: the strength of the NNN interaction and the maximum spin deviation.

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With either deceasing relative strength of the NNN interaction or increasing the maximum spin deviation both the odd-parity mode and the even-parity mode become more localized and the amplitude of the o!-center plane wave component increases [58]. Quasi-localized solutions are not found for o(1/4. 3.2.2. Amplitude dependence of the mode frequency The dependence of the frequency of the ILSR on its maximum spin deviation is shown in Fig. 13. Here the NNN/NN parameter o"1.0. The open circles and crosses denote the frequencies of the odd-parity ILSRs and the frequencies of the even-parity ILSRs, respectively. The frequencies are found by numerically solving Eq. (3.8) with periodic boundary condition. For small spin deviations, the frequency of an ILSR lies close to the Brillouin zone boundary frequency of the linear spin wave band. With increasing spin deviation the mode becomes more localized and its frequency drops further into the linear spin wave band. Note that for "xed maximum spin deviation, the frequency of the even-parity ILSR is lower than that of the odd-parity ILSR. Although there is no apparent distinction between odd-parity modes and even-parity modes with small maximum spin deviations, the di!erence between them increases as s increases. The dot-dashed line is the

 continuum approximation frequency obtained later in Section 3.3.2, which is in good agreement with the discrete results up to s "0.5.

 3.2.3. ILSR lifetime It is expected that an ILSR becomes unstable and delocalizes after su$cient time because the localized excitation is in resonance with the plane-wave spectrum. The lifetime of an ILSR has been investigated by means of molecular dynamics (MD) simulations by using the eigenvectors obtained

Fig. 13. Dependence of the frequency of stationary ILSR on its maximum spin deviation. The frequency is normalized by u , the frequency of the linear spin wave at the Brillouin zone boundary. The ferromagnetic chain contains 256 spins 8 with the parameter o"1.0. Open circles are the frequencies of odd-parity ILSRs, and crosses are the frequencies of even-parity ILSRs. The dot-dashed line is the continuum approximation given by Eq. (3.27) (after Ref. [58]).

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from numerically solving Eq. (3.8) to provide the initial conditions. Integrating the discrete equations numerically by using the fourth-order Runge}Kutta method with a time step of ¹ /200 8 where ¹ "p/4J S, the period of the linear spin-wave mode at the Brillouin zone boundary, 8  demonstrates that the lifetime of an ILSR depends on three factors (1) its parity, (2) the maximum spin deviation, and (3) the relative strength of the NNN interaction o. At su$ciently small amplitudes ILSRs of both parities are observed to last many hundreds of periods. With increasing amplitude the emission of plane-waves from the ILSRs becomes important so that the larger the amplitude the faster the decay. Symmetry is important since it is always observed that the even-parity mode is more unstable than the odd-parity one. The time evolution of the energy density distribution for an odd-parity ILSR with modest spin deviation (s "0.7) is

 shown in Fig. 14 where the energy density is measured from the ground state value. This oddparity mode preserves its initial shape over the complete simulation time of 1500¹ . Contrast this 8 result with that shown in Fig. 15, which presents the corresponding time evolution for an even-parity ILSR with the same spin deviation, where the even-parity mode starts to move and decay after about 750¹ . The strength of the NNN interaction also plays an important role in the 8 decay of an ILSR. As the NNN interaction increases the wave number of the spin-wave that is in resonance with the localized excitation moves away from the band edge (see Fig. 11) so that the coupling between the ILSR and the spin wave becomes weaker producing an increased lifetime. 3.3. Conditions for the occurrence of ILSRs The interrelation between the modulational instability of the extended band edge plane-waves and the existence of spatially localized excitations has been established for a number of nonlinear vibrational lattices [68}74]. Such stability analysis of extended plane-waves provides a useful way to predict under what conditions nonlinear localized excitation can occur. Here the existence

Fig. 14. Time evolution of the energy density distribution of the odd parity ILSR shown in Fig. 12a. The energy density shown here is measured from the ground state energy density and is in units of 2J S. Time is measured in units of ¹ .  8 Fig. 15. Time evolution of the energy density distribution of the even parity ILSR shown in Fig. 12b. Again the energy density is measured from the ground state energy density and is in units of 2J S. 

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conditions for an ILSR as determined both from linear stability analysis of extended nonlinear zone boundary mode and also from the continuum approximation are reviewed. 3.3.1. Modulational instability of extended nonlinear spin wave modes Owing to the translational invariance of the underlying lattices, for any "nite spin deviation, Eq. (3.4) also has time-periodic solutions of spatially extended nonlinear spin-wave modes with frequency u(q)"u (q)(1!s where the deviation s is not negligible. The nonlinear terms tend    to decrease the frequency and it is these extended spin-wave modes that are modulationally unstable under certain conditions and linear stability analysis can be used to determine the parameter space in which they are unstable. Assume an extended nonlinear spin-wave mode s>(t)"s e OL?\SR is perturbed so that L  s>(t)Ps>(t)"(s #b #it )e OL?\SR , (3.10) L L  L L where u is the frequency of the nonlinear extended spin wave and the perturbations b and t are L L real and much smaller than s in magnitude. To obtain the growth rate of modulation waves let  (3.11) b "be /L?\S R#c.c. , L and t "te /L?\S R#c.c. , (3.12) L where Q and u are the wave number and frequency of the modulation wave. Substituting

Eqs. (3.10) into (3.4) and linearizing in b and t gives two coupled linear equations







M M b   "0 , M M t   where the matrix elements of M are given by

(3.13)

u M "M " !2(1!s (sin Qa sin qa#o sin 2Qa sin 2qa) ,    2J S  M "!i2(1!s [(1!cos Qa) cos qa#o(1!cos 2Qa) cos 2qa] ,  

(3.14) (3.15)

and (3.16) M "!M !i (2s/(1!s )[(cos Qa!cos qa)#A(cos 2Qa!cos 2qa)] .     The dispersion curve of the modulation wave, u (q, Q), is determined by the condition that the

determinant of the matrix M is zero so that Eq. (3.13) has nontrivial solutions. For arbitrary q and Q, with the help of Eq. (3.5), one "nds







 1 u(q#Q)#u(q!Q) u (q#Q)!u (q!Q)  u !  (1!s "

 4 !2u (q) 2  u (q#Q)#u (q!Q) u (q#Q)#u (q!Q)    ; !s  .  !2u (Q) !2u (q)  



 



 (3.17)

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For long wavelength modulation Q;1, Eq. (3.17) can be further simpli"ed to





1 du du  Q Q!2su (q) , [u !Qv (q)(1!s]"  

  4 dq dq

(3.18)

where v (q)"du /dq is the group velocity of linear spin waves. It can be seen from Eq. (3.18) that   instability may only occur in the region where the curvature of the linear dispersion curve is positive. It is clear that Im+X (q, Q),'0 if and only if

0((du /dq)Q(2su (q) . (3.19)    Since the ILSR mode, if it exists, should bifurcate from the band edge mode, attention should be focused on the band edge spin wave. Setting qa"p in Eq. (3.5) gives u (p/a)"8J S and du /dq"4J Sa(4o!1) . (3.20)     Since the right-hand side (RHS) of the second equation of Eq. (3.20) is always negative for o4o  with o "0.25 regardless of the magnitude of the spin deviation s , the extended nonlinear band   edge mode is stable for this parameter range. However, as the relative strength of NNN coupling gets stronger so that o'o , the extended band edge mode becomes modulationally unstable to  long wavelength perturbations when the spin deviation exceeds the threshold





1 du  p  s" Q "(4o!1) . (3.21) 

 2u (p/a) dq N  Here use has been made of the fact that in a periodic ferromagnetic lattice of N spins the smallest wavevector is Q "2p/Na. In a real system s is essentially zero since N is of the order of 10. This

  instability region is also the region in which ILSRs can occur. The critical value of o agrees with  the numerical "ndings. The energy density distribution as determined by MD simulation for randomly perturbed extended nonlinear band edge spin wave mode for two values of the NNN strength at di!erent times is shown in Fig. 16. The dot-dashed lines represent the energy density distribution of the ground state. In Fig. 16a, the relative strength of the NNN interaction o"0.2 is below the critical value of o "0.25 while in Fig. 16b, o"0.6 is above the critical value. The initial spin deviations  are the same for both chains, and are given by s "(!1)L0.1#ds where the magnitude of random L L perturbation "ds " is less than 0.005. The behavior of the extended band edge mode is qualitatively L di!erent for the two cases. In a chain with o less than o , the energy density distribution remains  spatially extended throughout the simulation period, as shown in Fig. 16a. In a chain with o greater than o , Fig. 16b illustrates that the band edge spin wave is unstable and evolves into  temporal ILSR-like localized excitations. At t"360¹ , the initially uniformly distributed energy 8 has been concentrated into "ve ILSR-like excitations. This MD simulation demonstrates that the modulational instability of the extended band edge mode is a possible mechanism for the creation of ILSR excitations from extended modes. 3.3.2. Envelope solitons in the continuum approximation When the maximum spin deviation of an ILSR is small, so that it extends over a large number of lattice sites, the magnitude of the spin deviation s will vary slowly with site index n. Since the L

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Fig. 16. Decay of extended band edge spin waves into ILSRs via modulational instability. The time is measured in units of ¹ . The snapshots of energy density distributions are determined by MD simulations for randomly perturbed 8 extended band edge spin waves in chains containing 256 spins with periodic boundary condition. One ordinate unit is 0.05. The dot-dashed lines are the energy density distribution of the ground state: (a) The NNN interaction parameter o"0.2 is less than the critical value o ; (b) The NNN interaction parameter o"0.6 is greater than the critical value  o (after Ref. [58]). 

spatial symmetry of a small amplitude ILSR is close to that of the corresponding plane wave, the staggered variable

"(!1)Ls> (3.22) L L can be introduced where now is complex. This continuum approach can be used to determine L the existence condition of ILSRs and provide some properties of small amplitude ILSRs which are expected to be qualitatively correct even for ILSRs in the discrete limit. The analysis used here is not limited to the stationary modes but only assumes that the wavenumber associated with the ILSR is close to the band edge so that both the phase and the magnitude of vary slowly in space. L After substituting Eq. (3.22) in Eq. (3.4) and neglecting nonlinear terms higher than cubic a nonlinear SchroK dinger (NLS) type equation is obtained for (x, t), namely, iR /Rt"2J S(1!4o)a(R /Rx)#8J S !4J S" " . (3.23)    The NLS equation is integrable, and the condition for Eq. (3.23) to have a localized solution (one-soliton solution) is that the coe$cient of the second-order spatial derivative and the coe$cient of the cubic nonlinear term be of the same sign, that is, o'0.25. (The linear term can be removed by a gauge transformation.) Note that this critical value of o agrees with the value of o already  found from linear stability analysis. When o'o , Eq. (3.23) has both stationary and moving  localized solutions which are given by

(x, t)" sech






x!v t  e\ )V\SR>? , l 

(3.24)

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where p/a!K is the wave number of the carrier wave associated with the ILSR, the maximum

spin deviation and a a constant phase factor. A stationary mode can be obtained by setting K"0.  The envelope velocity v , the envelope width l , and the frequency of the traveling ILSR u are given    by v "!4J Sa(4o!1)K ,   l "((4o!1)/ )a 

(3.25) (3.26)

and u "8J S#2J S(4o!1)(Ka)!2J S  . (3.27)     The traveling velocity v of the envelope is just the group velocity of the linear spin wave of the  same wavevector which is v "du /dq"4J Sa(sin qa#2o sin 2qa) , (3.28)    for q"p/a!K with K small. In the continuum approximation for a given maximum spin deviation, the traveling ILSR and stationary ILSR have the same envelope shape with width inversely proportional to the maximum spin deviation and which increases with the NNN coupling strength. The "rst term on the right-hand side of Eq. (3.27) is just the frequency of the linear band-edge spin wave, and the third term is the anharmonic frequency shift. Since the frequency shift is negative, u is in the linear spin  wave band. The independence of the mode frequency on the strength of the NNN coupling is consistent with the fact that at the Brillouin zone boundary the spins precess in-phase with their NNNs. However, the NNN coupling does tend to broaden the envelope of the localized mode. In the discrete limit where the ILSR is highly localized so that the continuum approximation breaks down, one should expect these observations still to be qualitatively correct. It may seem surprising that the essential feature of an ILSR, the non-decaying plane-wave tail, does not appear in the solution although the continuum approximation describes the shape of the center of an ILSR when the maximum spin deviation is small and gives the correct threshold of o. The linear dispersion curve of the continuum model described by Eq. (3.23) can be obtained by setting "0 in Eq. (3.27), which has a parabolic shape with a gap at K"0. Since this dispersion

curve is just a local approximation of the real dispersion curve Eq. (3.5) in the neighborhood of zone boundary, in the continuum model the solution Eq. (3.24) describes a gap soliton. 3.4. Translating ILSRs So far the large amplitude stationary ILSR excitation is assumed to be described by an elementary excitation with a circular precession frequency u . If an ILSR is traveling along the  chain then the circular precession may be viewed as an internal degree of freedom and the translational motion as an external one. The separation of these degrees of freedom is a good approximation only when the wave number of the carrier wave is close to Brillouin zone boundary and hence the traveling velocity of the ILSR is small compared to the phase velocity of the carrier wave. To describe traveling ILSRs, solutions of the form s>(t)"s (t)e OL?\SR"(!1)L (t)e\ )L?>SR , L L L

(3.29)

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need to be examined where (t) is real and qa"p!Ka with Ka;1. The substitution of L Eq. (3.29) into Eq. (3.4) gives 1 d

L"(1! [( ! )sin Ka!o( ! )sin 2Ka] , L L\ L> L\ L> 2J S dt 

(3.30)

and (u /2J S) " [(1!  #(1!  #o((1!  #(1!  )] L> L\ L>   L L L\ #(1! [( # )cos Ka!o( # )cos 2Ka] . (3.31) L L\ L> L\ L> The translational velocity of the ILSR is determined by Eq. (3.30) while its envelope is determined by Eq. (3.31). 3.4.1. Properties of a single traveling ILSR The envelope of a traveling ILSR for the discrete limit can be found by numerically solving Eq. (3.31). Once the initial envelope shape of an ILSR is obtained MD simulations can be used to investigate its motion. Fig. 17 shows the time evolution of an ILSR with s "0.3, Ka"p/32 and

 o"1.0. Since the maximum spin deviation is fairly small, the initial envelope shape can be obtained from the continuum approximation, Eq. (3.24). In this case the characteristic o!-central plane wave pattern of a resonant mode, which would occur if Eq. (3.31) is solved, is negligible. It is observed that this small amplitude ILSR can travel freely through the lattice with the velocity given by Eq. (3.25), and that there is no decay or slowing down within numerical error. This simulation result con"rms the validity of the continuum approximation where small amplitude ILSRs are treated as gap solitons. As the maximum spin deviation increases this continuum approximation breaks down and one has to solve Eq. (3.31) numerically to obtain the initial spin deviations. Like a stationary ILSR, the moving ILSR has a weak plane wave tail in the o!-central region. Fig. 18 shows the time evolution of an ILSR with s "0.7, Ka"p/32 and o"1.0. Since the

 maximum spin deviation is not too large the ILSR can still travel through the lattice, but by the time 800¹ has passed (not shown) about 5% of the energy has decayed into the plane-wave 8 modes. The larger the amplitude and the faster the velocity of an ILSR, the larger the emission of plane-spin wave modes. 3.4.2. Collision between two ILSRs A fundamental property of solitons is that they pass through each other as non-interacting particles. Recent studies have shown that this is not the case for intrinsic localized modes in discrete lattices, and that both energy transfer between intrinsic localized modes and collision-induced decay into plane-wave modes are observed in computer simulations [74,75]. Here we examine interactions of both the small amplitude and large amplitude ILSRs in these FM chains. To launch two small amplitude ILSRs moving toward each other, two small amplitude ILSRs obtained from Eq. (3.24) are placed 256 sites apart in a chain of 512 spins with periodic boundary condition. The NNN interaction parameter o"1.0. The parameters for the two ILSRs are: left, s "0.2, Ka"!p/25.6; right, s "0.3, Ka"p/25.6. The collision is illustrated in Fig. 19.



 Before and after the collision the two modes move with uniform velocity and maintain their

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Fig. 17. Traveling ILSR with small maximum spin deviation in a 256 spin chain. The parameters are o"1.0, s "0.3

 and Ka"p/32. The energy density shown here is measured from the ground state energy density and is in units of 2J S.  Time is measured in units of ¹ . 8 Fig. 18. Traveling ILSR with intermediate maximum spin deviation. The parameters are o"1.0, s "0.7 and

 Ka"p/32. For clarity, only part of the 256 spin chain is shown. The energy density shown here is measured from the ground state energy density and is in units of 2J S. Time is measured in units of ¹ .  8

original shapes. More accurate integration shows that the energy transfer between them after one collision is less than 0.5% of the total energy. The interaction between ILSRs becomes more violent as the mode amplitude increases. As an example the collision between a large amplitude stationary ILSR and a small amplitude traveling ILSR for a chain of 256 spins is shown in Fig. 20. Each mode maintains its own shape before the collision, they interact strongly when they meet, and both become unstable after the collision. Only a fraction of the small amplitude traveling ILSR can pass through the stationary ILSR, and it decays quickly into plane-wave modes after the collision. Meanwhile the stationary ILSR also shakes away energy from its central region after the collision, although it still remains localized over the simulation interval examined. When both ILSRs have large amplitudes neither survives the collision. The observation reported in Ref. [75] that the collision between intrinsic localized vibrational modes tends to favor the growth of the larger excitation at the expense of smaller ones is not seen for the ferromagnetic chain with both NN and NNN interactions. The most likely explanation is that the NNN interaction produces a long-range coupling between spins.

4. Antiferromagnetic chain with on-site easy-axis anisotropy Since both intrinsic localized spin wave modes (ILSMs) in easy-plane ferromagnetic chains [41,59] and intrinsic localized spin wave resonances (ILSRs) in isotropic ferromagnetic chains [58] stem from the Brillouin zone boundary mode, they are not magnetic dipole-active. In order to generate or detect such ILMs it would seem that the direct interaction with electromagnetic radiation would be a desirable property. Since the intrinsic localized gap modes in diatomic lattices have been found to be IR-active we review here the available information on two sublattice spin

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systems. There is the added practical advantage that a large number of insulating magnetic materials are anisotropic antiferromagnets and many of the uniform antiferromagnetic resonance (AFMR) modes are IR-active. Thus, the possibility of the existence of ILSMs in the gap below the AFMR frequency combined with an associated magnetic dipole activity is of particular interest from the observational point of view. Takeno and Kawasaki [40,76] "rst proposed that the soft nonlinearity in a NN anisotropic exchange interaction of a one-dimensional Heisenberg antiferromagnetic chain could produce both symmetric and antisymmetric ILSMs in the gap below the uniform mode AFMR frequency. Later, magnetic gap solitons with "nite but small spin deviation were investigated in 1-D Heisenberg antiferromagnets [77,78]. These solitons have zero group velocity. In all of these works, the coherent state ansatz [79] was employed. As shown in the previous sections the classical torque equation provides a straightforward and transparent approach for exploring the nonlinear classical equations of motion and we continue to present this method. In this section, the nonlinear dynamics of classical antiferromagnetic ordered spins interacting via NN isotropic exchange and on-site anisotropy is examined with attention directed at the experimentally relevant dynamical properties of stationary and moving IR-active ILSMs. The introduction of on-site anisotropy rather than anisotropic exchange between neighbors enables one to make contact with the measured parameters for a variety of known magnetic insulators. 4.1. Equations of motion A perfect one-dimensional antiferromagnetic chain of N classical spins in which each spin interacts with its nearest neighbors via the Heisenberg exchange interaction and each spin feels an on-site anisotropy "eld is described by the Hamiltonian H"2J S ) S !D (SX) . (4.1) L L> L L L where both the exchange constant J and the single-ion anisotropy constant D are positive. In the ground-state adjacent spins point in opposite directions along the z-axis. The usual periodic boundary condition is imposed so that S "S . The anisotropy that is used in Eq. (4.1) is an L L>, e!ective anisotropy which may arise, for example, from the crystalline "eld interaction of the magnetic moments with their neighboring ions and/or from the long range dipolar interaction between the magnetic moments [80].

䉳

Fig. 19. Collision between two traveling small amplitude ILSRs in a chain of 512 spins with o"1.0. The parameters for the two ILSRs are: left, s "0.2, Ka"!p/25.6; right, s "0.3, Ka"p/25.6. The two ILSRs pass through each



 other as non-interacting particles as expected for soliton-like excitations. The energy transfer between them is less than 0.5% of the total energy after one collision. Fig. 20. Collision between a traveling small amplitude ILSR and a stationary large amplitude ILSR in a chain of 256 spins with o"1.0. The small amplitude ILSR is characterized by s "0.2, Ka"p/32, and the stationary ILSR by

 s "0.7, Ka"0. Both modes are unstable after the collision.



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For the Hamiltonian above, the classical equations of motion for a spin on nth site can be obtained from Eq. (3.2) where now the e!ective "eld, H, acting on the spin is L H(t)"! SLH"!2J(S #S )#2DSXe , (4.2) L L\ L> L X with e the unit vector along the positive z-axis. From Eqs. (3.2) and (4.2), one obtains the equation X of motion for s>, namely, L ids>/dt"!2JS[(sX #sX )s>!(s> #s> )sX]#2DSsXs> . (4.3) L L\ L> L L\ L> L LL In the small spin deviation limit, i.e., sX"(!1)L, the linear dispersion curve for spin wave is L obtained from Eq. (4.3) as X(q)"$((A#2)!4 cos qa ,

(4.4)

where X(q)"u(q)/2JS, A"D/J, q and a are the wave vector and the distance between adjacent spins, respectively. The two branches ($) correspond to the two di!erent directions of precession. A gap occurs below the q"0 antiferromagnetic resonance frequency X "X(0)"(A(A#4). $+0 Since the nonlinearity in both the exchange interaction and anisotropy "eld is soft ILSMs cannot appear above the top of the plane-wave spectrum in either isotropic or anisotropic antiferromagnetic chains. However, as the amplitude increases intrinsic localized spin wave gap modes (ILSGs) with frequencies below X can split o! from the bottom of the linear dispersion $+0 curve as shown in Fig. 21. The desired large amplitude-localized excitation is assumed to be described by an elementary excitation with a circular precession frequency u . If this mode is  traveling through the chain then the circular precession can be treated as an internal degree of freedom and the translational motion as an external one. Thus circularly polarized solutions of the form s>(t)"s (t)exp[i(qna!u t)] L L 

(4.5)

Fig. 21. Frequency of an intrinsic localized spin wave mode in an antiferromagnetic chain with on-site easy-axis anisotropy, relative to the linear spin wave dispersion curve. Only gap modes are allowed owing to the softness of the nonlinearity in both the exchange and the anisotropy "elds.

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are to be found where the envelope s (t)"[s (t)]H is time-dependent. The assumption is exact for L L stationary modes (q"0), and is a good approximation in cases where the localized mode moves slowly (with small q) and the magnitude of the spin deviation does not change rapidly from site to site. Hence we shall restrict q to the region near the center of Brillouin zone. Inserting Eq. (4.5) into Eq. (4.3) results in the following system of nonlinear equations for the transverse spin deviations: and

X s "(!1)L+((1!s #(1!s )s #[(s #s ) cos qa#As ](1!s, , L> L L\ L> L L L L\

(4.6)

1 ds L"(!1)L(s !s )(1!s sin qa . L> L\ L 2JS dt

(4.7)

Here the time variable t has been explicitly dropped from both equations. Eq. (4.6) determines the envelope of the mode while Eq. (4.7) determines the envelope velocity. The system of coupled nonlinear equations represented by Eq. (4.6) can support localized solutions. Note that no further approximation, such as the rotating wave approximation (RWA) necessary in nonlinear lattice vibration, is made in deriving Eq. (4.6). It should be emphasized again that Eqs. (4.6) and (4.7) are exact for stationary modes and they are a good approximation for slowly moving modes with broad envelopes. 4.2. Stationary intrinsic localized spin wave gap modes (I¸SGs) Setting q"0 in Eq. (4.7) yields a time-independent envelope, hence a stationary ILSG. Eq. (4.6) can be solved numerically for both the ILSG frequency and eigenvector with a appropriate symmetry imposed on the mode shape. Owing to the symmetry in Eq. (4.6) spin wave modes of the shapes proposed in Ref. [40] do not exist. However, it has been shown by others that both single-peaked and double-peaked intrinsic localized spin wave modes can be found with frequencies in the gap below X . $+0 4.2.1. Eigenvector of a single-peaked I¸SG Since a stationary ILSG, if it exists, bifurcates from the spatially uniform q"0 mode, both eigenvectors are expected to have the same pattern of sign alternation of spin deviations. In a singled-peaked ILSG the maximum spin deviation is at the center of the mode. Once the appropriate symmetry is imposed and the maximum spin deviation is "xed, both the eigenvector and eigenfrequency of an ILSG can be obtained numerically. As an illustration, the spin deviations of an ILSG versus site index in a chain of 128 spins is plotted in Fig. 22a for a maximum spin deviation s "0.7 and with anisotropy parameter A"1. The sign of the spin deviation alternates  from one spin to the next. Some qualitative statements can be made for ILSGs with di!erent maximum spin deviations and anisotropy parameters. For small s , the envelope of the ILSG  spreads over a large region of the lattice. As s increases, the degree of localization increases. For  "xed s , the degree of localization also increases with increasing anisotropy strength A. It is  observed that the frequency of an ILSG drops faster for a chain of larger anisotropy as s increases.  The frequency of a stationary single-peaked ILSG as function of s is plotted in Fig. 23 as open  circles for A"1.0.

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Fig. 22. Eigenvector shapes of stationary single-peaked and double-peaked ILSGs. Both modes are in a chain of 128 spins with anisotropy parameter A"1, and have the same maximum spin deviation s "0.7: (a) Single-peaked ILSG; (b)  double-peaked ILSG. The "lled circles identify up spins while the open circles identify down spins (after Ref. [116]). Fig. 23. Frequency of stationary ILSGs as function of maximum spin deviation. The antiferromagnetic chain consist of 128 spins with anisotropy strength A"1. The open circles are frequencies of single-peaked ILSGs, and the triangles are those of double-peaked ILSGs. The solid curve is obtained in the continuum limit.

4.2.2. Eigenvector of a doubled-peaked I¸SG Another kind of symmetrical localized mode that can exist in an easy-axis antiferromagnetic chain is a double-peaked mode, which is plotted in Fig. 22b for an ILSG with A"1 for a chain of 128 spins. This mode is di!erent from the single-peaked mode in that the maximum spin deviation is on the two nearest-neighbors of the mode center rather than on the mode center itself. The double-peaked mode has the same static properties as those of single-peaked mode except that the frequency of a double-peaked mode decreased faster with increasing maximum spin deviation than that of a single-peaked mode. For comparison, the frequency of the double-peaked mode as function of maximum spin deviation is plotted as triangles in Fig. 23. The solid curve in Fig. 23 is obtained from the continuum limit approximation to be discussed in Section 4.4. Although the two kinds of localized spin wave modes have similar static properties, we show that they have quite di!erent dynamical properties. 4.2.3. Power spectra of stationary ILSGs Once an ILSG eigenvector is obtained by numerically solving Eq. (4.6), it can be used as the initial condition in a molecular dynamics (MD) simulation to test the modes stability. A log power spectrum of the total transverse magnetic moment m>(t)" s>(t) of a single-peaked stationary LL ILSG is shown in Fig. 24a. The results show that there is a very strong peak at X /X "0.9073,  $+0 in the gap below the bottom of the plane-wave spectrum. The resulting stationary ILSG is

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Fig. 24. Power spectra of the transverse magnetic moment of stationary ILSGs in an easy-axis antiferromagnetic chain of 128 spins with A"1. Both modes have the same maximum spin deviation of 0.7: (a) Single-peaked ILSG; (b) double-peaked ILSG.

essentially monochromatic so the frequency obtained from the MD power spectrum can be compared to the frequency calculated from Eq. (4.6). The MD simulation frequencies found in this manner are plotted as open circles in Fig. 25 for single-peaked ILSGs resulting in excellent agreement over the entire range of s . MD simulations of double-peaked stationary ILSGs  demonstrate that these modes can also last for thousands of periods without any apparent decay in the absence of noise perturbation. The power spectrum of such a double-peaked stationary ILSG is plotted in Fig. 24b, which shows a single clean peak in the gap. The above MD simulations demonstrate that both single-peaked and double-peaked localized solutions found by the numerical procedure are true periodic orbits with a single frequency. A di!erent picture develops when the modes are randomly perturbed. The single-peaked mode is stable against noise perturbation whereas the double peaked one is unstable. A detailed discussion of the stability results is deferred until Section 4.5. 4.3. Moving gap modes Now let us turn to the case of traveling ILSGs. For certain nonlinear lattices it has been shown that traveling intrinsic localized vibrational modes can be excited, and the motion of such intrinsic localized excitations in discrete lattices can be very di!erent from the motion of soliton-like excitations in a continuum since discreteness breaks the continuous translational invariance [55,75,81}83]. Since the shape of an ILSG in a discrete lattice interchanges between single-peaked and double-peaked ones while it travels through the lattice, the symmetry classi"cation used for

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Fig. 25. Comparison between simulated and analytical frequencies of a stationary single-peaked ILSG in chains with di!erent anisotropy parameters over a wide range of maximum spin deviation s . The dot-dashed curves are frequencies  obtained by solving Eq. (4.6) with q"0. The open circles are MD simulation results. Top curve: A"0.8; Lower curve: A"2 (after Ref. [62]).

stationary ILSGs loses its signi"cance for traveling ILSGs. To launch a moving ILSG, the initial envelope is assumed to be single-peaked and can be found from Eq. (4.6) with non-zero q. The periodic boundary condition requires q to be of the form q"2pm/Na where m is an integer. For "xed maximum spin deviation the degree of localization increases as q increases. Once the initial envelope of the moving ILSG is obtained, numerical simulations can be used to investigate its motion. Fig. 26 shows the time evolution of four di!erent trajectories of moving ILSGs. Each curve in Fig. 26 presents the energy density distribution averaged over one ¹ , and is $+0 separated from the next by the time period of 20¹ . When both the relative strength of the $+0 anisotropy and the wave vector are small, as shown in the upper left panel (A"0.8, qa"2p/75) and the lower left panel (A"0.8, qa"p/15), the ILSG remains localized while moving, but detailed numerical studies show that a small spinwave tail is produced while the mode travels with uniform velocity. However, no noticeable slowdown is observed during the entire time interval as shown in the upper and lower left panels, respectively. As q grows, the traveling ILSG becomes increasingly unstable and quickly collapse into plane spin waves when q is su$ciently large (not shown). As either the strength of anisotropy A or the maximum spin deviation s grows,  it becomes increasingly di$cult for an ILSG to travel through the lattice. The upper right panel shows a case where the ILSG is pinned (A"2, s "0.5 and qa"2p/75). In order to keep  the ILSG (A"2) moving, one must either increase the wave vector (qa"p/15) as shown in the lower right panel or reduce the maximum spin deviation. Note that this traveling ILSG gradually slows down while its amplitude decreases. It is "nally pinned as illustrated in the lower right panel. Fig. 27a and Fig. 27b presents the power spectra of the transverse magnetic moment of two traveling ILSGs corresponding to the trajectories shown in the lower panels of Fig. 26. The peak

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Fig. 26. Comparison of traveling ILSGs with di!erent sets of parameters. The energy density is measured from the ground state: (a) Upper left panel, A"0.8, s "0.7 and qa"2p/75; (b) Lower left panel, A"0.8, s "0.7 and   qa"p/15; (c) Upper right panel, A"2,s "0.5 and qa"2p/75; (d) Lower right panel, A"2,s "0.5 and qa"p/15.   Each curve is separated from the next by the time period of 20¹ , and shows the mean energy density averaged over $+0 one ¹ . The lowest curve in each frame identi"es the starting time. The lattice contains 150 spins and the cyclic $+0 boundary condition is applied (after Ref. [62]). Fig. 27. Power spectra of the transverse magnetic moment of traveling ILSGs with various sets of parameters: (a) A"0.8, s "0.7 and qa"p/15; (b) A"2, s "0.5 and qa"p/15. In both cases the peak frequency in the power   spectrum su!ers a red shift from the internal frequency due to the motion of the ILSG (after Ref. [62]).

frequency of the power spectrum, u , is shifted from the internal frequency u by an amount of    *u"!qv, i.e., u "u !qv , (4.8)    since the ILSG is traveling at the velocity v. As expected the power spectra of traveling ILSGs are much broader than those of stationary ILSGs because the ILSGs are scattered by the discrete lattice due to the absence of continuous translational invariance. In both Fig. 27 a and Fig. 27 b a secondary peak at the plane wave u can be seen, which is more pronounced in Fig. 27b $+0 consistent with the observation that the traveling state in the lower right panel of Fig. 26 emits more spin waves per unit time. The above picture of a traveling localized mode is similar to that observed for the discrete nonlinear SchroK dinger lattice where the motion of the localized state can be described in terms of a Peierls}Nabarro (PN) barrier generated by the lattice discreteness [83,84]. For a traveling ILSG, the height of the PN energy barrier is the energy di!erence between the single-peaked ILSG and the double-peaked ILSG with the same frequency. This energy di!erence serves as a barrier against the motion of an ILSG. This barrier height increases as either the maximum spin deviation or the

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strength of anisotropy increases. Therefore an ILSG can be trapped by discreteness with increasing the anisotropy strength as observed in the right panels of Fig. 26. 4.4. Weak nonlinearity limit In previous sections numerical solutions to Eq. (4.6) have illustrated both stationary and traveling ILSG eigenvectors. Since, in the small spin deviation limits, the spatial size of such an ILSG becomes much larger than the lattice spacing the continuum approximation can be invoked to obtain approximate analytical solutions of ILSGs. These analytical solutions, though derived for ILSGs in the limit of weak anharmonicity, provide a qualitative understanding of how the anisotropy parameter and the maximum spin deviation change the properties of an ILSG with large spin deviation. 4.4.1. The reductive perturbation method Consider the two sublattices, A and B. Sublattice A contains all spins on even sites (up spins), while sublattice B contains all spins on odd sites (down spins). The appropriate new variables are

"s> and t "s> . (4.9) J J J J> As demonstrated by numerical simulations, both stationary and traveling ILSGs can be excited in easy-axis antiferromagnetic chains in the small amplitude limit although ILSGs with large amplitude would be trapped by the discreteness of the lattice. To represent the continuum approximation for ILSGs for a range of wavevectors, the reductive perturbation method [68,71] can be used in which the phase of the carrier wave is treated exactly and the envelope functions are treated in the continuum approximation. The solutions of Eq. (4.3) are expected to have the form

(t)" eH F (l, t)e IFJ , J HI H I t (t)" eH G (l, t)e IFJ, j, k"1, 2, 3,2 , J HI H I where e is a small parameter measuring the magnitude of spin deviations, and

(4.10)

h "q2la!ut . (4.11) J By construction the envelope functions F and G vary slowly with both position and time. HI HI Substituting Eq. (4.10) into Eq. (4.3), and equating powers of e and considering each harmonic separately, one obtains, to O(e), +(kX!A!2)F !(1#e\ IO?)G ,e IF"0 , I I I

(4.12)

+(1#e\ IO?)F #(kX#A#2)G ,e IF"0 . I I I

(4.13)

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Hence only linear terms contribute to this order, and Eqs. (4.12) and (4.13) can be satis"ed only if coe$cients of each harmonic are zero, which requires either kX"(A#2)!4 cos kqa and

F 2 cos kqa I" e\ IO? G X!A!2 I

(4.14)

or




F I "0 (4.15) G I for any k. Since the amplitudes of the fundamental harmonic (k"1) are non-zero Eq. (4.14) must hold for the case of k"1. Hence, the frequency of the carrier wave is the same as that of the plane spin wave of the same wavevector. One can show that the "rst equation in Eq. (4.14) cannot be satis"ed for any k'1. Eq. (4.15) therefore must hold for any k'1, that is, to O(e) no higher harmonics are generated. Collecting terms of O(e) in Eq. (4.3) and setting the coe$cient equal to zero gives

 

 

RG i RF I#2a I e IO?#(kX!A!2)F !(1#e IO?)G e IF"0 , (4.16) I I Rx 2JS Rt   I i RG RF I#2a I e IO?#(1#e IO?)F #(kX#A#2)G e IF"0 , (4.17) I I 2JS Rt Rx   I where t "et and x "ex with 2laPx. Because Eq. (4.15) holds for any k'1, the coe$cients for   kth harmonics involve only F and G . As the "rst equation in Eq. (4.14) is not satis"ed for any I I k'1, one can therefore conclude from Eqs. (4.16) and (4.17) that




F I "0 (k'1) . G I For k"1, we assume that F

(4.18) 

and G depend on x and t as   

m "x !ct , (4.19)    where c can be interpreted as the envelope velocity to "rst-order and is to be determined by the solvability condition. Substituting Eq. (4.19) into Eqs. (4.16) and (4.17) and with the help of the second equation in Eq. (4.14), one can obtain two coupled linear equations for the three variables, i.e., RF /Rm , F and G . Since the number of variables is larger than the number of equations,     the condition for nontrivial solution is that the two equations are identical hence, one "nds that c"v "2JS dX/dq"(4JSa/X) sin 2qa , (4.20)  where v is the group velocity of the linear spin wave with wavenumber q, and  iv /2JS#[(4a cos qa)/(X#A#2)]e\ O? RF X!A!2  , F !  (4.21) G "  1#e\ O?  Rm 1#e\ O?  where RF /Rm is an unknown function to be determined, and the envelope function F can be set    to zero for simplicity.

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Similarly collecting terms of O(e) in Eq. (4.3) and setting the coe$cient to zero gives









i RF RG RF RG RG I# I #2a I# I e\ IO?!2a I e IO? 2JS Rt Rx Rt Rx Rx      I 1 #F G GH e I\IF# [G (1#e\ IO?)#AF ] F FH e I\IF I I I I I I 2 I II II #(kX!A!2)F !(1#e\ IO?)G I I









e IF"0 ,

(4.22)



RG RF R F i RG RF I e IO? I# I #2a I# I e\ IO?#2a Rt Rx Rx 2JS Rt Rx      I 1 !G F FH e I\IF! [F (1#e\ IO?)#AG ] G GH e I\IF I I I I I I 2 I II II #(1#e\ IO?)F #(kX#A#2)G I I



e IF"0 .

(4.23)

where t "et and x "ex. By the same argument leading to Eq. (4.18), one can determine from   Eqs. (4.22) and (4.23) that




F I "0 (k'1) . G I

(4.24)

Eqs. (4.18) and (4.24) indicate that only fundamental harmonics appear up to the order of O(e). By reduction one can show that no higher harmonics can be generated up to any order which is consistent with the fact that no rotating wave approximation is needed in order to obtain numerically the eigenvector of an ILSG. The remaining part, i.e., the coe$cients of e F, yields two coupled nonlinear di!erential equations. These two di!erential equations can be simpli"ed by setting m "x !v t   

and q"t 

(4.25)

and seeking solutions of the form F (m , m , q) and G (m , m , q). With the help of the second       equation in Eqs. (4.14) and (4.21), one can combine the two nonlinear di!erential equations to obtain the following nonlinear SchroK dinger equation (NLS) for the envelope function F :  i

RF X RF #C(q) #P(q)F "F ""0 ,   Rm JS Rq 

(4.26)

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where C(q)"XdX/dq"4a cos 2qa!(v /2JS) ,  P(q)"2AX/(X#A#2) .

(4.27) (4.28)

4.4.2. Envelope solitons Eq. (4.26) has localized solutions (envelope solitons) when C(q)P(q)'0, and the solutions are given by F (m , q)"F sech  

 



P(q)  F (m !m) e (1POX $ O ,

  2C(q)

(4.29)

where both F and m are constants. Therefore to O(e) the spin deviation on even sites, (x, t), can

 be obtained in terms of x and t as





x!v t!x   e OV\S> SR , (4.30) ¸  where is the maximum spin deviation, ¸ the width parameter of the envelope given by

 2C(q)  1 (4.31) ¸"  P(q)

and

(x, t)" sech

 

*u"!(JSP(q)/2X)  (4.32)

is the nonlinear frequency shift. The spin deviation on odd sites, t(x, t), can then be found from Eq. (4.14) to be t(x, t)"![2 cos qa/(X#A#2)] e O? (x, t) .

(4.33)

This localized solutions given by Eqs. (4.30) and (4.33) is similar to the localized precession soliton in the collinear phase of an antiferromagnet, which was obtained from a continuum phenomenological description [14]. Although the functional forms of these solutions are slightly di!erent, they both exist in the gap of the linear spin wave spectrum. Envelope solitons with nonzero q still couple to electromagnetic radiation because they have a non-zero transverse magnetic moment while linear spin waves with nonzero q do not. The density of the transverse magnetic moment can be de"ned as






(x, t)#t(x, t) (x, t) 2 cos qa " 1! e O? . m>(x, t)" O 2a 2a X#A#2

(4.34)

Therefore the net transverse magnetic moment can be obtained by integrating m>(x, t) over the O lattice, i.e.,



M>(t)" O



 





p 2C(q)  nq¸  m>(x, t) dx" sech 2a P(q) 2 \



2 cos qa 1! e O? e OV\SHR , (4.35) X#A#2

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where the frequency of the net magnetic moment M>(t) rotating in the x}y plane is given by O uH"u![(JS)P(q)/u]  !qv . (4.36)

 Note that the frequency of a small amplitude ILSG decreases quadratically with its amplitude. The frequency of stationary ILSGs given by Eq. (4.36) with q"0 is plotted as the solid curve in Fig. 23 for comparison. Good agreement is obtained for ILSGs with maximum spin deviation less than 0.3 beyond which an apparent discrepancy occurs and the ILSG frequency drops faster than given by a quadratic law. It is clear from this "gure that single-peaked and doubled-peaked ILSGs converge in the small amplitude limit. Note that according to Eq. (4.35) the small-ampltiude stationary ILSGs (q"0) have a net magnetic moment independent of the maximum spin deviation. 4.5. Stability of stationary ILSGs In Section 4.2 the eigenvectors of stationary single-peaked and double peaked ILSGs have been presented. MD simulations in the absence of noise perturbation demonstrate that these localized spin wave modes are true periodic orbits with a single frequency. In realistic systems, however, noise perturbations are always present. It is therefore of experimental interest to investigate the dynamical behavior of ILSGs in the presence of such noise perturbations. Next we examine the linear stability analysis of ILSGs of both types. The eigenvector of a stationary ILSG satis"es a system of time-independent equation given by Eq. (4.6), which can be written formally in matrix form as ¸K sL "0 ,

(4.37)

where the eigenvector sL "(s , s ,2, s )2 ,   ,\ and ¸K is a tridiagonal matrix with the non-zero elements given by

(4.38)

¸ "X #(!1)L>((1!s #(1!s #A(1!s) , L> L LL  L\

(4.39)

¸ "¸ "(!1)L>(1!s . LL\ LL> L The time evolution of a perturbed ILSG can be written in the form

(4.40)

s>(t)"[s #u (t)]e\ SR , (4.41) L L L where the complex perturbation u (t) is separated from the stationary eigenvector. Substituting L Eq. (4.41) into the equation of motion given by Eq. (4.3) gives in matrix form





0 !¸K 1 d uL 0 " 2JS dt uL ' KK 0


 uL 0 uL '

.

(4.42)

Here the superscripts R and I represent the real and imaginary parts, respectively, and KK "¸K #M K .

(4.43)

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The matrix M K is also a tridiagonal matrix with non-zero elements given by s L M "(!1)L (s #s #As ) LL L> L (1!s L\ L s s L\ L M "(!1)L LL\ (1!s L\

(4.44) (4.45)

and s s L> L . (4.46) "(!1)L LL> (1!s L> The stability of an ILSG is determined by the eigenvalues, j's, of the system of equations given by Eq. (4.42). The perturbation grows exponentially if Re+j,'0 for one or more j's, where j"2JSK, and for any given ILSG eigenvector K can be obtained by solving the following equation: M

det[K#KK ¸K ]"0 .

(4.47)

When Eq. (4.47) is solved for single-peaked stationary ILSGs in a chain of 128 spins with periodic boundary condition, the real parts of the j's are essentially zero within numerical accuracy. The small positive values of some j's (Re+j,/u

10\ or less) for single-peaked ILSGs are $+0 explained by the inaccuracy of the ILSG eigenvectors and the numerical procedure used to solving Eq. (4.47). The double-peaked ILSG yields a di!erent picture. The maximum real part of j is plotted in Fig. 28 for double-peaked ILSG as function of the maximum spin deviation of the mode. The "lled circles are obtained for ILSGs in an antiferromagnetic chain with anisotropy parameter A"2 while the "lled squares are for the case of A"1. It appears that the double-peaked ILSG

Fig. 28. The maximum growth rate of perturbation as function of the maximum spin deviation for a double-peaked ILSG in an uniaxial easy-axis antiferromagnetic chain of 128 spins. The anisotropy parameters are A"2.0 for the "lled circles and A"1.0 for the "lled squares, respectively.

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becomes more unstable as either the maximum spin deviation or the anisotropy strength is increased. As the amplitude decreases, the maximum growth rate of the perturbation in a doublepeaked ILSG approaches zero, consistent with the fact that both single-peaked and double-peaked modes are governed by Eq. (4.26) in the continuum limit. Analytical analysis shows that single-peaked ILSGs are stable against noise perturbations while double peaked ILSGs are not. The question of how an unstable double-peaked ILSG evolves remains to be answered. To this end, let us consider the stability of stationary ILSGs in the presence of noise perturbation via MD simulations. Fig. 29 shows the time evolution of the two types of mode when the initial conditions of the eigenvectors include a random noise perturbation with maximum magnitude 0.5% of the largest spin deviation in the mode. Each curve in Fig. 29 presents the energy density distribution at a speci"c time. Fig. 29a demonstrates that the ILSG of single peak is stable since the shape of the mode is unchanged throughout the time interval, while Fig. 29b shows that the ILSG of double peaks is unstable and quickly evolves into a stable single-peaked mode. Fig. 30 shows the power spectra of the total transverse magnetic moment of both modes for di!erent initial conditions. In Fig. 30a, the power spectrum of the perturbed single-peaked ILSG

Fig. 29. Parity dependence of the stability of stationary ILSGs. The chain contains 150 spins and the anisotropy parameter A"2.0. Both ILSGs are initially randomly perturbed: (a) Time evolution of the energy density distribution of a single-peaked ILSG with maximum spin deviation of 0.7; (b) Time evolution of the energy density distribution of a double-peaked ILSG with maximum spin deviation of 0.58. The time t is measured in units of 2p/u (after Ref. $+0 [116]). Fig. 30. Power spectrum of the transverse magnetic moment of a stationary ILSG with di!erent symmetry under di!erent conditions. The antiferromagnetic chain consists of 150 spins with anisotropy parameter A"2.0: (a) Power spectra for a single-peaked ILSG with maximum spin deviation 0.7; (b) Power spectra for a double-peaked ILSG with maximum spin deviation of 0.58. Dot-dashed curves: unperturbed. Solid curves: perturbed with random noise. In both (a) and (b), the solid curve is shifted up by 4 decades from the dot-dashed curve for clarity (after Ref. [116]).

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(solid curve) contains a strong peak below the bottom of the plane wave spectrum corresponding to the local mode, as well as a much smaller peak at the q"0 AFMR frequency excited by the random noise perturbation. The peak of the solid curve matches the peak resulting from the exact eigenvector without perturbation (dot-dashed curve), indicating again that the mode with the single peak is stable. In Fig. 30b, the results for a double-peaked ILSG are shown. When a noise perturbation is included the clean power spectrum for a double-peaked mode (dot-dashed curve) is replaced by a new complex power spectrum (solid curve) peaked at a lower frequency corresponding to a new stable single-peaked ILSG since the perturbed unstable double-peaked ILSG evolves into a stable single-peaked one.

5. Antiferromagnetic chain with on-site easy-plane or biaxial anisotropy A large number of antiferromagnets actually are characterized by biaxial anisotropy while some have near gapless linear spin wave spectra which are often approximated by an easy-plane anisotropy [15,85]. In this section we examine stationary ILSMs in chains of classical spins coupled antiferromagnetically through nearest-neighbor exchange interactions with on-site biaxial anisotropy. The uniaxial easy-plane antiferromagnetic chain [57] then appears as a special case. 5.1. The model Hamiltonian We consider a one-dimensional antiferromagnetic chain of N classical spins which is described by the Hamiltonian H"2J S ) S !D (SV)!D (SW) , L L> V L W L L L L

(5.1)

where both the nearest-neighbor exchange constant, J, and the on-site anisotropy constants D and V D are positive and let D 5D . The z-axis is a hard axis and N is even. This chain is magnetically W V W ordered along the x-axis at low temperatures with spins pointing alternatively parallel or antiparallel to this axis. (The uniaxial easy-plane antiferromagnetic chain is recovered by setting D "D .) V W The e!ective magnetic "eld acting on nth spin is given by H(t)"! SLH"!2J(S #S )#2D SVe #2D SWe , L\ L> V L V W L W L

(5.2)

where e are unit vectors along positive x and y axes, respectively. Since spin waves in the model VW given by Eq. (5.1) are in general elliptically polarized the equations of motion for the x,y and z components of the classical spin vectors 1 dsV L "[(sX #sX )sW!(sW #sW )sX]!A sWsX , L\ L> L L\ L> L WLL 2JS dt

(5.3a)

1 dsW L"[(sX #sX )sV!(sV #sV )sX]#A sVsX L\ L> L L\ L> L VL L 2JS dt

(5.3b)

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and 1 dsX L"(sV #sV )sW!(sW #sW )sV!(A !A )sVsW , L\ L> L L\ L> L V W L L 2JS dt

(5.3c)

should be considered. The dimensionless variables are A "D /J and s "S /S where S is the VW VW L L magnitude of spin. With cyclic boundary conditions applied the eigenfrequencies of the linear spin waves are found from Eqs. (5.3) to be





A (1$cos qa) , X (q)"(A !A )(A #4)#4 sin qa# W ! V W V 2

(5.4)

where a is the lattice spacing between two adjacent spins and the dimensionless frequency X (q)"u (q)/2JS. Can an intrinsic localized spin wave mode exist above the top at the zone ! ! boundary of the plane wave spectrum? A necessary condition for an ILSM is that the substitution of q"p/2a#ii into Eq. (5.4) gives a real localized mode frequency but a complex frequency is found so this possibility can be excluded. The corresponding eigenvectors are given by





A #2(1Gcos qa) A #2(1Gcos qa) V ,!i,$ V ,$i , (5.5) X (q) X (q) ! ! where the upper (lower) sign is for the upper (lower) branch. Note that the polarization of spin waves for one branch is orthogonal to that of the other branch. A typical linear spin wave spectrum is plotted in Fig. 31 for the case of A OA . The two V W branches are degenerate at the Brillouin zone boundary. In this biaxial case a gap appears below the lower branch as a consequence of the breaking of rotational symmetry in the x}y plane. It is +sW , sX , sW , sX ,J    

Fig. 31. Linear spin wave spectrum for an antiferromagnetic chain with biaxial anisotropy. The anisotropy parameter is A "1.5 and A "1.0. The ILSR arrow identi"es the frequency of the intrinsic localized spin wave resonance while the V W ILSG arrow identi"es that of the intrinsic localized spin wave gap mode.

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clear from Eq. (5.4) that the gap X (0)"((A !A )(A #4) in the lower branch would disappear \ x W V for the hard axis uniaxial case where A "A . Since the small amplitude spin waves are elliptically V W polarized one may anticipate that the nonlinear resonant spin wave excitation which drops from the bottom of a branch will also be elliptically polarized. To "nd the eigenvector of a stationary ILSM one may use the ansatz sW(t)"sW cos ut, sX(t)"sX sin ut L L L L and sV(t)"(!1)L+1!(sW) cos ut!(sX) sin ut, . (5.6) L L L Here the squared terms in sW and sX cannot be neglected. Substituting Eq. (5.6) into Eqs. (5.3b) and L L (5.3c) one "nds, in the rotating wave approximation (RWA) where harmonics higher than the fundamental are ignored [30,34], the following coupled time-independent nonlinear equations: (!1)L>XsW"f (sX, sW)[sX #sX #A sX]#[f (sX , sW )#f (sX , sW )]sX , L L L L\ L> VL L\ L\ L> L> L

(5.7a)

and (!1)L>XsX"[f (sW , sX )#f (sW , sX )]sW#f (sW, sX)[sW #sW #(A !A )sW] , L L\ L\ L> L> L L L L\ L> V W L (5.7b) where






1 1 b !a , f (a, b)"(1!a)F ! , , 2, 22 1!a

(5.8)

and F(a, b, c, z) is the hypergeometric function [86]. With appropriate symmetry imposed Eqs. (5.7a) and (5.7b) can be solved for the eigenvector and eigenfrequency of a localized nonlinear spin wave mode. Below we consider only the single-peaked modes since double-peaked modes are unstable with respect to the noise. 5.2. Stationary intrinsic localized spin wave modes 5.2.1. Uniaxial easy-plane anisotropy For the uniaxial case where A "A "A the lower branch of spin wave spectrum is gapless and V W the only possible intrinsic localized spin wave modes are resonant modes (ILSRs) oscillating at frequencies below X (0)"2(A. Such in-band resonances occur because the upper and lower > branches have di!erent polarizations. A stationary localized resonance, if it exists, should bifurcate from the spatially uniform q"0 mode of the upper branch in Fig. 31. According to Eq. (5.5) the uniform mode eigenvector is (5.9) +sW , sX , sW , sX ,J+(A/2, !1, (A/2, 1, . L L L> L> For the ILSR one seeks the symmetric single peaked localized solution with the same pattern of sign alternation near the center of the resonance. Since the ILSR has a frequency within the lower branch of linear spin wave spectrum, far away from the mode center the localized solution would be

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mixed with the spatially uniform plane wave solution from the lower branch. Such mixing is demonstrated with the numerical solution of the set of coupled nonlinear equations given by Eqs. (5.7) for a chain of 64 spins with periodic boundary conditions. The intrinsic localized resonance does exist for a range of anisotropy parameters. As an illustration of the eigenvector the spin deviation versus site index of an ILSR is represented in Fig. 32 by the "lled circles for the parameters sX "!0.65 and A"1.0. The frequency of this  ILSR is found to be X "0.9301X (0) within the rotating wave approximation (RWA). As  > expected, near the center of the resonance the sign of the z component of the spin deviation alternates from one spin to the next while the sign of the y component does not change. Hence the time-periodic and spatially localized ILSR has an oscillating net magnetic moment in the y direction. Unlike the intrinsic localized gap modes of antiferromagnetic chains with easy-axis anisotropy described in Section 4, the spin deviations do not disappear with increasing distance from the mode center. Instead the localized excitation evolves into a weak plane wave pattern, as expected for a resonance. The plane wave pattern of the mode center has the eigenvector character of the q"0 mode from the upper branch. The seemly irregular o!-center region of sW far from the center L exhibits a smooth plane wave pattern under the transformation sWP(!1)LsW. This sign alternation L L of sW is a characteristic feature of the lower branch. The wavenumber q associated with the small L amplitude o!-center plane wave is obtained from the Fourier transform of sX in q space to be L q"0.6(p/2a), corresponding to a frequency of 0.9277X (0) in the lower branch, which is in good > agreement with the ILSR frequency given the fact that the small size of the lattice limits the accuracy of the wavenumber speci"ed.

Fig. 32. Shape of a stationary intrinsic localized spin wave resonance with the maximum spin deviation sX "!0.65 and  the anisotropy parameters A "A "1.0: (a) The spin deviation sX versus lattice site index n. The left side shows a factor V W L 5 expansion of the ordinate to display the plane wave character in the wings; (b) The spin deviation sW versus site index n. L The left side shows the same factor 5 expansion and a sign alternation to illustrate the resonant mode plane wave character. The sign alternation is a characteristic feature of the lower branch (after Ref. [57]).

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Fig. 33. Comparison between the rotating wave approximation(RWA) frequency and the MD simulation frequency for a stationary ILSR versus spin deviation. The anisotropy parameters A "A "1.0. The dot-dashed curve is obtained V W using the RWA, and the open circles are results calculated from the "rst 820 ¹ (0) MD simulation points of the net > magnetic moment MW(t) (after Ref. [57]). Fig. 34. Stability of the unperturbed ILSR shown in Fig. 32. The energy density shown here is measured from the ground state energy and averaged over one period. The time is measured in units of ¹ (0) (after Ref. [57]). >

The dot-dashed curve in Fig. 33 shows the ILSR frequency found in the RWA as a function of the maximum spin deviation sX . The frequency drops further into the lower spin wave band as  sX increases. These RWA frequencies can be compared to molecular dynamics (MD) simulation  frequencies. Since the RWA has been used to obtain the ILSR eigenvector the mode stability needs to be checked. In MD simulations the numerically determined eigenvector is used as the initial condition, i.e.,s "((!1)L(1!(sX), 0, sX), and the discrete equations of motion for the x}y}z spin L L L components are integrated numerically by using the fourth-order Runge}Kutta method with a time step of ¹ (0)/200 where ¹ (0)"p/JSX (0). > > > These molecular dynamics simulations show that an ILSR with modest spin deviation can last many hundreds of periods without apparent decay. For example, the time evolution of the ILSR energy density averaged over one period is plotted in Fig. 34. The parameters are the same as those in Fig. 32. No decay can be seen after 800 ¹ (0). When a noise perturbation ((0.1%) is added, the > ILSR in Fig. 34 moves after about 800 ¹ (0) while it remains localized as the perturbation > develops further. As the maximum spin deviation is increased, the amplitude of the plane wave component in both wings of this excitation increases leading to instability and delocalization after su$cient time as might be expected for a localized excitation which is nearly degenerate with some modes in the plane wave spectrum. Since the ILSR is a collective excitation, a calculation of the power spectrum of the total magnetic moment, M(t)" s (t), is a useful method with which to identify the relative strength of LL the di!erent frequency components of the excitation as well as to check the accuracy of the RWA.

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Fig. 35. Power spectrum of the net magnetic moment MW(t) of the stationary ILSR shown in Fig. 32. Besides the strong peak at 0.9265u (0) corresponding to the ILSR much weaker peaks appear in the power spectrum at u (0) and at 3u , > >  indicating that linear spin waves and the third harmonic are also excited due to the inaccuracy in the eigenvector resulting from the rotating wave approximation.

In the uniaxial case MX commutes with the Hamiltonian, and is therefore a constant of the motion. Because of the sign alternation in sX, MX is indeed zero. The total magnetic moment is thus linearly L polarized. Fig. 35 shows the log power spectrum of MW(t) for the ILSR plotted in Fig. 32. This power spectrum is calculated from the "rst 820 ¹ (0) MD data values. A strong peak appears at > X "0.9265X (0), which should be compared to the value 0.9301X (0) found in the RWA. Since  > > the eigenvector is not an exact eigenvector due to the RWA, linear spin waves are also excited. However, the strength of the power spectrum peak at X (0) is more than 3 orders of magnitude > weaker than the resonance peak. Peaks at the third and "fth harmonics are also present in the power spectrum, but their strengths are at least four orders of magnitude weaker than the peak corresponding to the fundamental ILSR frequency indicating that the RWA is a good approximation for this nonlinear system. The MD simulation frequency versus spin deviation is plotted in Fig. 33 as open circles and these values compare well with the RWA frequencies represented by the dashed line. The MD simulation frequency is slightly lower than the corresponding RWA frequency and the di!erence between the two becomes larger as the maximum spin deviation increases. The di!erence grows because more and more energy goes into higher harmonics but even so the overall agreement is satisfactory over the entire range. 5.2.2. Biaxial anisotropy When the rotational symmetry in the easy-plane is broken by setting A OA , a gap appears V W below the lower branch of the spin wave spectrum. Now both an intrinsic localized gap modes and a resonance may appear. The eigenvectors and frequencies of the ILSGs and ILSRs can be obtained by solving Eq. (5.7) including initial guesses of the appropriate pattern of sign alternation as given by Eq. (5.5) for q"0.

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Fig. 36. Shape of a stationary intrinsic localized spin wave gap mode with the maximum spin deviation sW "0.7 in  a chain of 128 spins characterized by A "1.5 and A "1.5: (a) The spin deviation sW versus lattice site index n; (b) The V W L spin deviation sX versus site index n. L Fig. 37. Dependence of the plane-wave wing amplitude on the center amplitude of ILSRs. The parameters for the antiferromagnetic chain are the same as in Fig. 36: (a) sX "0.335; (b) sX "0.480; (c) sX "0.581.   

When Eq. (5.7) is solved numerically to "nd the stationary ILSG eigenvector for a 128-spin antiferromagnetic chain characterized by A "1.5 and A "1.0, Fig. 36 is the result. Both y and V W z components of spin deviation vanish with increasing distance from mode center, as expected for a localized mode outside plane-wave spectrum. The ILSG in a biaxial easy-plane antiferromagnetic chain is elliptically polarized, and has a nonzero total magnetic moment oscillating along the hard axis. Fig. 37 presents the shapes of three stationary ILSRs in the same chain to illustrate the dependence of the wing amplitude on the center amplitude. Here only the z-components are plotted. Although the ILSRs have no nonzero net magnetic moment in the z direction, they do have a non-zero net magnetic moment oscillating along the y-axis, orthogonal to that associated with the ILSG. The wing amplitude increases rapidly with the center amplitude. It has been shown in the continuous  model that the amplitude of the `far-"elda radiation is exponentially small in the breather amplitude, and the decay is negligible for small amplitude breathers [66,87]. The numerical calculations described here suggests a similar dependence of the wing amplitude on the mode center amplitude for an ILSR although no analytical solution is available due to the complexity of the discrete system. MD simulations demonstrate that both ILSG and ILSR modes are long-lived. Since the frequency of an ILSG is outside the extended spin wave spectrum, its instability is caused entirely by the use of the rotating wave approximation. The resulting eigenvector becomes more inaccurate as the mode amplitude increases. On the other hand, since an ILSR is an in-band resonance, in

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addition to the rotating wave approximation, the coupling between the ILSR and extended spin waves of lower branch also contributes to its instability. The strength of the coupling is measured by the wing amplitude, which increases with the mode center amplitude and the density of state of the lower branch spin waves that are at the same frequency as the ILSR. Thus, the lifetime of an ILSR decreases much faster than a linear law with an increase in the mode amplitude. While ILSGs exist for the entire parameter range, no ILSR can occur for large anisotropy parameters when the frequency of the q"0 extended spin wave mode of the upper branch is in the neighborhood of the frequency of the zone boundary spin wave mode of the lower branch. This result may be understood qualitatively as follows. Since the polarization of an ILSR at the zone center is similar to that of the extended zone boundary spin wave modes of the lower branch, the ILSR is expected to interacted more strongly with the zone boundary spin waves. Furthermore, the lower branch has a large density of state at the zone boundary because of the #atness of the spectrum so there are more states with which to interact. The precise parameter range for the existence of ILSRs will be discussed in Section 5.3. Contact can be made between the anharmonic gap mode and resonant mode which have orthogonal net magnetic moments and the ILSG described in Section 4. Setting D "0 in Eq. (5.1) W reduces it to the Hamiltonian of an easy-axis antiferromagnetic chain with x-axis as the easy-axis. As D P0, the two dispersion curves shown in Fig. 31 converge into one, and the two elliptically W polarized orthogonal ILSR and ILSG result in a circularly polarized ILSG, which does not require the rotating wave approximation due to the uniaxial symmetry. 5.3. Existence conditions The numerical calculations described in Section 5.2 indicate that ILSGs can exist for any anisotropy parameters whereas ILSRs can only exist in a certain parameter regime. In this section the existence conditions are presented for small amplitude ILSGs and ILSRs for the continuum limit. The condition for the existence of small amplitude ILSGs also applies to ILSGs with large amplitude beyond the continuum approximation; however, it should be emphasized that the condition for the existence of small amplitude ILSRs does not guarantee the existence of large amplitude ILSRs. 5.3.1. Gap modes Since an anharmonic gap mode has the same pattern of sign alternation as the zone center extended spin waves of the lower branch, new variables can be introduced t "(!1)LsW,

"sX . L L L L Since Eq. (5.8) reduces to

(5.10)

(5.11) f (a,b)+1!a!b ,   within the continuum approximation up to the lowest nonlinear terms, that is, cubic terms, a nonlinear SchroK dinger equation for the envelope function t(x) can be obtained from Eqs. (5.7a) and (5.7b), namely, a

dt !at#bt"0 , dx

(5.12)

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where a is the spacing between two adjacent spins, [X (0)!X](A #4) V a" \ (A #4)!X V

(5.13)

and X[3X#X (0)]#(A #4)[X#3X (0)] \ V \ b" . 8[(A #4)!X](A #4) V V In this small amplitude limit, the envelope function (x) is given by the linear relation

(5.14)

X

(x)"! t(x) . (5.15) A #4 V Since Eq. (5.12) has localized solution if and only if a'0 and b'0, one can obtain the existence condition for the gap mode from Eqs. (5.13) and (5.14), this is, X(X (0). Thus, nonlinear gap \ modes always exist as long as there is a gap in the spin wave spectrum. The localized solution of Eq. (5.12) centered at x is  2a  x!x  . (5.16) sech a t(x)" b a









Since the frequency of a small-amplitude ILSG is just below X (0), one can determine from \ Eq. (5.13) the width of the localized mode lJ1/(*X and the central amplitude t J(*X where K *X is the frequency shift of the ILSG from X (0). An ILSG therefore becomes delocalized, \ approaching the extended spin wave, as its central amplitude decreases. 5.3.2. Resonant modes Since ILSRs, if they exist, have the same sign alternation as the extended zone center spin wave modes of the upper branch, the appropriate new variables now are t "sW,

"(!1)LsX . (5.17) L L L L In a similar fashion as described in the previous section, the envelope function t(x) also satis"es a nonlinear SchroK dinger equation given by Eq. (5.12) with parameters a and b replaced by X (0)!X A a" > X!A V V

(5.18)

and X(3X#A)#X (0)(X#3A) V > V . b" (5.19) 8A (X!A) V V The condition for the existence of localized solution is that both a and b are positive, which is equivalent to A (X(X (0) . V >

(5.20)

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Since from Eq. (5.4), X (0)"(A (A !A #4). For A *A , Eq. (5.20) can be satis"ed if and > V V W V W only if A (4 . W

(5.21)

The envelope function of an ILSR is given by Eq. (5.16) with the corresponding parameters a and b given by Eqs. (5.18) and (5.19). The plane-wave wings are neglected in this small amplitude limit. It can be easily seen from Eqs. (5.4) that (5.21) is equivalent to the condition that X (q) have > positive curvature at q"0. It should be emphasized again that Eq. (5.21) is the condition for the existence of small-amplitude ILSRs. Since the frequency shift of an ILSR from X (0) increases with > its amplitude as *XJt , and according to Eq. (5.20), the frequency of an ILSR is allowed to lie in K a narrow range just below X (0) as A approaches 4, no large-amplitude ILSRs can exist in this > V case. The existence condition of ILSRs given by Eq. (5.21) is consistent with observations in numerical searches of ILSRs. Like the ILSR in ferromagnetic chains examined in Section 3, the ILSR in easy-plane antiferromagnetic chains is strikingly di!erent from nonlinear resonant modes in other models [66,67] in that its fundamental frequency instead of higher harmonics is in the plane-wave spectrum. In addition, an intrinsic localized spin wave gap mode can exist in the gap below the lower branch of the spin wave spectrum when the uniaxial symmetry is broken. Since the ILSR and ILSG modes in an antiferromagnetic chain are elliptically polarized and have nonzero total magnetic moments orthogonal to each other, unlike the non-ir-active ILSR in isotropic ferromagnetic chains reviewed in Section 3, they can couple to far-IR radiation. The long-lived nonlinear excitations explored here approach continuously without threshold the corresponding spin waves of linear theory as their amplitudes decrease. They are thus strikingly di!erent from the topological sine-Gordon kink excitations found in 1D easy-plane magnets [20,21,88]. The key feature in the nonlinear dynamics problem of ILSR is the polarization di!erence between the two plane-wave branches. In numerical simulations studies of ILSRs the smaller the frequency of the q"0 mode in the upper branch, the less strongly coupled the resulting ILSR is to the other branch of the plane-wave spectrum.

6. Modulational instability of an extended nonlinear spin wave in an easy-axis antiferromagnet So far we have illustrated that both analytical studies and numerical simulations do predict the existence of intrinsic localized spin wave modes in various magnetic chains; however, there remains the fundamental question how best to excite such atomic scale large-amplitude excitations in homogeneous discrete lattices. Modulational instability (MI), which refers to the exponential growth of certain modulation sidebands of nonlinear plane waves propagating in a dispersive medium as a result of the interplay between nonlinearity and dispersion e!ects, has been studied in a variety of "elds [89}92]. In most of these cases, MI appears in continuous media where the propagation of nonlinear waves is usually governed by nonlinear SchroK dinger-type partial di!erential equations. Computer simulations and experiments [73,93,94] have demonstrated that one of the main e!ects of the modulational instability is the generation of localized pulses. For example, subpicosecond soliton-like optical pulses have been experimentally generated from a weakly

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modulated input via an induced modulation instability in single-mode optical "bers [93]. Given this observation, the modulational instability mechanism has recently has been proposed and examined as a possible way to produce energy localization in discrete lattices [69,70,72}74,94}96]. Although many aspects of MI in discrete systems are the same as those in continuous media, the discreteness can drastically modify the modulational instability parameter space as deduced from a continuum or even semi-discrete approximation [69]. The advantage of making use of modulational instability to create localized excitations in discrete lattices is that because of the lack of continuous translational symmetry the localized pulse generated by the nonlinear instability can be trapped by discreteness to form strongly localized long-lived excitations. In this section the focus is on the modulational instability of extended nonlinear spin waves in antiferromagnetic chains with easy-axis anisotropy. 6.1. Traveling nonlinear extended waves The one-dimensional antiferromagnetic chain to be investigated is described by the Hamiltonian given by Eq. (4.1). Because of the translational symmetry of the underlying lattice, the equation of motion, Eq. (4.4), can support nonlinear extended spin wave modes as well as stable intrinsic localized spin-wave modes in the gap below the standard antiferromagnetic resonance frequency. The traveling extended spin wave mode with wavevector q and frequency u is found by substituting into Eq. (4.3) the following circularly polarized trial solution: s> (t)"f e LO?\SR>F , L s> (t)"ge L>O?\SR>F , L>

sX "(1!f  , L sX "!(1!g . L>

(6.1)

Here h is a constant phase, and both spin deviations f and g are real and non-negligible. De"ning  the parameter r"2/(A#2) and inserting Eq. (6.1) into Eq. (4.3) gives for the ratio of the two amplitudes g r cos qa "! , f (1$(1!r cos qa)(1!f )

(6.2)

where the $ signs designate two degenerate branches. Owing to the symmetry between the up-spins and down-spins the solution with positive sign in the denominator is chosen so that "g/f "(1. Given the spin wave amplitude f, the frequency as a function of wavevector is X(q, f )"2(1!af #(A!2a cos qa)(1!f  ,

(6.3)

where a"!g/f ; hence, the frequency of an extended nonlinear spin wave depends on both its wavevector and its amplitude. For the small amplitude case f ;1, one "nds AX (q)  f , X(q, f )+X (q)!  X (q)#X (n/2a)  

(6.4)

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where X (q)"((A#2)!4 cos qa is the linear spin wave frequency. Eq. (6.4) indicates that the  nonlinear spin wave frequency decreases quadratically with increasing amplitude. 6.2. Modulational instability of extended waves 6.2.1. Linear stability analysis To study the modulational stability of the extended nonlinear spin waves a perturbed nonlinear spin wave of the form s> (t)"( f#b #it )e LO?\SR>F , L L L (6.5) s> (t)"(g#b #it )e L>O?\SR>F , L> L> L> is introduced where f, g and u are related by Eqs. (6.2) and (6.3), and the perturbations +b (t), and L +t (t), are real and are assumed to be small in comparison with the parameters of the carrier wave. L (Note that in this form the perturbation is added in a frame rotating with the exact periodic solution.) The advantage of using Eq. (6.5) is that it ensures that the resulting linearized equations of the perturbation have constant coe$cients instead of time-dependent coe$cients as would be obtained in the usual stability analysis of periodic solutions. Since the perturbations +b (t), and L +t (t), are arbitrary, this does not involve any approximation. Inserting Eq. (6.5) into Eq. (4.3) and L separating the real and imaginary parts, one obtains, up to linear terms of +b (t), and +t (t),, L L a system of coupled di!erential equations of +b (t), and +t (t),, which can be solved by expanding L L the perturbation in terms of Fourier components as







b b (Q) L "  e L/? , (6.6) t t (Q) / L  b b (Q) L> "  e L>/? . (6.7) t t (Q) / L>  This decomposition allows one to identify the time evolution of each individual component. Since +b (t), and +t (t), are real, L L bH(Q)"b (!Q) and tH(Q)"t (!Q) where (i"0, 1) . (6.8) G G G G Comparing the coe$cients for the same Fourier component gives







b (Q)  M M d b(Q)   "2JS dt t (Q) M M    t (Q)  where M 's are 2 by 2 matrices given by GH 0 M "  !2i(1!g sin Qa sin qa









b (Q)  b (Q)  , t (Q)  t (Q)  2i(1!f  sin Qa sin qa 0

(6.9)



,

(6.10)

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M "M ,  




(6.11)

2a(1!f  cos qa M "  !2(1!g cos Qa cos qa




M "!M #  

(A!2a) f  (1!f 



2(1!f  cos Qa cos qa , 2 ! (1!g cos qa a gf 2 cos Qa (1!g

cos qa

gf

!(A!2/a)g

!2 cos Qa (1!f 

(1!g

cos qa



(6.12)

.

(6.13)

The general solution of Eq. (6.9) is a superposition of terms having the time dependence e\ SKR where the u 's are the frequencies of the modulation wave relative to the extended nonlinear

spin wave and the !iu 's are the eigenvalues of the 4 by 4 matrix 2JSM. The stability of the

extended nonlinear spin wave mode is determined by the imaginary part of u , i.e., the extended

nonlinear spin wave is unstable when the Im+u ,'0, otherwise it's stable. De"ning the dimen sionless frequency j"u /2JS, the j's are obtained from

det"j(q, Q)I!iM""0 . (6.14) Eq. (6.14) determines the condition for the stability of an extended nonlinear spin wave with wavevector q with respect to the modulation with wavevector Q. Since M "M "0 when   Q"0, j(q, 0)"0 is always one of the eigenvalues of matrix M. Note that since the trace of the matrix M is zero regardless of the values of q and Q, the condition for stability is that all eigenvalues of M are imaginary. Otherwise there must be at least one eigenvalue having a positive real part. Furthermore, the symmetric modulation sidebands at q$Q have the same growth rate since j(q,!Q)"!jH(q, Q). Since there is no simple analytical form for the dispersion relation j(q, Q) for arbitrary q, Eq. (6.14) has to be solved numerically to determine the domains of instability in the (q, Q) plane. There are two important cases that can be solved analytically: the zone center and zone boundary spin waves. These two cases will be considered "rst before examining spin waves with arbitrary wavevectors. 6.2.2. The uniform mode The case of the q"0 extended nonlinear spin wave is particularly important since the condition for an instability also tells one when stationary ILSGs can exist [62]. Since M "M "0 when   q"0, the case of the zone center spin wave mode becomes particularly simple as Eq. (6.14) becomes det"j(0, Q)I#M M ""0 ,   which yields the following dispersion relation for the modulation wave:









B j (0, Q)" aC# !4E sin Qa$ ! a





 B  !4BC sin Qa , aC# a

(6.15)

(6.16)

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where 2 B" !(A#2)g!2a((1!g)(1!f ) , a

(6.17)

2 C"2a!(A#2) f ! ((1!g)(1!f ) , a

(6.18)

E"gf!((1!g)(1!f ) .

(6.19)

In the case of an isotropic chain (A"0), B"C"0 and E"!1 so that Eq. (6.16) reduces to j (0, X)"4 sin Qa , (6.20) ! which is the linear spin wave dispersion relation and is positive for any Q. This is to be expected since the q"0 mode in an isotropic chain is simply rotation of the whole lattice in spin space by an arbitrary amount and any small amplitude perturbation to this state is a superposition of linear spin waves. An instability can however occur for the zone center spin wave mode in an anisotropic antiferromagnetic chain. Since j (0, Q)50 holds for any Q, this branch is stable. To determine the > instability condition one need only focus on the j (0, Q) branch. It is clear from Eq. (6.16) that \ j (0,0)"0 and the instability occurs if and only if j (0, Q) becomes negative for some non-zero Q. \ \ This condition requires that 2E (aC#B/a) !BC . sin Qa( 4E

(6.21)

The RHS of Eq. (6.21) is always positive for any spin wave amplitude f, and is proportional to f  in the small f limit. In a "nite periodic lattice of size N the smallest wavevector is Q"2p/Na, thus there exists an amplitude threshold f &O(1/N) so that only zone center spin waves with amplitude  larger than f are unstable; however, in a 1-D solid N&10 and the amplitude threshold becomes  negligibly small. Thus, the zone center spin wave is always unstable to long wavelength modulation. Also it can be shown that the RHS of Eq. (6.21) can become greater than 1 for su$ciently large A and f so that the q"0 extended nonlinear spin wave is unstable to any perturbation. When the RHS of Eq. (6.21) is less than 1 the critical wavevector is given by Q a"arcsin 




 

2E (aC#B/a) !BC  , 4E

(6.22)

and the extended nonlinear spin wave mode is unstable to modulation with "Q"(Q .  Fig. 38 presents an example of the stability region of the q"0 extended nonlinear spin wave in the (Q, f ) plane for two di!erent anisotropy parameters. The anisotropy parameter A"1.0 in Fig. 38a demonstrating that the q"0 spin wave is stable to modulation waves with large wavevector. Fig. 38b shows the results for the more anisotropic case A"2.0 where the spin wave with large amplitude can become unstable with respect to any modulation wavevector.

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Fig. 38. Regions of modulational instability in the (Q, f ) plane for the q"0 extended spin wave: (a) The anisotropy parameter A"1.0. The large Q region is always stable regardless of the spin wave amplitude; (b) The anisotropy parameter A"2.0. As the strength of the anisotropy increases, the stable region shrinks. The spin wave with large amplitude becomes unstable to perturbation of any wavevector (after Ref. [102]).

In the small amplitude limit, Eqs. (6.16) and (6.22) can be simpli"ed to yield more transparent results. Keeping only lowest terms of f, Eq. (6.22) gives 1 Q a" (A/*X )f ,   1#*X /2X (0)  

(6.23)

where *X is the bandwidth of the linear spin wave band. In this small amplitude limit the critical  wavevector of the modulation wave is linear in the spin wave amplitude. Since the q"0 spin wave with a small amplitude is unstable only to long wavelength (Qa&f ) modulation, Eq. (6.16) can be simpli"ed in lowest order, i.e., f , to yield r 4(1!r) f  sin Qa# sin Qa . j (0, Q)"! \ 1!r 1#(1!r

(6.24)

The RHS of Eq. (6.24) is 0 at Q"0 and negative for 0("Q"(Q , hence j (0,Q) is purely  \ imaginary in the small Q region. The maximum growth rate is found from Eq. (6.24) to be






1 A f Im+j (0, Q), " \

 2 1#*X /2X (0)   for the modulation wavevector Q "Q /(2 .

 

(6.25)

(6.26)

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6.2.3. The zone boundary mode At the Brillouin zone boundary q"p/2a and the spin deviation at odd sites g"0, hence a"0. However, the ratio cos qa/a is well de"ned, i.e., cos qa 1 #(1!f  p P as qP . r a 2a

(6.27)

Inserting Eq. (6.27) into Eq. (6.14) gives an equation for j(p/2a, Q), namely,




j




p p , Q #a j , Q #a "0 ,  2a  2a

(6.28)

where the coe$cients a and a are   8A a "16(1!f )sin Qa# f (1#(1!f )sin Qa  r

(6.29)

and 4 a "! (1#(1!f  )#8(1!f sin Qa ,  r

(6.30)

respectively. The dispersion relation for the perturbation wave j(p/2a, Q) is described by




p !a $(a!4a   . j ,Q " ! 2a 2

(6.31)

Since a (0 and a 50, the stability condition becomes a!4a 50, which leads to     4 4 (1#(1!f  ) ! (1#(1!f  )! f  sin Qa50 . r r r





(6.32)

The extended nonlinear spin waves at the Brillouin zone boundary are stable to perturbations by any wavevector since the inequality given by Eq. (6.32) holds for any Q and f since sin Qa41. 6.2.4. Instability region for spin waves of arbitrary wavevector Although for nonlinear spin waves of arbitrary wavevector q the dispersion relation j(q, Q) has to be obtained by numerically solving Eq. (6.14), still it is of value to consider some qualitative properties of the eigenvalues. Since det"jI!iM""det"(jI!iM )#(jI!iM )M (jI!iM )\M " , (6.33)      and iM , M and M are real 2 by 2 matrices, Eq. (6.14) is a fourth order polynomial of j with    real coe$cients. Thus the j(q, Q)'s are therefore either real or form complex conjugate pairs. In regions where the nonlinear extended spin waves are stable Im+j(q, Q),"0 for any of the four j(q, Q)'s, while at least one of the four j(q, Q)'s has positive imaginary part in unstable region. Fig. 39a shows a typical plot of the regions of modulational instability in the (Q, q) plane which are determined by the values of Im+j(q, Q), for an anisotropic antiferromagnetic chain. The dot-dashed lines separate the regions of stability (I and III) and region of instability (II). For a given

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Fig. 39. (a) Diagram of regions of modulational instability in the (Q, q) plane for extended spin waves of amplitude f"0.2 in an easy-axis antiferromagnetic chain. The anisotropy parameter A"2.0. Regions I and III are stable regions, and region II is unstable. Spin waves with wavevector larger than p/4a are stable to a perturbation of any wavevector. (b) The real and imaginary part of the two relevant j(q, Q)'s along the long-dashed line CD shown in (a). Solid lines: real parts; dot-dashed line: imaginary parts. The two j(q, Q)'s converge at instability boundaries.

spin wave amplitude f, the area of the unstable region grows with increasing anisotropy parameter A. The lower boundary moves towards the direction of large Q, while the upper boundary approaches q"p/4a. However, spin waves with q'p/4a are stable against any perturbation independent of the values of Q and f. On the other hand, as the chain becomes more and more isotropic with A approaching zero, the area of the unstable region shrinks until it disappears completely for the isotropic chain. Numerical solutions of Eq. (6.14) demonstrate that among the four j(q, Q)'s, two of them are always real in the (Q, q) plane, and therefore are irrelevant with respect to the spin wave instability. To see how the relevant j(q, Q)'s evolve from a stable region to an unstable one, the real and imaginary parts of the two j(q, Q)'s are plotted in Fig. 39b for "xed Q along the long-dashed line CD shown in Fig. 39a. The two j(q, Q)'s converge at the instability boundaries to form degenerate double roots while they form a complex conjugate pair in the unstable region (II). Thus the instability boundaries are determined by the condition that Eq. (6.14) have double roots. That the existence of an intrinsic localized mode is always accompanied by an instability of the corresponding extended nonlinear waves has been shown in a number of studies for various lattice dynamical models [69,72,74,95]. The fact that ILSMs can occur only in the gap below the standard antiferromagnetic resonance frequency at q"0 while no ILSM exists at the Brillouin zone boundary [62] is in agreement with Fig. 37 where only extended spin waves with small q are

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unstable to a long wavelength modulation while the zone boundary spin waves are stable to modulation of any wavelength. 6.3. Comparison between numerical simulations and analytical results In an easy-axis antiferromagnetic chain linear stability of an extended nonlinear spin wave with wavevector q modulated by a small-amplitude wave of wavevector Q is determined by the dispersion relation j(q, Q) which can be obtained from Eq. (6.14). Although such linear stability analysis can determine the important domain in parameter space and predict quantitatively how the amplitude of a modulational sideband evolves at the onset of the instability, such analysis is based on the linearization around the unperturbed carrier wave. Since the linear approximation must fail at large time scales as the amplitude of the unstable sideband grows exponentially and since it neglects additional combination waves generated through wave-mixing processes which can become signi"cant at large time scales if its wavevector falls inside an instability domain, such linear stability analysis cannot determine the long-time evolution of a modulated extended nonlinear spin wave. This requires the application of molecular dynamics simulations. For such numerical simulations for easy-axis chains with various anisotropy parameters the initial conditions involve coherently modulated extended nonlinear spin waves of the form





b s> (0)" f# [b (Q)e L/?#c.c#i(t (Q)e L/?#c.c.)] e LO? ,  L 2 





b s> (0)" g# [b (Q)e L>/?#c.c#i(t (Q)e L>/?#c.c.)] e L>O? , L>  2 

(6.34)

where c.c. denotes the complex conjugate, (b (Q), b (Q), t (Q), t (Q)) is a normalized eigenvector of     the M matrix, and b is a small parameter measuring the relative strength of the modulation wave to the carrier wave, typically &0.01. The amplitudes f and g are related by Eq. (6.2). Since "b#it"O"bH#itH", the two satellites at q$Q have di!erent strengths except when q"0. Given s>(0), the z-components of spins can be obtained from L (6.35) sX(0)"(!1)L(1!"s>(0)" . L L Once an initial condition is given the time evolution of a modulated spin wave can be investigated with MD simulations. In order to monitor the time evolution of individual Fourier components, one requires the complete spatial Fourier transform of spin deviations






N N ,\ ! (p4 . (6.36) m(p,t)" s>(t)e\ LNp,, L 4 4 L The growth rate of each individual Fourier component can be obtained by the least-squares "tting of "m(p, t)" over the "rst few periods during which time it is expected to grow at the rate of 2 Im+j(q, Q),. A chain of 128 spins with periodic boundary conditions has been used as a speci"c example. The anisotropy parameter is taken to be A"1.0, and the spin wave amplitude f"0.2. Figs. 40 and 41 shows the long time evolution of the carrier wave with wavevector q"15p/64a modulated by

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Fig. 40. Time evolution of the main Fourier components of an extended spin wave with q"15p/64a and f"0.2 modulated by a small amplitude wave with Q"17p/64a. The components are at q (solid curve), q#Q (dot-dashed curve) and q!Q (long-dashed curve). The anisotropy parameter is A"1.0, and time is measured in units of ¹ , the period $+0 of the uniform mode. Fig. 41. Time evolution of the complete Fourier spectrum of the extended spin wave described in Fig. 40. After a su$ciently long time, combination modes appear due to wave mixing processes.

small amplitude waves with wavevectors Q"$17p/64a which falls in the unstable region. The exponential growth of q$Q satellite sidebands at the initial stage of instability is obvious as can be seen in the log-linear plot of Fig. 40. Fig. 41 shows the time evolution of the complete Fourier spectrum where additional combination waves generated from wave-mixing processes can be seen after about 300 ¹ as the instability develops further. $+0 The growth rates as a function of the modulation wavevector for the running carrier waves with various wavevectors are plotted in Fig. 42. The solid curves represent analytical results obtained from diagonalizing the matrix M while the "lled circles are MD simulation results. The excellent agreement between these two sets of results demonstrates that the linear stability analysis does give a quantitatively correct description of the instability onset. Fig. 42 shows that while the carrier waves with small q are unstable to long wavelength modulation (small Q), a carrier wave of large q (q"15p/64a) is stable to long wavelength modulations but unstable to some short wavelength modulations (large Q). This "nding should be contrasted with those found in Ref. [69] for a monatomic Klein}Gordon chain where the small Q region is always the unstable region as long as an instability occurs for the corresponding carrier wave. Note that because combination waves are neglected in the linear analysis the prediction of stability does not necessarily rule out the occurrence of instability in the long time evolution. This point is illustrated by the long time evolution of a perturbed carrier wave with wavevector q"15p/64a plotted in Figs. 43 and 44 where the modulation wavevectors Q"$p/8a lie in the stable region as shown in Fig. 42a. The Fourier component corresponding to the carrier wave

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Fig. 42. Comparison of analytical and MD simulation results of the growth rate of modulational waves for carrier waves with various wavevectors. The parameters are A"1.0 and f"0.2. X is the standard antiferromagnetic resonance $+0 frequency. The wavevectors of carrier waves are: (a) 15p/64a; (b) 7p/32a; (c) p/8a, and (d) 0. The solid curves are analytical results while the "lled circles are MD simulation results (after Ref. [102]).

Fig. 43. Long time instability induced evolution of speci"c fourier components by combination excitations. The components are at q (solid curve), q#Q (dot-dashed curve), q!Q (long-dashed curve), q#2Q (doted curve) and q!2Q (short-dashed curve). The wavevector and amplitude of the carrier wave are q"15p/64a and f"0.2, respectively. The modulation wavevector Q"p/8a lies in the stable region, as can be seen in Fig. 42a. Time is measured in units of ¹ . $+0 Clearly, the q$Q components are initially stable while the combination modes at q$2Q are unstable. Fig. 44. Time evolution of the complete Fourier spectrum for the MD simulation described in Fig. 43. At long times combination modes appear due to wave mixing processes.

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remains the same for a period of approximately 180¹ before the instability occurs. From the $+0 time evolution of the spatial Fourier components at wave vectors q, q$Q and q$2Q plotted in Fig. 43, it can be seen that the q$Q components do not grow until after t"180¹ , just as $+0 predicted in the linear stability analysis; however, the q$2Q components which are neglected in the linear stability analysis grow signi"cantly after 180¹ so that the carrier wave becomes $+0 unstable. The time evolution of the complete Fourier spectrum in q space is plotted in Fig. 44. The generation of combination modes becomes evident after su$cient time. This simulation study demonstrates that the combination modes at q$2Q, q$3Q generated by the nonlinearity, though their magnitudes are smaller than that of the q$Q by at least a factor b at t"0, may fall in the instability region and play an important role at su$ciently large time scales. Hence, the condition for stability for large time scales is that the main satellite modulation and also all combination modes must not lie in the regions of instability. Note that unlike other models, such as the Klein}Gordon lattice and Fermi}Pasta}Ulam lattice, the nonlinearity in the uniaxial easy-axis antiferromagnetic chains does not generate combination waves at $2q,$3q,2, etc. The stability condition is given by




mod q$nQ,



p , unstable regions, n"1,2,2 , a

(6.37)

which is quite restrictive so that only carrier waves with wavevector q'p/4a are stable at long times. As the anisotropy parameter increases the antiferromagnetic chain e!ectively appears more discrete and according to the analytical results the area of the instability region in the (Q, q) plane also grows so that the upper boundary of the instability region in Fig. 39 approaches q"p/4a. Consider a chain with a larger anisotropy parameter A"2.0 but amplitude of the extended nonlinear carrier waves still f"0.2. The growth rates of the amplitude of modulation waves for carrier waves with a wide range of wavevectors are plotted in Fig. 45. The MD simulation results ("lled circles) are in excellent agreement with the analytical results. The instability region steadily grows with increasing carrier wave wavevector, and the carrier wave with q"15p/64a is unstable to modulation by any wavevector until the carrier wave wavevector increases beyond q"p/4a where it becomes stable. 6.4. Modulational instability recurrence The MD simulation examples have demonstrated that the linear stability analysis correctly describes the initial stage of instability and the numerical simulation results are in excellent agreement with the analytical ones. On the other hand, since the linear stability analysis is based on the linearization around the initial extended nonlinear spin wave modes one should not expect the linear analysis to be valid when the instability is fully developed, as demonstrated by the numerical simulations. It is generally believed that the initial state dominated by a single mode would eventually evolve into a nearly chaotic state after su$ciently long time since the direction of energy #ow should favor equipartition among the numerous modes available so that it occupies maximum volume in phase space. In the intermediate stage, however, the time evolution of the unstable mode can exhibit both regular and irregular behavior. Studies of a number of monatomic lattices and

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Fig. 45. Growth rate of modulational waves as a function of modulation wavevector in an antiferromagnetic chain with large anisotropy. The parameters are A"2.0 and f"0.2, and X is the standard antiferromagnetic resonance $+0 frequency. Solid curves are analytical results while "lled circles are MD simulation results. The wavevector of carrier waves are: (a) 15p/64a; (b) 7p/32; (c) 3p/16; (d) p/8; (e) p/16, and (f ) 0. Note that the carrier wave with q"15p/64a is unstable to perturbation of any wavevector (after Ref. [102]).

continuum models [94,96,97] have shown that under appropriate but rather restrictive conditions the linearly unstable carrier wave can behave quite regularly, showing interesting recurrence phenomenon (MI recurrence) over long time scales. Since the condition of MI recurrence is so restrictive even in monatomic lattices it is not clear whether MI recurrence can be observed in more complex systems, such as the antiferromagnetic chains considered here. To illustrate the MI recurrence phenomenon in easy-axis antiferromagnetic chains, an antiferromagnetic chain of 128 spins with anisotropy parameter A"1.0 has been examined. The q"0 extended spin wave of amplitude f"0.2 is modulated at wavevector Q"$p/32a with b"0.01. Figs. 46 and 47 show the time evolution of the Fourier components of this modulated nonlinear plane spin wave. In Fig. 46 the unstable carrier wave (solid curve) and the modulation waves (dot-dashed curve) both show a quasi-periodic behavior with a period of approximately 180¹ . Fig. 47 shows the time evolution of complete Fourier spectrum of the $+0 modulated wave. Besides the q and q$Q components, the combination modes q$2Q, q$3Q, etc., are also apparently visible in the spectrum although their magnitudes are much smaller than that of the main satellites. The underlying physics of MI recurrence is the mode coupling between the unstable carrier wave and its satellite modes generated from the modulation. The satellite modes grow exponentially and the energy is transferred from the carrier wave to the satellite modes until the growth is saturated; then the energy is returned back to the carrier wave. Such a process can repeat itself over a long time scale under appropriate conditions. Although the MI recurrence phenomenon is ubiquitous

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Fig. 46. Time evolution of the main Fourier components illustrating modulational instability recurrence. The components are at q (solid curve) and q$Q (dot-dashed curve). The parameters are A"1.0, q"0, f"0.2 and Q"p/32a. Time is measured in units of ¹ . Note that the two sidebands are symmetric when q"0. $+0 Fig. 47. Time evolution of the complete Fourier spectrum for the MD simulation described in Fig. 46. At long times combination modes can be clearly seen.

since it appears in many di!erent models, the condition for the quasi-periodic recurrence phenomenon to occur is quite restrictive, especially in the discrete lattices of interest here. Numerical simulations both for monatomic chains [96] and for `diatomica antiferromagnetic chains demonstrate that MI recurrence has a strong dependence on the wavevectors of the carrier and modulation waves. Extensive numerical simulations have shown that the quasi-periodic behavior of the system can be easily destroyed as the wavevector of either the carrier wave or modulational wave is changed even by the smallest amount allowed in the periodic lattice. This is because the quasi-periodic behavior requires that the energy is con"ned between the carrier wave and a small number of modulation satellites; however, in general, the energy can leak into additional combination modes and this process tends to be irreversible.

7. Production of intrinsic localized spin wave modes and the CW driving of antiferromagnetic instabilities Although the ILMs in a variety of discrete nonlinear lattices are reminiscent of impurity modes in linear lattices which can be probed by conventional radiation sources, their generation and detection demand di!erent approaches because of the homogeneity of the underlying lattices. A number of approaches have been proposed to generate and detect ILMs. For instance, it has been suggested that ILMs can be thermally excited in molecular crystals. The signature of their occurrence would be the transition of thermal relaxation from exponential law to non-exponential law with increasing temperature [98]. Another approach for generating ILM excitations is to use

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an optimal control scheme with a pre-designed sequence of laser pulses to indirectly excite vibrational ILM in crystals [99]. However, the best method for the generation and detection of these large-amplitude localized excitations in crystals is still an open question. Furthermore, realistic damping has been ignored, which could severely a!ect the practical feasibility of the proposed schemes. This section is concerned with reviewing the possibility of using the modulation instability for the generation and detection of ILSMs in antiferromagnetic systems in general and then applying these methods to particular materials with realistic parameters. The dissipation of spin waves in magnetic materials is usually weak compared to that of lattice vibrations in crystals. For example, the ratio of the linewidth to the antiferromagnetic resonance frequency C/u&10\ in bulk MnF [100] and FeF [101] is quite a bit smaller than the corresponding linewidth to TO mode   frequency C/u&10\ for lattice vibrations. With this in mind some simulation studies are described for antiferromagnetic materials where intrinsic localized spin wave modes (ILSMs) [40,57,62,76] are created via modulational instability [69,102] when the uniform mode is driven with a large amplitude CW ac "eld. One class of systems, namely layered antiferromagnets, stands out as particularly interesting since some of these can be represented by a one dimensional system with reasonable accuracy. Within this class the lowest-lying uniform spin wave mode of the layered antiferromagnet (C H NH ) CuCl has been found to be unstable and it appears that this spin     system can be driven su$ciently hard with a laboratory CW microwave "eld so that intrinsic localized spin wave modes would be produced. 7.1. Creation of intrinsic localization 7.1.1. Redistribution of energy for a lossless system Numerical simulations have demonstrated that the energy initially concentrated in one unstable plane wave mode will "nally #ow to all available modes in Fourier space, e.g., the energy is delocalized in Fourier space. Since a delocalized state in Fourier space can be either a localized state or a delocalized state in the corresponding real space, depending on the relative phases between Fourier components, the time evolution in Fourier space alone does not tell one the complete process of energy redistribution. In a su$ciently long time the system will "nally reach equipartition of energy since entropy should grow during the system's time evolution so that it approaches a state where the energy is evenly distributed not only among modes in Fourier space but also on lattice sites in real space. This "nal arrangement does not exclude the possibility of energy localization at intermediate stages since one of the main e!ects of modulational instability is the creation of localized excitations from spatially extended excitations [93]. This modulationalinstability-induced energy localization has been proposed to be a useful mechanism for the formation of intrinsic localization [69,93}95,99]. First we review how the energy initially concentrated in one mode is redistributed in an antiferromagnetic chain in the absence of dissipation. The time evolution of a large amplitude zone center mode perturbed by random noise in both Fourier space and real space is plotted in Fig. 48. The chain consists of 128 spins with anisotropy parameter A"2.0 and amplitude of the zone center spin wave f"0.2 with the amplitude of noise perturbation small compared to that of the carrier wave. In Fig. 48a the time evolution of the complete Fourier spectrum shows that the q"0 mode remains stable for a short period of time (about 80¹ ) then quickly decays into other $+0

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Fig. 48. Decay of an extended nonlinear spin wave into intrinsic localized spin wave excitations on a lossless chain via its modulational instability. The anisotropy parameter is A"2.0. Initially the q"0 extended spin wave with amplitude f"0.2 is perturbed by random noise. Time is measured in units of ¹ , and energy is measured from ground state in $+0 units of 2JS: (a) Time evolution of the perturbed spin wave in Fourier space; (b) Time evolution of the energy density distribution in real space (after Ref. [102]).

Fourier components so that the energy becomes delocalized in Fourier space. In Fig. 48b, the time evolution of the energy density distribution in real space shows a di!erent picture. The initial uniformly distributed energy becomes localized as the instability develops so that a number of localized excitations are created and they appear to be trapped by the discreteness of the lattice. Numerical experiments with di!erent anisotropy parameters and carrier wave amplitudes demonstrate that although localized excitations can be created in this way their lifetimes depend strongly on the anisotropy parameter of the lattice and the amplitude of the initial carrier wave. Since the anisotropy here is on-site it is not only a measure of the nonlinearity but also an e!ective measure of the discreteness of the lattice. As the anisotropy parameter A or the carrier wave amplitude decreases the lifetime of localized excitations decreases. An energy}energy correlation function can be used to obtain a more quantitative characterization [94,103], namely, C (n, t)"N #





e(m, t) e(m#n, t) K [ e(m, t)] K

,

(7.1)

where 122 indicates the average over initial conditions. For a uniform energy distribution, such as our initial conditions, C (n) is uniformly distributed, while when localized excitations appear # C (n) should reduce to a central spike. Since the total energy is a conserved quantity in the absence # of dissipation, the degree of localization can be measured by the height (or the width) of the central spike.

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Fig. 49. The height of the central peak of energy}energy correlation function as a function of time. Initially the q"0 extended spin wave with amplitude f"0.2 is perturbed by random noise. Solid curve: A"2.0. Dot-dashed curve: A"1.0. Each curve is averaged over 20 initial conditions. The oscillation in the dot-dashed curve indicates ILSGs generated via MI in a chain with small anisotropy parameter are short lived (after Ref. [102]).

The height of the central spike of the energy}energy correlation function as a function of time is plotted in Fig. 49 for two antiferromagnetic chains with anisotropy parameters A"1.0 and 2.0, respectively. In both cases, the carrier waves have the same amplitude and wavevector, i.e., q"0 and f"0.2, and each curve is averaged over 20 initial conditions. Note that the solid curve (A"2.0) is qualitatively di!erent from the dot-dashed curve (A"1.0). In the case of the larger anisotropy parameter, the height of the central spike in the energy}energy correlation function increases with time during the simulation period, which indicates that localized excitations are generated and grow with time. Although localized excitations are also generated in a lattice with a smaller anisotropy parameter, such as for MnF , they are short-lived and the energy appears to  be readily exchanged back and forth between localized excitations and extended spin waves demonstrating once again that both discreteness and strong anharmonicity appear to be essential for the creation of long-lived localized excitations. 7.1.2. Redistribution for a dissipative uniaxial antiferromagnet Next we consider the in#uence of weak dissipation where the equations of motion become dS /dt"S ;H!eS ;(S ;H) . L L L L L L

(7.2)

The new second term on the RHS represents Landau}Gilbert damping [85] which preserves the spin length. Here e is a small parameter measuring the damping strength. For the case of weak dissipation the amplitude decay rate C of plane spin waves is, from Eq. (7.2), C(q)"2JS(A#2)e#O(e f ) .

(7.3)

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The condition that the maximum MI growth rate be greater than the damping rate gives a dissipation-imposed amplitude threshold f . From Eqs. (6.25) and (7.3), one obtains  *X 2(A#2)S  e. 1# (7.4) f "  2 X (0) A  It should however be pointed out that Eq. (7.4) does not guarantee the formation of ILSMs from the MI since the formation of ILSMs is a dynamical process in which the competing e!ects of nonlinearity and dispersion reach a delicate balance. The characteristic time scale of this nonlinear process can be obtained from the nonlinear frequency shift given in Eq. (6.4), namely,






2p X (q)#X (p/2a)  ¹ " "  ¹ . ,* "*u" $+0 Af

(7.5)

With the parameters A"2.0 and f"0.2 used in the numerical simulations, one "nds ¹ + ,* 93¹ . The conclusion is that ILSMs can be created from the MI only when e!ects of $+0 nonlinearity and dispersion are much stronger than the dissipation e!ect [104], hence the necessary condition becomes C(q)¹ ;1 . (7.6) ,* MD simulations with the perturbed q"0 extended large amplitude spin wave with amplitude f"0.2 as initial condition have been carried out and the time evolution of the energy distribution examined for two di!erent dissipation values. The energy density displayed in Fig. 50a for C/u (0)"10\ and Fig. 50b for C/u (0)"10\ is multiplied by eCR for ease in viewing. Interesting  

Fig. 50. The in#uence of weak dissipation on ILSMs' formation from modulational instability. The energy density is multiplied by eCR for ease in viewing. Initially the q"0 extended spin wave with amplitude f"0.2 is perturbed in a lattice with A"2.0: (a) C/u (0)"10\; (b) C/u (0)"10\ (after Ref. [102]).  

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general di!erences are found between the two cases. The ILSMs in the weaker dissipation case (Fig. 50a) are much more localized and appear to be pinned less strongly than for the case of no dissipation previously shown in Fig. 48b. On the other hand, the ILSMs in the stronger dissipation case (Fig. 50b) are somewhat delocalized and hence more mobile. This di!erence results from the competition between the MI-induced energy localization and the delocalizing dissipation e!ect. In the weaker dissipation case (C¹ +0.058) the MI process can take place before dissipation ,* becomes signi"cant, while in the stronger dissipation case (C¹ +0.58) the dissipation e!ect ,* prevents energy from being strongly localized by decreasing the amplitude and hence reducing the e!ective strength of the nonlinearity. 7.2. Uniaxial FeF



Studies reviewed in Section 4 for an antiferromagnetic chain with single-ion easy-axis anisotropy have demonstrated that ILSMs can exist in the gap below the AFMR uniform mode frequency for any D'0, and that the ratio of the anisotropy "eld to the exchange "eld, H /H , is a crucial  # parameter which determines the localization properties of these new modes [40,62,102,105]. For a speci"c maximum spin deviation, the larger the ratio H /H , the more strongly localized is the  # spin wave mode, with the mode frequency moving further into the gap. For the two standard easy axis antiferromagnets MnF and FeF very di!erent localization properties are to be expected.   Because of its relatively weak anisotropy "eld, H /H "10\ [85] broad intrinsic localized spin  # wave gap modes (ILSGs) created on a short time scale should be generated in MnF , which would  be di$cult to identify in experiment due to the coexistence of extended nonlinear spin waves. On the other hand, because FeF has a much larger anisotropy value, H /H "0.345 [101], strong   # localization should be produced and remain for a much longer time scale, which because of the relatively large frequency shifts should be more easily separated from the extended spin waves associated with the AFMR. 7.2.1. CW driver The parameters for the chain are chosen to match those of FeF , which are given in Table 1.  Since only the ratios between parameters matter in these computer simulations D/J"0.69 is used in Eq. (4.1). For a dissipative chain of classical spins, the equation of motion is given by Eq. (7.2) but now the gyromagnetic ratio c is explicitly included and multiplies the terms on the RHS of that

Table 1 Model parameters for FeF [101], FeCl [107] and (C H NH ) CuCl [109]. H is the exchange "eld, H is the       #  anisotropy "eld tensor, H" F(n) (even (odd) n) is the dipolar "eld tensor arising from the same (di!erent)   L sub-lattice, and u (0)/c are the AFMR "elds for resonance. Both H and H have only diagonal elements, which are !    listed here. All parameters are in units of Oersteds

FeF  FeCl  (C H NH ) CuCl    

H #

H 

HQ  

H  

u (0)/c !

5.55;10 1.41;10 829

1.91;10 1.57;10 +69, 974, 0,

Neglected Neglected +!247, 500,!253,

Neglected Neglected +75,!150, 75,

5.01;10 1.71;10 1915, 494

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equation in order to make contact with experimental measurements. For FeF e in Eq. (7.2) is  chosen such that C/u "4;10\ to agree with the intrinsic linewidth of the AFMR mode $+0 *H"20 Oe at 4.2 K [101]. The e!ective "eld acting on the nth spin is given by (7.7) H(t)"! sLH#H (t) ,  L where H is the Hamiltonian in the absence of driving "eld, and H is the "eld strength of the  circularly polarized driving "eld in the x}y plane. Although one could imagine modulating the driving "eld in various ways because the extended nonlinear spin wave frequency red shifts with increased as "eld strength, the MD simulations described here correspond to the simplest possible case: a CW source with a "xed driving frequency. The appropriate perturbation is H (t)"H (cos u t e !sin u t e ) .    V  W

(7.8)

7.2.2. Molecular dynamics simulations Molecular dynamics (MD) simulations have been used to estimate the magnitude of the ac "eld required to create ILSGs via the modulations instability of large amplitude extended spin waves for an easy-axis antiferromagnetic chain of 256 spins with periodic boundary conditions [106]. In the MD simulations the discrete equations of motion for the xyz spin components, given by Eq. (7.2), are integrated numerically using the fourth-order Runge}Kutta method. For a particular ac "eld strength, the driving frequency is set to a value slightly below the AFMR frequency of the uniform mode, u . The optimal driving frequency should maximize the total energy fed into the $+0 antiferromagnetic chain so that the system attains the maximum possible nonlinear contribution. The time evolution of the energy per spin in a chain driven by a CW ac "eld with "xed strength H "4.0;10\u /c at three di!erent frequencies is shown in Fig. 51. In each case the chain  $+0 has the same initial con"guration, that is, the spins are randomly tilted from their ground state con"guration with an average spin deviation 1dS 2"0.005. The driving frequencies u /u are L  $+0 0.994 (dot-dashed line), 0.995 (solid line), and 0.996 (dashed line), respectively. The time is measured in units of ¹ "2p/u . These three MD simulations demonstrate that the driving frequency $+0 $+0 is a crucial parameter and that u "0.995u is the optimal driving frequency for this  $+0 particular ac "eld strength. For this optimal case, the energy in the chain increases smoothly with time during the "rst 240 ¹ , and then becomes saturated at longer times with the deviations $+0 produced by irregular #uctuations. Although the details of the MD simulation results depend on the initial spin con"guration the evolution of energy per spin does not show qualitative di!erence between what is shown here and other random initial con"gurations. The time evolution of the energy density distribution in the analog 1-D FeF system is plotted in  Fig. 52 for the optimal case of H "4.0;10\u /c and u "0.995u . After the driving  $+0  $+0 "eld is turned on the energy density distribution increases smoothly with time and remains plane wave-like until the instability triggered by the random initial condition begins to manifest itself after about 240 ¹ when the energy tends to build up in the system and the extended spin waves $+0 become unstable. With continued development of the instability the extended q"0 spin wave decays into a few ILSG excitations which slowly move around the lattice. Since the ILSG excitations have lower frequencies than the uniform mode their coupling to the ac driving "eld is weak and the energy in the chain reaches this steady state as shown in Fig. 51 as the solid line. (The "ne wiggles shown here result from the fact that up spins for ILSGs in uniaxial easy-axis

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Fig. 51. Time evolution of energy per spin in a periodic uniaxial easy-axis antiferromagnetic chain of 256 spins with parameters corresponding to those of FeF , driven by a circularly polarized ac CW "eld. The strength of the circularly  polarized ac "eld is H "4.0;10\u /c, and the frequencies of ac CW "elds u /u are as follows: 0.994  $+0  $+0 (dot-dashed line), 0.995 (solid line), and 0.996 (dashed line). Fig. 52. Energy distribution versus time showing ILSMs generated in the analog 1-D FeF model. The particular driving  frequency u "0.995u and H "4.0;10\u /c. The energy density is in arbitrary units, and the time is  $+0  $+0 measured in units of ¹ . $+0

antiferromagnets have a larger deviation than adjacent down spins and hence have higher anisotrophy energy [62].) Since both the ILSG and the uniform mode AFMR have net transverse magnetic dipole moments it is instructive to look at the resulting power spectrum of M>(t) to di!erentiate between the two signatures. This spectrum which is calculated from the data during the time interval between 300 and 1938 ¹ is shown in Fig. 53. Since the driving frequency is 0.995 times the $+0 AFMR frequency almost all of the complex structure seen here is associated with the generation of ILSGs from the uniform mode. The power spectra calculated from MD simulations with di!erent random initial con"gurations show qualitatively similar results. Although these numerical simulations demonstrate that the ILSMs can be generated in a chain with the FeF parameters via the modulational instability mechanism by driving the unstable  extended spin waves with a CW ac source, the optimum ac "eld parameter required to produce this e!ect corresponds to H "200 Oe at a driving frequency of u "52.34 cm\. Such intense CW   sources do not yet exist in this frequency region. Also it should be noted that this "eld strength represents an actual lower estimate since it is known that localization is more di$cult to produce in a 3-D system than in a 1-D one [39]. 7.3. Uniaxial FeCl



This layered antiferromagnet has the spins oriented in ferromagnetic sheets along the hexagonal c-axis with successive sheets antiparallel to each other [107]. The "eld parameters for this system

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Fig. 53. Power spectrum of the transverse magnetic moment M>(t) showing the ILSMs generated in the analog 1-D FeF model. The parameters are the same as for Fig. 52. The power spectrum is calculated from the simulation data  during the time period between 300 and 1938 ¹ . $+0

are given in Table 1. The nonlinear dynamical properties of the spins in FeCl as driven by an ac  "eld could be expected to be quite di!erent from those just found for FeF , since now the ratio of  H /H "11, that is, A"22, and the e!ective exchange "eld is only 2.5% of the FeF value.  #  However, as we have already seen in Section 7.1 damping adds another constraint to the production of localization upon driving the uniform mode to large amplitudes. Since there is no reason for the dissipation of the spins in FeCl to be any smaller than the value previously found in FeF , one   can assume that FeCl and FeF have the same ratio of C/u . Furthermore, it is to be expected   $+0 that the uniform mode in FeCl must be driven to a nonlinear regime such that the product  C¹ <1 and that this has the same value as for FeF . From Eq. (7.5) one then obtains for the ,*  threshold spin wave amplitude f +0.58f . Next, to estimate the required strength of the CW $$ $! ac "eld, use can be made of the expression fJH /C to obtain H +40 Oe, which is only a factor   5 smaller than that required for the previous system. The strength of this CW source is still too large to be viable in the submillimeter wave region. 7.4. Biaxial (C H NH ) CuCl      As the localization strength of an ILSM is really determined by the ratio of H /H it should be  # possible to apply the same method to excite ILSMs with H comparable to H but for antiferro # magnets with the AFMR frequency in the GHz region where powerful sources are available. Since the exchange "eld H is scaled down by a factor of 100 one would expect the required strength of # the driving "eld to decrease by roughly the same factor. The necessary `largea ac "eld is now only a few Oersteds. Because of such practical constraints a well-known layered antiferromagnet (C H NH ) CuCl [108,109] becomes a reasonable candidate [106]. The structure of this com    pound is face-centered orthorhombic. The interactions between the spin 1/2 copper ions within the ab-plane are strongly ferromagnetic while there exists a very weak antiferromagnetic interaction between neighboring copper ions in adjacent layers. At ¹"1.4 K the interlayer antiferromagnetic exchange "eld is H "829 Oe and the intralayer ferromagnetic exchange "eld H "5.5;10 Oe. # # Thus H /H "1.51;10\;1, and below the NeH el temperature (¹ "10.2 K) the low-frequency # # ,

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spin dynamics can be described quite accurately by a 1-D two-sublattice antiferromagnet with spins in the same layer pointing in the same direction. Since a given layer contains many similarly oriented Cu spins it can be modeled by one classical spin. 7.4.1. Nonlinear dipolar anisotropy To describe the low frequency spin dynamics of this two sublattice antiferromagnet, an e!ective one-dimensional Hamiltonian of the form 1 (7.9) H"2J S ) S # S ) D ) S # S ) F(n!n) ) S , L LY $ L L> L L 2 LLY L L is used where S is the e!ective spin of nth layer, and is treated as a classical vector with unit length. L The antiferromagnetic exchange constant J '0, and the anisotropy tensor D arises from the $ anisotropic ferromagnetic exchange interaction between spins belonging to the same layer. The third term describes the magnetic dipole}dipole interactions. Since the frequencies of the spin waves in (C H NH ) CuCl lie in the GHz region, this dipolar term should play a signi"cant role.     The e!ective dipolar interaction tensors F(n!n) can be obtained by summing contributions from spins belonging to the nth layer, that is






3r r 1 I! H H , (7.10) F(n!n)"F("n!n")"(gk ) r r H HZLY  H where r represents the vector pointing from a spin denoted by 0 in the nth layer to a spin denoted H by j in the nth layer, and the summation runs over all spins in the nth layer. (Note that j"0 should be excluded from the summation when n"n.) The largest component F(0) represents the dipolar interaction between spins belonging to the same layer. Owing to the symmetry of the lattice [19], both the anisotropy tensor D and the e!ective dipolar interaction tensors F(n!n) have only non-zero diagonal elements. From Eqs. (7.7) and (7.9) the e!ective magnetic "eld acting on the nth spin is H(t)"!2J (S #S )!2D ) S ! F(n!n) ) S #H (t) . (7.11) L $ L\ L> L LY  LY The model parameters are set to be J "207 Oe and D"diag+34.5, 487, 0, Oe so that the z-axis $ is the easy-axis, and the antiferromagnetic exchange "eld and the anisotropy "eld match those measured for (C H NH ) CuCl . These are listed in Table 1. The e!ective dipolar interaction     tensor F(n!n) has been obtained by summing contributions from spins belonging to the nth layer [109]. Since the Hamiltonian does not posses uniaxial symmetry there are two antiferromagnetic spin wave branches, u (q), which are given by ! u (q)"[4J (1$cos qa)#2(D !D )#F (q)$F (q)!F(0)#F(0)] ! $ V X V V X X ;[4J (1Gcos qa)#2(D !D )#F(q)GF(q)!F(0)#F(0)], (7.12) $ W X W W X X where a is the distance between adjacent layers, and F(q)" F(2l)e J? and F(q)" F(2l#1)e J>?. J J

(7.13)

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Fig. 54. Spectrum of the antiferromagnetic spin waves in (C H NH ) CuCl as calculated from Eq. (7.12) with the "eld     parameters given in Table 1. The positive curvature of the lower branch at the center of the Brillouin zone permits intrinsic localized spin wave modes in the gap.

The model parameters have been chosen to yield the experimentally measured antiferromagnetic resonance frequencies, that is, u (0)/c"1915 Oe and u (0)/c"494 Oe. At the zone boundary, the > \ two branches are degenerate with u (p/2a)/c"1584 Oe. The dispersion curves for the two low ! lying antiferromagnetic branches are plotted in Fig. 54. Now consider the instability properties of these antiferromagnetic spin wave branches. Because of the negative curvature of the upper dispersion curve branch at q"0 the existence of intrinsic spin wave resonances (ILSRs) can be ruled out [57]; however, the positive curvature of the lower branch at the zone center permits the existence of ILSGs to occur in the gap below u (0). \ 7.4.2. MD simulation results To study with MD simulations the creation of ILSGs in this system where magnetic dipoledipole anisotropy is important, a CW microwave "eld given by Eq. (7.8) has been applied to a chain of 256 spins with periodic boundary conditions [106]. Since the AFMR frequency is in such a low frequency region where the phonon density of state is small the coupling between the Cu ion and lattice is expected to be weak. The damping parameter e is set to be 10\ so that the ratio of C/u is of the same order as that in FeF . At time 0 the spin associated with each layer is randomly  oriented from the easy axis by a small amount with 1dS 2"0.005. L The smallest driving "eld strength that will induce signi"cant localization can be searched for in the same way as described in Section 7.2. With the driving "eld strength given, the driving frequency should be chosen to maximize the energy going into the system. After a few trials, the

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frequency of the driving "eld is set to u /c"487.5 Oe for the ac "eld of strength H "0.987 Oe.   The MD simulation results showing the evolution of the energy density distribution for this optimal case are plotted in Fig. 55. The amplitude of extended nonlinear spin wave grows smoothly for about 70 ¹ (2p/u (0)) while for longer times the extended mode decays into slowly moving $+0 \ localized excitations. MD simulations with di!erent random initial con"gurations show qualitatively similar results. In Fig. 55, the ILSGs extend over roughly 20}30 lattice sites. The characterize the spatial size of ILSGs in a more quantitative way, one can calculate the energy-energy correlation function as de"ned in Eq. (7.1). For an extended uniform spin wave, C (n)"1. As the system is driven into # strongly nonlinear region, the extended spin wave becomes unstable and a central spike grows in the energy-energy correlation function. The average size of ILSGs can be de"ned as the FWHM of the central peak. For the run shown in Fig. 55, the FWHM of the central peak averaged between 100 and 200 ¹ equal to 20.8a. $+0 The ILSGs for this anisotropic chain have non-zero net magnetic moment, M (t)" SW(t), W L polarized in the y direction. Fig. 56 displays the power spectrum of M (t) for the MD run shown in W Fig. 55. The power spectrum is calculated from the data during the time period from 90 and 512 ¹ to exclude the contribution from initial extended spin waves. The dot-dashed curve $+0 identi"es the frequency of the driving "eld. The lower frequency components shown here are generated by the production of ILSMs which are produced by the decay of unstable extended spin

Fig. 55. Energy density distribution versus time showing ILSMs generated in the analog 1-D (C H NH ) CuCl model.     The strength of the applied ac circularly polarized CW "eld is H "0.987 Oe, and the frequency is the corresponding  optimal frequency u /c"487.5 Oe. The energy density is in arbitrary units, and time is measured in units of  ¹ (2p/u (0)). $+0 \ Fig. 56. Power spectrum of the net magnetic moment M (t) showing the ILSMs generated in the analog 1-D W (C H NH ) CuCl model. The parameters are the same as for Fig. 55. The power spectrum is calculated from the     simulation data during the time period from 90 to 512 ¹ , and is normalized so that the integrated strength is unity. $+0 The dot-dashed line identi"es the driving frequency.

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Fig. 57. Energy density distribution versus time showing ILSMs generated in the analog 1-D (C H NH ) CuCl model.     All parameters are the same as those in Fig. 55 except that the frequency of ac CW "eld is u /c"485 Oe, which is the  optimal frequency corresponding to the new applied ac "eld strength H "1.4 Oe. The energy density is in arbitrary  units, and time is measured in units of ¹ . $+0 Fig. 58. Power spectrum of the net magnetic moment M (t) showing the ILSMs generated in the analog 1-D W (C H NH ) CuCl model. The parameters are the same as for Fig. 57. The power spectrum is calculated from the     simulation data during the time period from 90 to 512 ¹ , and is normalized so that the integrated strength is unit. $+0 The dot-dashed line identi"es the driving frequency.

waves. The center of gravity of the power spectrum is calculated to be 451$4.8 Oe for 10 MD runs with di!erent random initial con"gurations. To demonstrate the e!ect of the ac "eld strength on the ILSG spectrum, the power of the CW microwave source is doubled so that the ac "eld strength is increased by a factor of (2 to H "1.4 Oe. As a consequence of the red shift resulting from the increasing driving power, the  optimal driving frequency is moved down further to u /c"485 Oe. Figs. 57 and 58 show the MD  simulation results. As demonstrated in Fig. 57, the e!ect of the instability begins to show up earlier owing to the increase in the driving "eld strength, and the ILSGs are more localized with the FWHM of the central peak of the energy-energy correlation function [102] averaged between 100 and 200 ¹ equal to 17.8a compared to 20.8a in Fig. 49a. The power spectrum of M (t) for the $+0 W corresponding run is displayed in Fig. 58. Compared to Fig. 56 where the shift of the center of gravity of the frequency spectrum from the driving frequency is 9%, in Fig. 58 there are more low-frequency components producing a corresponding shift of 10% since on average the ILSGs generated in this case have larger amplitudes and hence lower mode frequencies. Although spin wave resonances in the paramagnetic phase of (C H NH ) CuCl have been     extensively studied, the measurement of the AFMR in the antiferromagnetic phase is relatively rare and the exact intrinsic linewidth of the AFMR is not available. In the formation process of ILSGs, there are two competing factors, that is, nonlinear instability and damping. The nonlinear instability results in the decay of extended nonlinear wave into localized excitations while the

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Fig. 59. Power spectra of the net magnetic moment M (t). All parameters are the same as those in Fig. 55 except for the W damping coe$cient and the optimal driving frequency: (a) e"10\ and u /c"487.5 Oe; (b) e"10\ and  u /c"490 Oe. The power spectrum is calculated from the simulation data during the time period from 90 to 512 ¹ ,  $+0 and is normalized so that the integrated strength is unity. The dot-dashed lines identify the driving frequencies.

damping prevents the formation of localized excitations by reducing the strength of the nonlinearity. To investigate how damping might a!ect the formation of ILSGs in (C H NH ) CuCl ,     computer simulations have been carried out with di!erent e's. Plotted in Fig. 59 are the power spectra of M (t) for e"10\ and 10\, respectively. At e"10\, the e!ects of nonlinearity and W dispersion are still much stronger than the dissipation e!ect. In this case, the extended large amplitude spin wave breaks into localized modes producing a ILSG band in the power spectrum, as shown in Fig. 59a. As the damping coe$cient is increased further to e"10\ in Fig. 59b, the power spectrum exhibits a dominant peak at the driving frequency. In this case, the characteristic time scale for the formation of ILSGs from extended large amplitude spin wave is longer than the damping time, preventing the extended spin wave from breaking into ILSGs. The study via MD simulations appears to demonstrate that ILSGs can be created via modulational instability by driving unstable extended nonlinear spin waves in a realistic antiferromagnetic material with a conventional microwave source. Since the ILSMs have frequencies below the extended spin wages and are magnetic dipole-active, it is anticipated that their signature can be directly probed by microwave homodyne detection methods.

8. Conclusions 8.1. Summary A variety of nonlinear features associated with intrinsic localization in simple 1-D periodic ferromagnets and antiferromagnets have been presented in this review. Since the discrete nature of

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Table 2 Qualitative comparison of intrinsic localized mode properties for 1-D crystal and magnetic lattices Ions

Spins

Comparison

Monotomic Intersite interactions One degree of freedom

Ferromagnetic Onsite#intersite interactions Two degrees of freedom

Di!erent Di!erent

nn two-body potential SoftPno intrinsic localization

nn exch.#on site anisotropy SoftPintrinsic localization

Di!erent

Diatomic Two ILGM parities (odd stable)

Two sublattice antiferromagnet Two ILSM parities (odd stable)

Similar

Diatomic#nn interactions Gap mode

Antiferromagnet#nn interactions Uniaxial anisotropyPgap mode Biaxial anisotropyPgap mode#reson

Similar Di!erent

Diatomic ILGM Electric dipole active q"0 optic mode stable BZ mode unstable

Antiferromagnetic ILSG Magnetic dipole active q"0 optic mode unstable BZ mode stable

Similar Di!erent Di!erent

Odd potential terms P local dc distortion

Only even terms No local dc distortion

Di!erent

the lattice plays a crucial role in de"ning the properties of these excitations only two subsections of this review (Sections 4.4 and 5.3) have dealt with the continuous limit. There is some value in comparing and contrasting the nano-scale behavior of the magnetic excitations described here with those already identi"ed in simulation studies of the excitations in nonlinear crystal lattices. For the vibration of a monatomic 1-D crystal lattice only intersite interactions are relevant while even for a simple ferromagnet there are onsite interactions to consider because of the local anisotropy "eld. When NN two body potentials are introduced into the monatomic vibrational problem the resulting anharmonicity is soft so intrinsic localized modes can not form. For the 1-D magnetic case the nonlinearities produced by both the NN exchange interaction and the anisotropy "eld are also soft but now the on-site anisotropy can produce a gap at the bottom of the spin wave spectrum and localized modes may appear in this gap. For such modes to be strongly localized the anisotropy energy and the exchange energy would need to be comparable. The di!erences between these two kinds of nonlinear systems are summarized in the "rst two rows of Table 2. The application of a dc magnetic "eld provides another di!erence between the two cases. As described in Section 2 for the special case of a 1-D ferromagnetic chain with NN interactions and easy plane anisotropy with a large enough magnetic "eld applied perpendicular to that plane, ILSMs can appear above the top of the linear spin wave spectrum. Numerical methods have been used to study the collisions between these objects as well as the collision of these objects with magnetic defects. The results described here can also be expected to apply to some magnetic superlattices systems.

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A multilayer stack consisting of alternating ferromagnetic "lms and non-magnetic "lms has weak ferromagnetic coupling between the ferromagnetic layers as well as a small inplane anisotropy. The application of a su$ciently strong external magnetic "eld perpendicular to the stack will then reorient the spin alignment to the "eld direction with the added feature that spin precession will not produce an intra-"lm demagnetizing "eld [61]. The simplest way to make possible intrinsic localization in a 1-D ferromagnetic chain without anisotropy or a magnetic "eld is to include both NN and NNN isotropic exchange coupling. This possibility was reviewed in Section 3. It was shown that when the strength of NNN exchange interaction relative to the NN interaction exceeds a speci"c threshold value then intrinsic localized spin wave resonances (ILSRs) of both odd and even parity may appear coincident with the frequencies of the linear spin wave spectrum. Numerical studies demonstrate that the lifetime of an ILSR depends on the mode parity, the maximum spin deviation, and the relative strength of the NNN interaction to the NN one. In the small amplitude continuum approximation the traveling ILSR and stationary ILSR have the same envelope shape: its width is inversely proportional to the maximum spin deviation and increases with increasing NNN coupling strength. The properties of a translating ILSR depend on the size of the spin deviation. If the maximum spin deviation is modest the ILSR can travel through the lattice, but it is scattered by the discreteness of the lattice and decays into plane-spin waves over su$cient distances. The larger its amplitude, or the larger its velocity, the larger is the emission of plane-wave modes. For colliding ILSRs soliton-like behavior is found for small spin deviations in that the ILSRs preserve their shapes after collision and the energy transfer between them is negligible but for large amplitudes neither ILSR can survive the collision. The 1-D diatomic lattice with realistic NN two body potentials has been used to identify where and how intrinsic localization might appear in the vibrational spectrum. It was found that because of the soft anharmonicity intrinsic localized gap modes (ILGMs) may drop out of the optic branch and appear in the gap between the optic and acoustic plane-wave branches. The two sublattice 1-D antiferromagnet has somewhat similar features as outlined in rows 3 and 4 of Table 2. The case of NN exchange interactions and on-site easy-axis anisotropy has been described in some detail in Section 4. Here truly localized modes can be produced with frequencies in the gap below the standard AFMR frequency. The amplitude of such an ILSG is either single or double peaked and for both cases the ILSG frequency decreases as its amplitude grew. The degree of localization increases as either the maximum spin deviation or the ratio of the anisotropy constant to the exchange coupling constant increases. In the small spin deviation limit both types of ILSGs became identical envelope solitons. Although single- and double-peaked ILSGs are observed to have similar static properties, both analytical study and MD simulations reveals that only the singlepeaked ILSGs are stable in the presence of a noise perturbation whereas a randomly perturbed double-peaked ILSG evolves into a single peaked one. The instability of a double peaked ILSG increases with its amplitude and also with the relative strength of anisotropy "eld. Although ILSMs can exist at any site owing to the homogeneity of the lattice, the lack of continuous translational invariance in a discrete lattice prevents ILSMs of large amplitudes from moving from site to site and they become more easily pinned as the anisotropy to exchange "eld ratio increases. The pinning of ILSGs can be understood in terms of a Peierls}Nabarro (PN) barrier created by the lattice discreteness. The height of the PN energy barrier is the energy di!erence between the single-peaked ILSG and the double-peaked ILSG at the same frequency and this barrier increases

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with the mode amplitude and the onsite anisotropy "eld. Unlike the ILSR in a NN and NNN exchange coupled ferromagnetic chain, an ILSG in an easy-axis antiferromagnetic chains can interact with far infrared radiation since the excitation has a net transverse magnetic moment thus these gap modes may be more relevant from the experimental point of view. Another magnetic case, which is similar to that for localized vibrational modes in the 1-D diatomic lattice, occurs for an antiferromagnetic chain with uniaxial on-site easy-plane anisotropy which was reviewed in Section 5. For this system one linear spin wave mode goes to u"0 at q"0 while the other mode remains at "nite frequency so there is no spin wave gap and only an intrinsic localized spin wave resonance (ILSR) is possible. Although ILSGs can always exists in gapped antiferromagnetic chains, an ILSR can exist only when the upper branch dispersion curve has positive curvature at the center of Brillouin zone, with frequency lying in a constrained range. As its amplitude decreases such a nonlinear excitation approaches continuously without threshold the corresponding spin wave mode of linear theory and is strikingly di!erent from the topological sine-Gordon kink excitations found in 1D easy-plane magnets [20,21,88]. The key feature in the nonlinear dynamics of an ILSR is the polarization di!erence between the two plane wave branches. The smaller the frequency of the q"0 mode in the upper plane wave branch, the less strongly coupled the resulting ILSR is to the other branch of the plane wave spectrum. When biaxial symmetry replaces the uniaxial one then ILSM can exist in the gap below the lower branch spin wave spectrum with properties similar to those described in Section 4. This overlap between the spin wave and vibrational problems is summarized in rows 3 and 4 of Table 2. Since the ILSR and ILSG modes in an antiferromagnetic chain are elliptically polarized and have nonzero transverse magnetic moments orthogonal to each other, unlike the non-ir-active ILSR in isotropic ferromagnetic chains studied in Section 3, they can coupled to far ir radiation as can ILGMs in diatomic lattices (see row 5 of Table 2). To determine how the intrinsic localized modes manifest themselves in physical systems the modulational instability of extended nonlinear spin waves in easy-axis antiferromagnetic chains, both analytically in the frame of linear stability analysis and numerically by means of molecular dynamics simulations, has been reviewed in Section 6. The analysis is equivalent to that for a `diatomica lattice with two degrees of freedom per site but an important di!erence with the vibrational problem is that it involves both on-site and intersite nonlinearity. Because of this di!erence the simplest case of NN interactions produces di!erent instability criteria for the crystal and spin systems as outlined in row 5 of Table 2. Stability analysis shows that the instability of an ILSM is determined by its symmetry. In contrast to the monatomic Klein}Gordon chain [69] where plane waves with wavevectors in the lower half of Brillouin zone are always unstable to long wavelength perturbations, in easy-axis antiferromagnetic chains spin waves with wavevectors close to the zone center are stable to both long wavelength and short wavelength perturbations but unstable to perturbations of moderate wavelengths. However, since the amplitude threshold for the instability of long wavelength spin waves is inversely proportional to the lattice size it tends to zero for macroscopic systems. Numerical simulations reveal that combination waves generated via wave-mixing processes can have signi"cant e!ect on the spin wave stability at large time scales. Section 7 focuses on one of the main e!ects of the modulational instability which is the creation of localized pulses. Weak dissipation imposes a "nite amplitude threshold even for in"nite chains and, in addition, ILSMs become mobile during formation because of the reduced strength of the nonlinearity. This section develops the intimate connection between modulational instability and

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the dynamical localization of spin waves for real systems and demonstrates with a particular application to layered antiferromagnets that magnetic dipole-active ILSMs can be created via modulational instability by driving unstable extended nonlinear spin waves with an appropriate microwave radiation source. Since the ILSMs have frequencies lower than the extended spin waves, they produce an ILSM band in the absorption spectrum below the driving frequency. This signature of the ILSMs may be directly probed by homodyne detection methods. The opportunity of experimentally identifying ILMs appears at hand since the same experimental arrangement can be used both for their generation and their detection. For ILSMs which are not magnetic dipole-active, such as ILSRs in isotropic ferromagnetic chains, other techniques such as neutronscattering, may be required. Perhaps the biggest di!erence between ILMs in the crystal and spin problems is that identi"ed in row 6 of Table 2. It comes about because the odd potential terms present in the two body interaction for vibrational dynamics are absent for the NN exchange and on site anisotropy interactions in spin dynamics. Associated with such anharmonic potential terms is the production of a local dc distortion concomitant with the excitation of an ILGM. Since this dc strain "eld is long range some care is required in MD simulations, especially for higher dimensional cases. Thus an MD simulation for the ILM magnetic lattice is much simpler than that for a vibrational lattice with realistic potentials. 8.2. Other systems and future prospects Although ferrimagnets have not yet been considered they represent perhaps the closest magnetic example of the vibrational diatomic lattice system. Based on the antiferromagnetic and diatomic vibrational studies of ILMs in 1-D lattices which have been reported to date, the properties of magnetic ILMs in a ferrimagnet with NN antiferromagnetic exchange interactions between neighboring spins of unequal magnitude are easy to visualize. Since the linear dynamics is now represented by an acoustic ferrimagnetic branch and an optical exchange resonance branch for the case where a frequency gap may exist between these branches then an ILSGM would be possible here. The introduction of anisotropy and applied dc external "eld would increase the richness of these systems. The recently analyzed 1-D coupled rotor lattice [110}112] may at "rst appear closer to the spin models treated here than the dynamics of other 1-D discrete lattice models. But one feature that makes the rotor model standout is the coexistence of rotational and librational motions so that in the extreme case, an intrinsic localized rotor mode can consist of only one rotational center plus librational wings which can never occur for the spin model that we have reviewed here. In addition, since the equations of motion for the rotor model involve both "rst and second derivatives, it is not possible to make direct contact between spin and rotor dynamics. Nevertheless, just like all other ILM-bearing models, the existence of an ILM in the rotor model is closely related to an instability of some extended plane wave states. Because the coupled rotor lattice can be mapped onto a Josephson junction array [113] there is the opportunity to carry out experimental tests exploiting this analogy but the lattice scale will necessarily be much coarser than for the atomic lattice considered in this review. The quantization of ILSMs represents a di$cult question yet to be answered. Recent vibrational studies have shown that energy focusing (ILMs) is prevalent not only in classical but also in

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quantum discrete nonlinear lattices. However, these studies are either based on quantum models that are constructed to conserve the phonon number operator [114] or based on exact numerical diagonalization of Hamiltonian matrices in truncated phonon spaces [115]. In the "rst case the model is analytically tractable but can hardly be expected to describe any realistic physical system. In the second case, the number of particles in the chain (N) is limited to be less than 8 owing to the rapid increase of matrix dimension with N even for a truncated phonon space, and as yet no study of size-dependence has been carried out. On the other hand, it is known that the amplitude threshold for modulational instability is proportional to 1/N. This result suggests that the occurrence of nonlinear localization in small chains might have a strong size dependence so that the results obtained for a particular small chain may not be reliable for comparison with experiments. Magnetic lattices may be expected to provide an important alternative for the study of quantum ILMs since the number of states (2S#1) of each spin is usually much smaller than the number of allowed Einstein phonon states at each site (&17 in Ref. [115]). The current computational power of parallel computers should allow magnetic chains with a large range of sizes to be studied through the exact diagonalization approach without truncation. The simplest models suitable for the study of quantum ILMs might be Heisenberg ferromagnets, either isotropic ferromagnetic chains with both nearest- and next-nearest-neighbor exchange interactions or ferromagnetic chains with nearest-neighbor exchange interaction and easy-axis on-site anisotropy.

Acknowledgements We thank J.P. Sethna and R.H. Silsbee for helpful conversations and N.I. Agladze for the production of some of the "gures. This work is supported in part by NSF-DMR-9631298, ARO-DAAH04-96-1-0029 and NSF ECS-9612255. Some of this research was conducted using the resources of the Cornell Theory Center, which receives major funding from the National Science Foundation and New York State.

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