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Supersolid magnetic phase realization in strongly anisotropic easy-plane antiferromagnet with Ising-like exchange interaction in the transverse magnetic field
Journal of Magnetism and Magnetic Materials 348 (2013) 68–74
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Supersolid magnetic phase realization in strongly anisotropic easy-plane antiferromagnet with Ising-like exchange interaction in the transverse magnetic field Ph.N. Klevets n, O.A. Kosmachev, Yu.A. Fridman V.I. Vernadsky Taurida National University, Academician Vernadsky Ave., 4, 95007 Simferopol, Ukraine
art ic l e i nf o
a b s t r a c t
Article history: Received 1 May 2013 Received in revised form 22 July 2013 Available online 17 August 2013
Phase states of strongly anisotropic easy-plane spin-1 antiferromagnet with Ising-like exchange interaction have been investigated at transverse orientation of the external magnetic field. It is shown that the supersolid magnetic phase does not realize at this orientation of the magnetic field, while it does at the longitudinal orientation of the external magnetic field. The strong single-axis exchange anisotropy has essential influence on the dynamic and the static properties of the system. & 2013 Elsevier B.V. All rights reserved.
Keywords: Antiferromagnet Ising-like exchange interaction Easy-plane anisotropy Phase transition Supersolid magnetic phase
1. Introduction Nowadays magnets with competing exchange interactions and strong single-ion anisotropy attract considerable attention, related with the search for new quantum states. Generally, such magnets are systems in which localized magnetic moments (or spins) interact via competing exchange interactions, giving rise to various types of degeneracy of the ground state. At certain circumstances, this can result in the formation of spin liquid states, or in the realization of the rather new quantum state – supersolid magnetic phase (it is also called the intermediate or biconical phase) [1–3]. The simplest example of a magnet with competing exchange interactions is a two-sublattice antiferromagnet in which the exchange interaction between the magnetic ions in sublattice differs from the intersublattice exchange interaction. The interest to antiferromagnets has especially increased recently, due to the possibility of high-frequency (terahertz) spin excitations by femtosecond laser pulses [4,5]. There are numerous researches of the antiferromagnets. For example, the degrees of freedom for 1D antiferromagnet were investigated in [6], the isotropic antiferromagnet within the frameworks of the quantum sigma-model was investigated in [7], the properties of single-axes antiferromagnets in the external magnetic field were studied in [8,9], and review
n
Corresponding author. Tel.: þ 38 095 3301055. E-mail addresses: [email protected], [email protected] (Ph.N. Klevets). 0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.08.020
[10] provides a deep and thorough description of the properties of strongly anisotropic antiferromagnets. It was shown in [2,3,11,12] that various phases can realize in two-sublattice antiferromagnet: spin liquid, supersolid phase, magnetic plateaus. These phases can also be observed in a particular case of two-sublattice Ising-like antiferromagnet [12]; however, the regions of their existence and the types of the phase transitions can be essentially different. The realization of the states, listed above, essentially depends on the value and type of single-ion anisotropy [2,3,11,12]. At this, the easy-plane single-ion anisotropy plays essential role [10]. Generally, all investigations of the magnetic systems, related with the search for new phases, can be divided into two groups: the systems with strong single-ion anisotropy (which energy is comparable or even higher than the energy of exchange interaction), and the systems with weak single-ion anisotropy (which energy is lower than the energy of exchange interaction). In the first case, the strong single-ion anisotropy of the easy-plane type leads to the quantum reduction of the spin, and, consequently, the modulus of magnetization is not constant anymore, but depends on the relationship between the values of the external magnetic field, the single-ion anisotropy, and the exchange interaction [13,14]. At rather large values of the anisotropy constant, the effect of spin reduction leads to the realization of the spin state with zero value of average spin (per one lattice site) even in a non-zero (but sufficiently weak) magnetic field, i.e., the quadrupolar phase realizes in the system [14]. In the second case, the magnetization of the system remains almost constant in magnitude (it deviates
Ph.N. Klevets et al. / Journal of Magnetism and Magnetic Materials 348 (2013) 68–74
from M0 on a small value proportional to D/2J negligible at small values of the single-ion anisotropy D), and just changes its direction at varying the value and orientation of the external magnetic field. That is, the quadrupolar phase does not realize in case of the weak single-ion anisotropy. The aim of the present work is to investigate the phase states and the phase transitions in two-sublattice spin-1 antiferromagnet with Ising-like exchange interaction and strong easy-plane singleion anisotropy in the external magnetic field. Although the Isinglike exchange interaction depends only on the Z-component of spin operator, the spin operator itself has all three components, ! x y z S ¼ S ; S ; S . Previously, we have already investigated this system, but only in the external magnetic field perpendicular to the basal plane [15] (so-called, longitudinal magnetic field [13]). The Hamiltonian of such a system has the following form: 2 1 z z ^ ¼1 ∑ J ∑ J~ ni nj Szni Sznj þ D∑ Szni H∑Szni H ni mi Sni Smi þ 2ni ;mi 2ni ;nj =ði a jÞ ni ni
ð1Þ
where i and j take the values 1 and 2 for the first and the second sublattice, respectively; J n1 m1 ¼ J n2 m2 o0 are the constants of insublattice Ising-like exchange interactions corresponding to the ferromagnetic ordering in sublattices; J~ 4 0 is the constant of n1 n2
the anisotropic inter-sublattice Ising-like exchange interaction of Skni
the antiferromagnetic type; is the projection of spin operator of the ith sublattice at the nth site (k ¼x, y, z); D 40 is the constant of easy-plane single-ion anisotropy (XOY is the basal plane); H is the external magnetic field in energy units, and is oriented perpendicularly to the basal plane. As it was shown in [15], various phases can realize in such a system depending on the relationship between the values of the external magnetic field, the exchange constants, and the easyplane single-ion anisotropy. In case of the strong magnetic field, the ferromagnetic (FM) phase realizes in the system. In case of the weak magnetic field, the influence of strong easy-plane single-ion anisotropy leads to the realization of the quadrupolar (QU) phase. And in the most interesting and at the same time the more complex case of the intermediate magnetic field the supersolid (SS) magnetic phase can realize. If we introduce the coordinate systems, related with the sublattices, then the magnetization of the first sublattice in the SS phase makes some angle ϑ1 with the direction of the external magnetic field (i.e., the Z-axis), and the magnetization of the second sublattice – angle ϑ2 with the Z-axis. For simplicity, we suppose that magnetizations of both sublattices lie in the XOZ plane. In case of the longitudinal magnetic field, the angles of magnetization orientations depend on both values of the exchange interactions and are different. Moreover, the possibility of the SS phase realization is determined by the anisotropic character of the exchange interactions (J and J~ ). The results, obtained for the magnetic field perpendicular to the basal plane, are in very good qualitative and quantitative agreement with other theoretical investigations [2,12,13]. Also, nevertheless, the system, considered in [15], is just a model, there are real magnetic systems which characteristics are very close to that considered and which are a promising objects for experimental observation of the supersolid magnetic phase. This is, for example, NiCl24SC(NH2)2 in which Ni þ þ ions have spin equal to unity, and where, due to the influence of strong easy-plane singleion anisotropy, the ground state is singlet with spin projection Sz ¼ 0 [13]. Also, results obtained are applicable to the quasi-twodimensional system of Ba2CoGe2O7, where the magnetic spin-3/2 Co2 ions form layers of strongly anisotropic square lattices [16–19]. However, there is a gap in the investigations for the case of the external magnetic field, oriented along the basal plane (so-called, transverse magnetic field). Therefore, we will try to fill this gap
69
with the current investigation, and will compare system's behavior for two orientations of the external magnetic field: parallel and perpendicular to the easy-plane. According to [13], the magnetic ! field parallel to the basal plane is called “transverse” ( H ? OZ), despite the magnetic field perpendicular to the easy-plane which ! is called “longitudinal” ( H J OZ). Further, we will also use this terminology. We also consider the low temperature case (T « TN, TN is the Neel temperature). In this case, we can apply the mean-field approximation [12,20]. It is quite adequate, because the considerations may be restricted to the account of only the ground state at low-temperatures [6,14]. Besides, as we consider the case of strong single-ion anisotropy which influence leads to the quantum effects, we need to take into account this interaction exactly. The exact account of single-ion anisotropy is ensured by the application of the diagram technique for the Hubbard operators [14,15,20–23]. Unfortunately, detailed explanations of this mathematical approach can be found only in Russian; therefore, we dared to provide some explanations on the Hubbard operators' technique in the Appendix. 2. Transverse magnetic field Consider the sytem behavior in the external magnetic field parallel to the easy-plane. In this case, the Hamiltonian of the system can be presented as follows: 1 ^ ¼ 1 ∑ J H Sz Sz ∑ J Sz Sz þ D∑ðSzni Þ2 H∑Sxni 2ni ;mi ni mi ni mi 2ni ;nj ni nj ni nj ni ni
ð2Þ
ði a jÞ
Consider two cases: strong and weak magnetic field. 2.1. Strong magnetic field Consider the case when the Zeeman energy is considerably higher, than the energies of the exchange interactions and the single-ion anisotropy. In this case, the magnetic moments are collinear and are oriented along the field, i.e., the FM phase realizes in the system. It is convenient to carry out the further consideration in the rotated coordinate system in which the quantization axis (the Z-axis) coincides with the orientation of the sublattices' magnetizations, i.e., with the direction of the external magnetic field. In this case, the Hamiltonian (2) takes the following view: 1 ^ ¼ 1 ∑ J H Sx Sx ∑ J~ Sx Sx þ D∑ðSxni Þ2 H∑Szni 2ni ;mi ni mi ni mi 2ni ;nj ni nj ni nj ni ni
ð3Þ
ði a jÞ
As the terms describing the energy of the exchange interactions do not depend on the projection of spin operator on the quantization axis in the Hamiltonian (2), the single-site Hamiltonian is rather simple: ^ 0 ðni Þ ¼ D∑ðSx Þ2 H∑Sz H ni ni ni
ni
ð4Þ
Solving the Schrödinger equation with the Hamiltonian (4), we can determine the energy levels of a magnetic ion: EðiÞ 71 ¼
D 8 κ ; EðiÞ 0 ¼D 2
ð5Þ
and the eigenvectors of the single-site Hamiltonian: Ψ ð1Þ ¼ cos θj1i sin
θj1i; Ψ ð0Þ ¼ j0i; Ψ ð1Þ ¼ sin θj1i þ cos θj1i
ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2ffi 2 H D where κ ¼ H þ 2 ; cos 2θ ¼ κ ; sin 2θ ¼ 2κ The Hubbard operators are constructed on the basis of the eigenvectors (6) [14,15]. The relationship between the spin operators
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and the Hubbard operators is given by Sz ¼ cos 2θ X 11 X 11 þ sin 2θ X 11 þ X 11 ; pffiffiffi pffiffiffi S þ ¼ 2 cos θ X 10 þ X 01 þ 2 sin θ X 10 X 01 ;
† S ¼ S þ : ð7Þ
Using Eq. (7), it can be easily shown that the single-site Hamiltonian (4) is diagonal in the representation of the Hubbard operators: ^ 0 ðni Þ ¼ ∑ EðiÞ X MM . H M ¼ 1;0;1
M
ni
The spectra of elementary excitations in the FM phase can be found by carrying out the bosonization the Hubbard operators [24]: h i ε21 ðkÞ ¼ D2 þ κ D2 þ κ 2J k jJ~k j 12Dκ ;
ε22 ðkÞ ¼ 4H2 þ D2 ;
ð8Þ
where Jk and |J~ k | are the Fourier transforms of the corresponding exchange integrals. We can find the field of the phase transition from the FM phase using the requirement of the gap vanishing in the low-frequency spectrum ε1(k): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t 2 D2 D 3 D2 ? D2C DC D þ H FM ¼ 2DC 2DC D2 DC 4 2 8 16
ð9Þ
where DC ¼ J 0 jJ~ 0 j=2. As it will be shown below, the critical value of the anisotropy constant DC corresponds to the phase transition from the antiferromagnetic (AFM) phase into the QU phase. As follows from Eq. (7), the average value of sublattices' magnetization (per one site) is given by D E D E z
S ¼ cos 2θ X 11 X 11 ; ð10Þ and in the low-temperature limit it has the following form: z
H S cos 2θ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 H þ D2
ð11Þ
It follows from the last expression that sublattices' magnetizations reach their maximum only in the isotropic case. This result coincides with [14] and can be explained by the fact that the operator of single-ion anisotropy does not commute with the projection of the spin operator on the Z-axis (see Eq. (4)). As a result, this component of the spin operator is not the integral of motion, and, consequently, the saturation cannot be reached at finite magnetic field. This phase state is ferromagnetic, but is not saturated. 2.2. Weak magnetic field Now let us consider the case of the weak magnetic field when the energy of the external magnetic field is considerably lower than the energy of the single-ion anisotropy and the exchange interactions. In this case, the influence of strong easy-plane singleion anisotropy can lead to the realization of the QU phase. The single-site Hamiltonian is given by Eq. (4) in this case too. This means that all expressions, obtained above for the strong magnetic field, are also valid for the weak magnetic field. Thus, the magnon spectra in the QU phase are given by Eq. (8). The only difference is that D=2κ -1 (because H-0) in the QU phase. This means that the average value of the magnetization becomes zero only at zero magnetic fields, because Eq. (11) is valid at any value of the magnetic field differing from zero. Consequently, H Q?U ¼ 0 determines the transition field from the QU phase. In order to answer the question what phase state realizes in-between the nonsaturated FM phase and the QU phase, we need to investigate the system in the interval of fields:
? 0 o H oH FM . However, at first, let us consider in more details the case of zero magnetic fields. The single-site Hamiltonian has the following view at zero magnetic fields: 2 ^ 0 ðni Þ ¼ D∑ Sz þ ð1Þi þ 1 H∑Szni ð12Þ H ni ni
ni
where H ¼ ðJ 0 jJ~ 0 j=2Þ Sz is the mean field. It is seen from Eq. (12) that either the AFM phase or the QU phase can realize in the system depending on the relationship between the constants of exchange interaction and easy-plane anisotropy. Solving the Schrödinger equation with the Hamiltonian (12), one can find the energy values of a magnetic ion: E 7 1 ¼ D 8 H; E0 ¼ 0
ð13Þ
while the eigenvectors of the Hamiltonian (12) coincides with the Eq. (6), if we take into account that cos 2θ ¼ 1 at zero magnetic fields. In this case, the relationship between the spin operators and the Hubbard operators takes the simple view: pffiffiffi þ Sz ¼ X 11 X 11 ; S þ ¼ 2 X 10 þ X 01 ; S ¼ S þ : ð14Þ Implementing the procedure of Hubbard operators' bosonization [24], we can easily derive the spectra of elementary excitations valid at small values of the anisotropy constant (D « J0, |J~ 0 |) in the AFM phase:
ε1 ðkÞ ¼ HD;
ε2 ðkÞ ¼ 2H;
ð15Þ
the spectra in the QU phase are as follows:
ε1 ðkÞ ¼ DJ k þ jJ~k j=2; ε1 ðkÞ ¼ J k jJ~ k j=2;
ð16Þ
and are valid at large values of the anisotropy constant (D » J0, |J~ 0 |). From the requirement of gaps vanishing in the low-frequency magnon spectra we can find the value of the easy-plane anisotropy at which the phase transitions from the AFM phase into the QU phase and vice versa occur: from the AFM phase – spectra (15): DAFM ¼ J 0 jJ~ 0 j=2; from the QU phase – spectra (16): DQ U ¼ J 0 jJ~ 0 j=2 These values coincide, and, consequently, the phase transition the AFM phase – the QU phase is of the second order, and can occur only at zero magnetic fields when DC ¼ J 0 jJ~ 0 j=2
ð17Þ
It should be noted that Eq. (17) coincides with the expression for DC, introduced above in the expression for the critical field of the phase transition from the nonsaturated FM phase (9). Now let us proceed to the investigation of the spin state which ? realizes in the interval of fields 0 o H o H FM . Averaging of Eq. (7), obtained above for arbitrary fields, shows that sublattice's magnetization always differs from zero at the nonzero external field; however, its modulus changes with the magnetic field increasing ? from zero up to H FM . Let us investigate the free energy of the sublattices to determine the phase state of the system. The single-site Hamiltonian for the first and second sublattices has the following form: 2 ^ 0 ðni Þ ¼ D∑ Sy H þ ð1Þi H cos φi ∑Szni H sin φi ∑Sxni ð18Þ ni ni
ni
ni
while deriving Eq. (18), we supposed that, generally, sublattices' magnetizations are not collinear and make different angles φ1 and φ2 with the quantization axis (the Z-axis). Implementing the diagonalization procedure [24], one finds the energy levels of a magnetic ion for the first and the second sublattice H ffiffi sin φi ð cos αi þ sin αi Þ sin 2βi Eð1iÞ ¼ ð1Þi H2 cos φi cos 2αi 1 þ cos 2βi 2p 2
D D sin 2αi 1 þ cos 2βi þ 1 cos 2βi ; 4 4
Ph.N. Klevets et al. / Journal of Magnetism and Magnetic Materials 348 (2013) 68–74
1 D ð1Þi þ 1 H cos φi cos 2αi þ ð1 þ sin 2αi Þ 4 2 h þ ð1Þi þ 1 H cos φi cos 2αi
In connection with this, it is reasonable to consider two limit cases:
iÞ Eð0;1 ¼
H D þ ð1 þ sin 2αi Þ cos 2β i þ pffiffiffi sin φi ð cos αi þ sin αi Þ sin 2βi 2 2 7
1n ð1Þi 3H cos φi cos 2αi 4
D D þ ð13 sin 2αi Þþ ð1Þi þ 1 H cos φi cos 2αi þ ð1 þ sin 2αi Þ cos 2βi 2 2
H þ pffiffiffi sin φi ð cos αi þ sin αi Þ sin 2β i cos 2γ i 2 D 8 ð1Þi þ 1 H cos φi sin 2αi cos 2αi sin β i 2
H þ pffiffiffi sin φi ð cos αi sin αi Þ cos βi sin 2γ i ; 2 where the parameters of unitary transformations given by the following system of equations: pffiffiffi 2H sin φi ð cos αi þ sin αi Þ ; tan 2βi ¼ iþ1 D cos φi 2 ð1 þ sin 2αi Þ þ ð1Þ tan βi ¼
ð19Þ
α, β, and γ are
H ð1Þi þ 1 H cos φi cos 2αi 3 cos 2βi pffiffiffi sin φi ð cos αi þ sin αi Þ sin 2βi 2
ð20Þ
ð21Þ
the equilibrium angle of sublattice's magnetization orientation is determined from the following condition: ðiÞ
∂F i ∂E1 ¼ ¼0 ∂ φi ∂ φi
The standard procedure of determining the equilibrium state of the magnet is to minimize the total free energy of the system. However, the ground state is not well-defined in antiferromagnets, and the total energy of the antiferromagnet does not correspond to the equilibrium state. Therefore, we investigate the minimum of the free energy density of the system which, in general, has the following form: F ¼ T ln Z, where Z ¼ ∑ expðEM =TÞ is the partition function. The general view of the free energy density is the same for both sublattices. As we consider the low-temperature case, we can restrict our consideration with the account of the lowest energy level E1, then the free energy density has the following view: Dpffiffiffiffiffiffiffiffiffiffiffi2ffi 1s Hsi cos φi Fi ¼ 2
The lowest energy level is E1, and, as we consider the lowtemperature case, the free energy of the ith sublattice is simply F i ¼ Eð1iÞ
? 1. Magnetic field H tends to the critical value H FM . With increasing
external magnetic field, sublattices' magnetizations Sz will tend to unity. In this qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 case, it is convenient to choose the small value η2 ¼ 1 Sz as the order parameter. It can be shown
that sublattice's magnetization Sz ¼ cos 2α [14], then the order parameter near the FM phase is given by η2 ¼ sin 2α. 2. Magnetic field H tends to the critical value H Q?U ¼ 0. In this case,
it is convenient to choose sublattice's magnetization, s ¼ Sz ¼ cos 2α, as the order parameter, because its modulus tends to zero near the QU phase.
M ¼ 1;0;1
ð1Þi þ 1 H cos φi sin 2αi D2 cos 2αi ; Hffiffi p sin φi ð cos αi sin αi Þ 2
D D þ sin 2αi 3 cos 2βi 1 þ cos 2β i sin 2γ i 2 2 npffiffiffi ¼2 2H sin φi ð cos αi sin αi Þ cos βi þ h i o ð1Þi þ 1 2H cos φi sin 2αi D cos 2αi sin βi cos 2γ i :
71
ð22Þ
jointly resolving Eqs. (19)–(22), we find the equilibrium angles of sublattices' magnetizations orientations: 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 1 @D D 1@ D D 2A þ þ H ; cos φ2 ¼ þ H2 A cos φ1 ¼ H 4 4 H 4 4 ð23Þ obviously, φ1 ¼ φ2 which means that the intermediate state of the Ising-like antiferromagnet with strong easy-plane single-ion anisotropy in the transverse magnetic field is not a supersolid phase (as in the longitudinal magnetic field), but the spin-flop (SF) phase. Please note that the angles of magnetizations' orientations in sublattices do not depend on the exchange interaction in case of the transverse magnetic field (compare with the previous case of the longitudinal magnetic field – see [15]) which results in the equal absolute values of angles of magnetizations' orientations in sublattices. Taking the advantage of the Landau theory of the phase transitions, we can determine the fields of the phase transitions from the SF phase into the FM and the QU phase. To do this, we need to investigate the free energy density of the system F by expanding it into series of the corresponding order parameter which must tend to zero at the lines of the phase transitions (stability lines) of the FM and the QU phases, respectively.
ð24Þ
As it was shown in [14], the angle α varies from 0 to π/4 in the considered state which means the decrease of the magnetization modulus from 1 to 0. Thus, not only magnetization's direction changes with the decreasing magnetic field in the intermediate state, but also its modulus decreases due to the influence of the easy-plane single-ion anisotropy, and it becomes zero in the limit H-H Q?U . This effect is called the quantum reduction of the spin and is due to the influence of the easy-plane anisotropy. Consider the free energy density when the external magnetic field is close to the critical transition field into the QU phase. Expanding Eq. (24) into series by the magnetization s and restricting our consideration by the second order terms of s, we obtain the expression for the free energy density near the QU phase: 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 D @ D 2 DA D 2 F ¼ s þ s þH ð25Þ 2 4 4 4 The field of the phase transition into the QU phase can be easily obtained from Eq. (25): ? H SFQ U ¼0
ð26Þ
As you can see, Eq. (26) coincides with the transition field from the QU phase into the SF phase – H Q?U ¼ 0. The free energy density analysis shows that the phase transition from the SF phase into the QU phase is of the first order, and Eq. (26) determines the line at which SF phase becomes unstable during the phase transition into the QU phase. Consider now the case when magnetic field is close to the critical transition field into the FM phase. Expanding (26) into series by the small parameter η, we obtain the expression for the free energy density near the FM phase: D F¼ 4
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 2 2 D D D D η4 2 45 2 5η þ H D þH þ H2 5 4 2 8 4 4 4 4 4
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Fig. 1. Dependence of magnetizations' orientations in sublattices on the value of the external transverse magnetic field.
Fig. 2. Dependence of sublattice's magnetization on the value of external transverse magnetic field.
2 D 4 4
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 D η6 þH 2 5 4 16
ð27Þ
from which it is easy to find the field of the phase transition into the FM phase: pffiffiffiffiffiffiffiffi ? H SFFM ¼ 3=2D ð28Þ ? , then the coefficients near η 2 is negative which If H oH SFFM indicates to the first order phase transition from the SF phase into the FM phase, and the line, determined by Eq. (28), is the line where SF phase becomes unstable during the phase transition into the FM phase. Fig. 1 shows the dependence of magnetization orientation in sublattices depending on the value of the transverse magnetic field. The dependence of the sublattice's magnetization on the value of the external magnetic field is shown in Fig. 2. As you can see, the phase transition from the FM phase into the SF phase occurs with hysteresis which also indicates the first order phase transition.
3. Conclusions Carried out investigations have shown that three phase states are possible in the Ising-like antiferromagnet with strong single-ion
anisotropy at varying of the external magnetic field. In case of the transverse magnetic field, these are: the non-saturated ferromagnetic (FM) phase, the spin-flop (SF) phase, and the quadrupolar (QU) phase. The supersolid (SS) magnetic phase which realizes in case of the longitudinal magnetic field cannot realize in case of the transverse magnetic field – the SF phase realizes instead. It was shown before [14] that the same phase states realize in the Heisenberg strongly anisotropic antiferromagnet; however, the Ising-like antiferromagnet, considered in the present work, exhibits some interesting peculiarities. Let us consider the results obtained with that, obtained previously in [14,15]. In case of the longitudinal magnetic field, the phase transitions the QU phase – the SS phase is of the second order in strongly anisotropic Heisenberg antiferromagnet [14]. While, as our investigations have shown [15], this phase transition is of the first order in the Ising-like antiferromagnet. At this, most likely, the SF phase realizes in the region of the QU and the SS phases co-existence. The type of the phase transition the SS phase – the FM phase is also of the first order; however, orientation of the magnetic moment of one of the sublattices remains constant in the region of the FM and the SS phases co-existence, and abruptly orients along the external magnetic field only at the phase transition line. At this, the magnetization modulus depends on the value of the field in the SS phase – it increases with the increasing external field, and reaches saturation at the phase transition into the FM phase [15]. In case of the transverse magnetic field, the SS phase does not realize in the Ising-like antiferromagnet, despite the case of Heisenberg antiferromagnet [14,25]. The SF phase realizes instead of the SS phase. Such behavior can be explained by the influence of the easyaxis exchange anisotropy in the Ising-like antiferromagnet. Both phase transitions, the SF phase – the FM phase and the SF phase – the QU phase, are of the first order. At this, as well as in the case of the longitudinal orientation of the magnetic field, the magnetization modulus depends on the magnitude of the external magnetic field in the SF and the FM phases; however, magnetization reaches saturation only at infinite magnetic field, because the operator of single-ion anisotropy does not commute with the Z-component of spin operator, and, consequently, this component of spin moment is not the integral of motion. Also, the phase transition from the FM phase into the SF phase occurs with hysteresis. To summarize the results obtained for the longitudinal and transverse orientations of the external magnetic field, let us compare the influence of the effective fields of the exchange interactions on the sublattices for different orientations of the external magnetic field. In case of the longitudinal magnetic field, the effective fields of the exchange interactions are collinear to the external magnetic field. This result in the competition between the external magnetic field and the effective exchange fields [15] – the effective fields of the exchange interactions amplify the influence of the external magnetic field in one of the sublattices, and weak it in another sublattice. This leads to different absolute values of angles of magnetizations' orientations in sublattices, and, consequently, to the realization of the SS phase. In case of the transverse external magnetic field, the effective fields of the exchange interactions act perpendicularly to the external magnetic field. At this, the absolute values of angles of magnetizations' orientations coincide for different sublattices, as they do not depend on the exchange interactions anymore (see Eq. (25)). Consequently, this leads to the realization of the SF phase. Thus, the realization of the SS phase essentially depends on the orientation of the external magnetic field, and the SS phase cannot realize in case of the transverse external magnetic field. Besides, magnons' dispersion law essentially depends on the magnetic field orientation. In case of the longitudinal magnetic field, magnon spectra are dispersionless [15], while in the transverse
Ph.N. Klevets et al. / Journal of Magnetism and Magnetic Materials 348 (2013) 68–74
magnetic field the spectra (8) and (16) depend on the wave vector k. And if such form of the magnon spectrum is common for highfrequency magnons, it requires some explanations for the lowfrequency magnons. In case of the longitudinal magnetic field, the independence of the low-frequency magnon spectra on the wave vector k in the FM and the QU phases is related with the absence of the transverse component of the spin operators (Sx and Sy) in the Hamiltonian of the system. Probably, the magnon spectra will depend on the wave vector in the intermediate SS phase, but their determination in this phase is beyond the scope of the present investigations. In case of the transverse magnetic field, the Hamiltonian of the Ising-like antiferromagnet always contains non-zero transverse components of spin operator which is exhibited in the dependence of Eqs. (8) and (16) on k. As it was mentioned in the Introduction, there are real magnetic systems in which the supersolid magnetic phase can be found experimentally. These are, for example, NiCl24SC(NH2)2 and Ba2CoGe2O7 [13,16–19]. The results, obtained in the present investigations, show that in case of the system with the Ising-like exchange interactions and strong easy-plane single ion anisotropy, the supersolid magnetic phase cannot be observed in the transverse magnetic field. The case of the longitudinal magnetic field perpendicular to the basal plane is the most favorable for experimental observation of the supersolid magnetic phase. This must be taken into account by experimentalists while trying to find experimental evidence of the supersolid magnetic phase existence.
Acknowledgments
non-diagonal Hubbard operators. These solutions allow obtaining the relationship between the material parameters of the system and the parameters of rotation. After the unitary transformations, mentioned above, the singlesite Hamiltonian takes the diagonal form: H 0 -H~ 0 ¼ UH 0 U þ ¼ ∑ En X nn ; n
where En are the energy levels of a magnetic ion. The next step is to transcribe the Hubbard operators through the pseudo-Hubbard operators which is case of S ¼1 have the following relationship with Bose operators of creation and annihilation: 11 00 þ X~ n ¼ 1anþ an bn bn ; X~ n ¼ anþ an ; 11 10 þ þ ¼ bn bn ; X~ n ¼ 1anþ an bn bn an X~ n 01 X~ n ¼ anþ 11 11 01 10 þ þ þ X~ n ¼ 1anþ an bn bn bn ; X~ n ¼ bn ; X~ n ¼ anþ bn ; X~ n ¼ bn an
Here a are the Bose-operators corresponding to the transition of a magnetic ion from the state E1 into the state E0 and vice versa, and b correspond to the transition from the state E1 into the state E 1 and vice versa. α The Lie algebra of the pseudo-Hubbard operators X~ n is the α same as the Lie algebra of the Hubbard operators X n . The direct calculations prove the following equality: h ij pq i iq pj ¼ δjp X~ n δqi X~ n : X~ n ; X~ n
The authors are grateful to Prof. Ivanov B.A. for fruitful discussion. The work was supported by the State Fund for Fundamental Research, Grant F33.2/002.
Appendix We have taken the advantage of the method of the Hubbard operator’s bosonization while obtaining the magnon' spectra. The application of this approach allows obtaining the exact expressions for the spectra of elementary excitations for a rather complex system, like, for example, that, investigated in the present work. Unfortunately, this mathematical technique is explained in details only in Russian; therefore, we have decided briefly explain its key features here. The procedure of spectra derivation is fulfilled in two steps. At first, the single-site Hamiltonian is diagonalized and re-written through the Hubbard operators describing the transition of a magnetic ion from the jM i state into the jM′i state: X M′M n
Ψ n ðM′Þ Ψ n ðM Þ [21]. Next, the Hubbard operators X α are trann α scribed through the pseudo-Hubbard operators X~ . n
Generally, the single-site Hamiltonian is not diagonal in the basis of the eigenvectors of the Sz operator. The generalized u-v transformation is required to diagonalize it [26]. At this, the singlesite Hamiltonian is rotated in the spin space according to the following rule: nn nm mm n mn U þ sin nm φ ¼ exp nmφX φ X mn¼ 1 þ cos φ 1 X þ X φ exp iμ X exp iμ X ;where φ is the parameter of rotation. At this, the law of transformation of the Hubbard operators during the unitary rotation is given by the following expression: þ X pq -U nm φ X pq U nm φ X pq φ : After these rotations, one obtains the system of equations on the parameters φi which solutions must be restricted with that satisfying the condition of vanishing of the coefficients at
73
α
However, X~ n operators are not identical to the Hubbard operators X αn . These operators cannot be equated in principle, because the Hubbard and the pseudo-Hubbard operators act in spaces of different dimensionality: the dimensionality of the physical space of X αn operators equals three (at S ¼1); while the Hilbert space of a and b operators is infinitely dimensional. Besides, there are other differences, in particular: 10 þ 11 þ pq þ 01 11 X ¼ X qp ; X~ a X~ ; X~ a X~ n
n
11 X 01 ¼ X 01 n Xn n ;
n
n
01 11 X~ n X~ n
n
n
01 þ ¼ anþ anþ bn bn a X~ n
Generally, the Hubbard operators, and, consequently, the single-site Hamiltonian, cannot be transcribed through any combinations of the Bose operators. At the same time, one can construct the Bose analog of the Hamiltonians (1) and (2), i.e., the operator acting in the infinitely dimensional Hilbert space; at this, certain part of its coefficient functions equals the coefficient functions of the Hamiltonians (1) and (2). Transcribing the Hamiltonians (1) and (2) through the Bose operators and restricting ourselves with the terms quadratic in the operators of creation and annihilation, we obtain the following general expression: þ
H ð2Þ ¼ ∑ ðE0 E1 Þakþ ak þ ∑ ðE1 E1 Þbk bk k
k
n 1 þ þ ∑ Ak akþ ak þ ∑ A~ k akþ ak þ A~ k ak ak 2 k k n 1 ~ þ þ þ þ ∑ Bk bk bk þ ∑ B k bk bk þ B~ k bk bk : 2k k
Diagonalizing this Hamiltonian with standard Bogolubov's u–v transformation, one obtains: þ
H ð2Þ ¼ ε0 þ ∑ εα ðkÞαkþ αk þ ∑ εβ ðkÞβ k βk ; k
where the coefficients at of quasi-particles.
k
α and β operators determine the spectra
74
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References [1] L. Balents, Nature (London) 464 (2010) 199. [2] Judit Romhányi, Frank Pollmann, Karlo Penc, Physical Review B 84 (2011) 184427. [3] D. Peters, I.P. McCulloch, W. Selke, Physical Review B 79 (2009) 132406. [4] T. Satoh, S.-J. Cho, R. Iida, T. Shimura, K. Kuroda, H. Ueda, Y. Ueda, B.A. Ivanov, F. Nori, M. Fiebig, Physical Review B 105 (2010) 077402. [5] A.V. Kimel, B.A. Ivanov, R.V. Pisarev, P.A. Usachev, A. Kirilyuk, Th. Rasing, Nature Physics 5 (2009) 727. [6] B.A. Ivanov and A.K. Kolezhuk, Physical Review Letters 74 (1995) 1859. [7] E.G. Galkina, A.Yu. Galkin, B.A. Ivanov, Low Temperature Physics 34 (2008) 522. [8] V.M. Loktev, V.S. Ostrovskiy, Physics of the Solid State 20 (1978) 3086. [9] V.M. Loktev, V.S. Ostrovskiy, Ukrainian Journal of Physics 23 (1978) 1708. (in Ukrainian). [10] V.M. Loktev, V.S. Ostrovskiy, Fizika Nizkih Temperatur 20 (1994) 983. (in Russian). [11] P. Sengupta, C.D. Batista, Physical Review Letters 98 (2007) 227201. [12] Luis Seabra, Nic Shannon, Physical Review B 83 (2011) 134412. [13] V.M. Kalita, I. Ivanova, V.M. Loktev, Physical Review B 78 (2008) 104415. [14] Y.A. Fridman, O.A. Kosmachev, P.N. Klevets, European Physical Journal B 81 (2011) 185.
[15] Ph. N. Klevets, O.A. Kosmachev, Yu.A. Fridman, Journal of Magnetism and Magnetic Materials 330 (2013) 91. [16] A. Zheludev, T. Sato, T. Masuda, K. Uchinokura, G. Shirane, B. Roessli, Physical Review B 68 (2003) 024428. [17] H.T. Yi, Y.J. Choi, S. Lee, S.-W. Cheong, Applied Physics Letters 92 (2008) 212904. [18] H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, Y. Tokura, Physical Review Letters 105 (2010) 137202. [19] S. Miyahara, N. Furukawa, Journal of the Physical Society of Japan 80 (2011) 073708. [20] A. Yu., O.A. Fridman, A.K. Kosmachev, Kolezhuk, B.A. Ivanov, Physical Review Letters 106 (2011) 097202. [21] R.O. Zaitsev, Journal of Experimental and Theoretical Physics 41 (1975) 100. [22] F.P. Onufrieva, Journal of Experimental and Theoretical Physics 67 (1988) 346. [23] F.P. Onufrieva, Journal of Experimental and Theoretical Physics 53 (1981) 1241. [24] V.V. Val'kov, T.A. Val'kova, Application of the indefinite metrics to the transition from atomic to Bose (Bose-Fermi) representation of quantum Hamiltonians, Pre-print IF SO AN USSR No. 644F, Krasnoyarsk, 1990 (in Russian). [25] V.M. Kalita, V.M. Loktev, Low Temperature Physics 31 (2005) 619. [26] V.V. Val'kov, Theoretical and Mathematical Physics 76 (1988) 766.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Supersolid magnetic phase realization in strongly anisotropic easy-plane antiferromagnet with Ising-like exchange interaction in the transverse magnetic field Ph.N. Klevets n, O.A. Kosmachev, Yu.A. Fridman V.I. Vernadsky Taurida National University, Academician Vernadsky Ave., 4, 95007 Simferopol, Ukraine
art ic l e i nf o
a b s t r a c t
Article history: Received 1 May 2013 Received in revised form 22 July 2013 Available online 17 August 2013
Phase states of strongly anisotropic easy-plane spin-1 antiferromagnet with Ising-like exchange interaction have been investigated at transverse orientation of the external magnetic field. It is shown that the supersolid magnetic phase does not realize at this orientation of the magnetic field, while it does at the longitudinal orientation of the external magnetic field. The strong single-axis exchange anisotropy has essential influence on the dynamic and the static properties of the system. & 2013 Elsevier B.V. All rights reserved.
Keywords: Antiferromagnet Ising-like exchange interaction Easy-plane anisotropy Phase transition Supersolid magnetic phase
1. Introduction Nowadays magnets with competing exchange interactions and strong single-ion anisotropy attract considerable attention, related with the search for new quantum states. Generally, such magnets are systems in which localized magnetic moments (or spins) interact via competing exchange interactions, giving rise to various types of degeneracy of the ground state. At certain circumstances, this can result in the formation of spin liquid states, or in the realization of the rather new quantum state – supersolid magnetic phase (it is also called the intermediate or biconical phase) [1–3]. The simplest example of a magnet with competing exchange interactions is a two-sublattice antiferromagnet in which the exchange interaction between the magnetic ions in sublattice differs from the intersublattice exchange interaction. The interest to antiferromagnets has especially increased recently, due to the possibility of high-frequency (terahertz) spin excitations by femtosecond laser pulses [4,5]. There are numerous researches of the antiferromagnets. For example, the degrees of freedom for 1D antiferromagnet were investigated in [6], the isotropic antiferromagnet within the frameworks of the quantum sigma-model was investigated in [7], the properties of single-axes antiferromagnets in the external magnetic field were studied in [8,9], and review
n
Corresponding author. Tel.: þ 38 095 3301055. E-mail addresses: [email protected], [email protected] (Ph.N. Klevets). 0304-8853/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmmm.2013.08.020
[10] provides a deep and thorough description of the properties of strongly anisotropic antiferromagnets. It was shown in [2,3,11,12] that various phases can realize in two-sublattice antiferromagnet: spin liquid, supersolid phase, magnetic plateaus. These phases can also be observed in a particular case of two-sublattice Ising-like antiferromagnet [12]; however, the regions of their existence and the types of the phase transitions can be essentially different. The realization of the states, listed above, essentially depends on the value and type of single-ion anisotropy [2,3,11,12]. At this, the easy-plane single-ion anisotropy plays essential role [10]. Generally, all investigations of the magnetic systems, related with the search for new phases, can be divided into two groups: the systems with strong single-ion anisotropy (which energy is comparable or even higher than the energy of exchange interaction), and the systems with weak single-ion anisotropy (which energy is lower than the energy of exchange interaction). In the first case, the strong single-ion anisotropy of the easy-plane type leads to the quantum reduction of the spin, and, consequently, the modulus of magnetization is not constant anymore, but depends on the relationship between the values of the external magnetic field, the single-ion anisotropy, and the exchange interaction [13,14]. At rather large values of the anisotropy constant, the effect of spin reduction leads to the realization of the spin state with zero value of average spin (per one lattice site) even in a non-zero (but sufficiently weak) magnetic field, i.e., the quadrupolar phase realizes in the system [14]. In the second case, the magnetization of the system remains almost constant in magnitude (it deviates
Ph.N. Klevets et al. / Journal of Magnetism and Magnetic Materials 348 (2013) 68–74
from M0 on a small value proportional to D/2J negligible at small values of the single-ion anisotropy D), and just changes its direction at varying the value and orientation of the external magnetic field. That is, the quadrupolar phase does not realize in case of the weak single-ion anisotropy. The aim of the present work is to investigate the phase states and the phase transitions in two-sublattice spin-1 antiferromagnet with Ising-like exchange interaction and strong easy-plane singleion anisotropy in the external magnetic field. Although the Isinglike exchange interaction depends only on the Z-component of spin operator, the spin operator itself has all three components, ! x y z S ¼ S ; S ; S . Previously, we have already investigated this system, but only in the external magnetic field perpendicular to the basal plane [15] (so-called, longitudinal magnetic field [13]). The Hamiltonian of such a system has the following form: 2 1 z z ^ ¼1 ∑ J ∑ J~ ni nj Szni Sznj þ D∑ Szni H∑Szni H ni mi Sni Smi þ 2ni ;mi 2ni ;nj =ði a jÞ ni ni
ð1Þ
where i and j take the values 1 and 2 for the first and the second sublattice, respectively; J n1 m1 ¼ J n2 m2 o0 are the constants of insublattice Ising-like exchange interactions corresponding to the ferromagnetic ordering in sublattices; J~ 4 0 is the constant of n1 n2
the anisotropic inter-sublattice Ising-like exchange interaction of Skni
the antiferromagnetic type; is the projection of spin operator of the ith sublattice at the nth site (k ¼x, y, z); D 40 is the constant of easy-plane single-ion anisotropy (XOY is the basal plane); H is the external magnetic field in energy units, and is oriented perpendicularly to the basal plane. As it was shown in [15], various phases can realize in such a system depending on the relationship between the values of the external magnetic field, the exchange constants, and the easyplane single-ion anisotropy. In case of the strong magnetic field, the ferromagnetic (FM) phase realizes in the system. In case of the weak magnetic field, the influence of strong easy-plane single-ion anisotropy leads to the realization of the quadrupolar (QU) phase. And in the most interesting and at the same time the more complex case of the intermediate magnetic field the supersolid (SS) magnetic phase can realize. If we introduce the coordinate systems, related with the sublattices, then the magnetization of the first sublattice in the SS phase makes some angle ϑ1 with the direction of the external magnetic field (i.e., the Z-axis), and the magnetization of the second sublattice – angle ϑ2 with the Z-axis. For simplicity, we suppose that magnetizations of both sublattices lie in the XOZ plane. In case of the longitudinal magnetic field, the angles of magnetization orientations depend on both values of the exchange interactions and are different. Moreover, the possibility of the SS phase realization is determined by the anisotropic character of the exchange interactions (J and J~ ). The results, obtained for the magnetic field perpendicular to the basal plane, are in very good qualitative and quantitative agreement with other theoretical investigations [2,12,13]. Also, nevertheless, the system, considered in [15], is just a model, there are real magnetic systems which characteristics are very close to that considered and which are a promising objects for experimental observation of the supersolid magnetic phase. This is, for example, NiCl24SC(NH2)2 in which Ni þ þ ions have spin equal to unity, and where, due to the influence of strong easy-plane singleion anisotropy, the ground state is singlet with spin projection Sz ¼ 0 [13]. Also, results obtained are applicable to the quasi-twodimensional system of Ba2CoGe2O7, where the magnetic spin-3/2 Co2 ions form layers of strongly anisotropic square lattices [16–19]. However, there is a gap in the investigations for the case of the external magnetic field, oriented along the basal plane (so-called, transverse magnetic field). Therefore, we will try to fill this gap
69
with the current investigation, and will compare system's behavior for two orientations of the external magnetic field: parallel and perpendicular to the easy-plane. According to [13], the magnetic ! field parallel to the basal plane is called “transverse” ( H ? OZ), despite the magnetic field perpendicular to the easy-plane which ! is called “longitudinal” ( H J OZ). Further, we will also use this terminology. We also consider the low temperature case (T « TN, TN is the Neel temperature). In this case, we can apply the mean-field approximation [12,20]. It is quite adequate, because the considerations may be restricted to the account of only the ground state at low-temperatures [6,14]. Besides, as we consider the case of strong single-ion anisotropy which influence leads to the quantum effects, we need to take into account this interaction exactly. The exact account of single-ion anisotropy is ensured by the application of the diagram technique for the Hubbard operators [14,15,20–23]. Unfortunately, detailed explanations of this mathematical approach can be found only in Russian; therefore, we dared to provide some explanations on the Hubbard operators' technique in the Appendix. 2. Transverse magnetic field Consider the sytem behavior in the external magnetic field parallel to the easy-plane. In this case, the Hamiltonian of the system can be presented as follows: 1 ^ ¼ 1 ∑ J H Sz Sz ∑ J Sz Sz þ D∑ðSzni Þ2 H∑Sxni 2ni ;mi ni mi ni mi 2ni ;nj ni nj ni nj ni ni
ð2Þ
ði a jÞ
Consider two cases: strong and weak magnetic field. 2.1. Strong magnetic field Consider the case when the Zeeman energy is considerably higher, than the energies of the exchange interactions and the single-ion anisotropy. In this case, the magnetic moments are collinear and are oriented along the field, i.e., the FM phase realizes in the system. It is convenient to carry out the further consideration in the rotated coordinate system in which the quantization axis (the Z-axis) coincides with the orientation of the sublattices' magnetizations, i.e., with the direction of the external magnetic field. In this case, the Hamiltonian (2) takes the following view: 1 ^ ¼ 1 ∑ J H Sx Sx ∑ J~ Sx Sx þ D∑ðSxni Þ2 H∑Szni 2ni ;mi ni mi ni mi 2ni ;nj ni nj ni nj ni ni
ð3Þ
ði a jÞ
As the terms describing the energy of the exchange interactions do not depend on the projection of spin operator on the quantization axis in the Hamiltonian (2), the single-site Hamiltonian is rather simple: ^ 0 ðni Þ ¼ D∑ðSx Þ2 H∑Sz H ni ni ni
ni
ð4Þ
Solving the Schrödinger equation with the Hamiltonian (4), we can determine the energy levels of a magnetic ion: EðiÞ 71 ¼
D 8 κ ; EðiÞ 0 ¼D 2
ð5Þ
and the eigenvectors of the single-site Hamiltonian: Ψ ð1Þ ¼ cos θj1i sin
θj1i; Ψ ð0Þ ¼ j0i; Ψ ð1Þ ¼ sin θj1i þ cos θj1i
ð6Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D2ffi 2 H D where κ ¼ H þ 2 ; cos 2θ ¼ κ ; sin 2θ ¼ 2κ The Hubbard operators are constructed on the basis of the eigenvectors (6) [14,15]. The relationship between the spin operators
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and the Hubbard operators is given by Sz ¼ cos 2θ X 11 X 11 þ sin 2θ X 11 þ X 11 ; pffiffiffi pffiffiffi S þ ¼ 2 cos θ X 10 þ X 01 þ 2 sin θ X 10 X 01 ;
† S ¼ S þ : ð7Þ
Using Eq. (7), it can be easily shown that the single-site Hamiltonian (4) is diagonal in the representation of the Hubbard operators: ^ 0 ðni Þ ¼ ∑ EðiÞ X MM . H M ¼ 1;0;1
M
ni
The spectra of elementary excitations in the FM phase can be found by carrying out the bosonization the Hubbard operators [24]: h i ε21 ðkÞ ¼ D2 þ κ D2 þ κ 2J k jJ~k j 12Dκ ;
ε22 ðkÞ ¼ 4H2 þ D2 ;
ð8Þ
where Jk and |J~ k | are the Fourier transforms of the corresponding exchange integrals. We can find the field of the phase transition from the FM phase using the requirement of the gap vanishing in the low-frequency spectrum ε1(k): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u t 2 D2 D 3 D2 ? D2C DC D þ H FM ¼ 2DC 2DC D2 DC 4 2 8 16
ð9Þ
where DC ¼ J 0 jJ~ 0 j=2. As it will be shown below, the critical value of the anisotropy constant DC corresponds to the phase transition from the antiferromagnetic (AFM) phase into the QU phase. As follows from Eq. (7), the average value of sublattices' magnetization (per one site) is given by D E D E z
S ¼ cos 2θ X 11 X 11 ; ð10Þ and in the low-temperature limit it has the following form: z
H S cos 2θ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 H þ D2
ð11Þ
It follows from the last expression that sublattices' magnetizations reach their maximum only in the isotropic case. This result coincides with [14] and can be explained by the fact that the operator of single-ion anisotropy does not commute with the projection of the spin operator on the Z-axis (see Eq. (4)). As a result, this component of the spin operator is not the integral of motion, and, consequently, the saturation cannot be reached at finite magnetic field. This phase state is ferromagnetic, but is not saturated. 2.2. Weak magnetic field Now let us consider the case of the weak magnetic field when the energy of the external magnetic field is considerably lower than the energy of the single-ion anisotropy and the exchange interactions. In this case, the influence of strong easy-plane singleion anisotropy can lead to the realization of the QU phase. The single-site Hamiltonian is given by Eq. (4) in this case too. This means that all expressions, obtained above for the strong magnetic field, are also valid for the weak magnetic field. Thus, the magnon spectra in the QU phase are given by Eq. (8). The only difference is that D=2κ -1 (because H-0) in the QU phase. This means that the average value of the magnetization becomes zero only at zero magnetic fields, because Eq. (11) is valid at any value of the magnetic field differing from zero. Consequently, H Q?U ¼ 0 determines the transition field from the QU phase. In order to answer the question what phase state realizes in-between the nonsaturated FM phase and the QU phase, we need to investigate the system in the interval of fields:
? 0 o H oH FM . However, at first, let us consider in more details the case of zero magnetic fields. The single-site Hamiltonian has the following view at zero magnetic fields: 2 ^ 0 ðni Þ ¼ D∑ Sz þ ð1Þi þ 1 H∑Szni ð12Þ H ni ni
ni
where H ¼ ðJ 0 jJ~ 0 j=2Þ Sz is the mean field. It is seen from Eq. (12) that either the AFM phase or the QU phase can realize in the system depending on the relationship between the constants of exchange interaction and easy-plane anisotropy. Solving the Schrödinger equation with the Hamiltonian (12), one can find the energy values of a magnetic ion: E 7 1 ¼ D 8 H; E0 ¼ 0
ð13Þ
while the eigenvectors of the Hamiltonian (12) coincides with the Eq. (6), if we take into account that cos 2θ ¼ 1 at zero magnetic fields. In this case, the relationship between the spin operators and the Hubbard operators takes the simple view: pffiffiffi þ Sz ¼ X 11 X 11 ; S þ ¼ 2 X 10 þ X 01 ; S ¼ S þ : ð14Þ Implementing the procedure of Hubbard operators' bosonization [24], we can easily derive the spectra of elementary excitations valid at small values of the anisotropy constant (D « J0, |J~ 0 |) in the AFM phase:
ε1 ðkÞ ¼ HD;
ε2 ðkÞ ¼ 2H;
ð15Þ
the spectra in the QU phase are as follows:
ε1 ðkÞ ¼ DJ k þ jJ~k j=2; ε1 ðkÞ ¼ J k jJ~ k j=2;
ð16Þ
and are valid at large values of the anisotropy constant (D » J0, |J~ 0 |). From the requirement of gaps vanishing in the low-frequency magnon spectra we can find the value of the easy-plane anisotropy at which the phase transitions from the AFM phase into the QU phase and vice versa occur: from the AFM phase – spectra (15): DAFM ¼ J 0 jJ~ 0 j=2; from the QU phase – spectra (16): DQ U ¼ J 0 jJ~ 0 j=2 These values coincide, and, consequently, the phase transition the AFM phase – the QU phase is of the second order, and can occur only at zero magnetic fields when DC ¼ J 0 jJ~ 0 j=2
ð17Þ
It should be noted that Eq. (17) coincides with the expression for DC, introduced above in the expression for the critical field of the phase transition from the nonsaturated FM phase (9). Now let us proceed to the investigation of the spin state which ? realizes in the interval of fields 0 o H o H FM . Averaging of Eq. (7), obtained above for arbitrary fields, shows that sublattice's magnetization always differs from zero at the nonzero external field; however, its modulus changes with the magnetic field increasing ? from zero up to H FM . Let us investigate the free energy of the sublattices to determine the phase state of the system. The single-site Hamiltonian for the first and second sublattices has the following form: 2 ^ 0 ðni Þ ¼ D∑ Sy H þ ð1Þi H cos φi ∑Szni H sin φi ∑Sxni ð18Þ ni ni
ni
ni
while deriving Eq. (18), we supposed that, generally, sublattices' magnetizations are not collinear and make different angles φ1 and φ2 with the quantization axis (the Z-axis). Implementing the diagonalization procedure [24], one finds the energy levels of a magnetic ion for the first and the second sublattice H ffiffi sin φi ð cos αi þ sin αi Þ sin 2βi Eð1iÞ ¼ ð1Þi H2 cos φi cos 2αi 1 þ cos 2βi 2p 2
D D sin 2αi 1 þ cos 2βi þ 1 cos 2βi ; 4 4
Ph.N. Klevets et al. / Journal of Magnetism and Magnetic Materials 348 (2013) 68–74
1 D ð1Þi þ 1 H cos φi cos 2αi þ ð1 þ sin 2αi Þ 4 2 h þ ð1Þi þ 1 H cos φi cos 2αi
In connection with this, it is reasonable to consider two limit cases:
iÞ Eð0;1 ¼
H D þ ð1 þ sin 2αi Þ cos 2β i þ pffiffiffi sin φi ð cos αi þ sin αi Þ sin 2βi 2 2 7
1n ð1Þi 3H cos φi cos 2αi 4
D D þ ð13 sin 2αi Þþ ð1Þi þ 1 H cos φi cos 2αi þ ð1 þ sin 2αi Þ cos 2βi 2 2
H þ pffiffiffi sin φi ð cos αi þ sin αi Þ sin 2β i cos 2γ i 2 D 8 ð1Þi þ 1 H cos φi sin 2αi cos 2αi sin β i 2
H þ pffiffiffi sin φi ð cos αi sin αi Þ cos βi sin 2γ i ; 2 where the parameters of unitary transformations given by the following system of equations: pffiffiffi 2H sin φi ð cos αi þ sin αi Þ ; tan 2βi ¼ iþ1 D cos φi 2 ð1 þ sin 2αi Þ þ ð1Þ tan βi ¼
ð19Þ
α, β, and γ are
H ð1Þi þ 1 H cos φi cos 2αi 3 cos 2βi pffiffiffi sin φi ð cos αi þ sin αi Þ sin 2βi 2
ð20Þ
ð21Þ
the equilibrium angle of sublattice's magnetization orientation is determined from the following condition: ðiÞ
∂F i ∂E1 ¼ ¼0 ∂ φi ∂ φi
The standard procedure of determining the equilibrium state of the magnet is to minimize the total free energy of the system. However, the ground state is not well-defined in antiferromagnets, and the total energy of the antiferromagnet does not correspond to the equilibrium state. Therefore, we investigate the minimum of the free energy density of the system which, in general, has the following form: F ¼ T ln Z, where Z ¼ ∑ expðEM =TÞ is the partition function. The general view of the free energy density is the same for both sublattices. As we consider the low-temperature case, we can restrict our consideration with the account of the lowest energy level E1, then the free energy density has the following view: Dpffiffiffiffiffiffiffiffiffiffiffi2ffi 1s Hsi cos φi Fi ¼ 2
The lowest energy level is E1, and, as we consider the lowtemperature case, the free energy of the ith sublattice is simply F i ¼ Eð1iÞ
? 1. Magnetic field H tends to the critical value H FM . With increasing
external magnetic field, sublattices' magnetizations Sz will tend to unity. In this qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 case, it is convenient to choose the small value η2 ¼ 1 Sz as the order parameter. It can be shown
that sublattice's magnetization Sz ¼ cos 2α [14], then the order parameter near the FM phase is given by η2 ¼ sin 2α. 2. Magnetic field H tends to the critical value H Q?U ¼ 0. In this case,
it is convenient to choose sublattice's magnetization, s ¼ Sz ¼ cos 2α, as the order parameter, because its modulus tends to zero near the QU phase.
M ¼ 1;0;1
ð1Þi þ 1 H cos φi sin 2αi D2 cos 2αi ; Hffiffi p sin φi ð cos αi sin αi Þ 2
D D þ sin 2αi 3 cos 2βi 1 þ cos 2β i sin 2γ i 2 2 npffiffiffi ¼2 2H sin φi ð cos αi sin αi Þ cos βi þ h i o ð1Þi þ 1 2H cos φi sin 2αi D cos 2αi sin βi cos 2γ i :
71
ð22Þ
jointly resolving Eqs. (19)–(22), we find the equilibrium angles of sublattices' magnetizations orientations: 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 1 @D D 1@ D D 2A þ þ H ; cos φ2 ¼ þ H2 A cos φ1 ¼ H 4 4 H 4 4 ð23Þ obviously, φ1 ¼ φ2 which means that the intermediate state of the Ising-like antiferromagnet with strong easy-plane single-ion anisotropy in the transverse magnetic field is not a supersolid phase (as in the longitudinal magnetic field), but the spin-flop (SF) phase. Please note that the angles of magnetizations' orientations in sublattices do not depend on the exchange interaction in case of the transverse magnetic field (compare with the previous case of the longitudinal magnetic field – see [15]) which results in the equal absolute values of angles of magnetizations' orientations in sublattices. Taking the advantage of the Landau theory of the phase transitions, we can determine the fields of the phase transitions from the SF phase into the FM and the QU phase. To do this, we need to investigate the free energy density of the system F by expanding it into series of the corresponding order parameter which must tend to zero at the lines of the phase transitions (stability lines) of the FM and the QU phases, respectively.
ð24Þ
As it was shown in [14], the angle α varies from 0 to π/4 in the considered state which means the decrease of the magnetization modulus from 1 to 0. Thus, not only magnetization's direction changes with the decreasing magnetic field in the intermediate state, but also its modulus decreases due to the influence of the easy-plane single-ion anisotropy, and it becomes zero in the limit H-H Q?U . This effect is called the quantum reduction of the spin and is due to the influence of the easy-plane anisotropy. Consider the free energy density when the external magnetic field is close to the critical transition field into the QU phase. Expanding Eq. (24) into series by the magnetization s and restricting our consideration by the second order terms of s, we obtain the expression for the free energy density near the QU phase: 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 D @ D 2 DA D 2 F ¼ s þ s þH ð25Þ 2 4 4 4 The field of the phase transition into the QU phase can be easily obtained from Eq. (25): ? H SFQ U ¼0
ð26Þ
As you can see, Eq. (26) coincides with the transition field from the QU phase into the SF phase – H Q?U ¼ 0. The free energy density analysis shows that the phase transition from the SF phase into the QU phase is of the first order, and Eq. (26) determines the line at which SF phase becomes unstable during the phase transition into the QU phase. Consider now the case when magnetic field is close to the critical transition field into the FM phase. Expanding (26) into series by the small parameter η, we obtain the expression for the free energy density near the FM phase: D F¼ 4
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 2 2 D D D D η4 2 45 2 5η þ H D þH þ H2 5 4 2 8 4 4 4 4 4
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Fig. 1. Dependence of magnetizations' orientations in sublattices on the value of the external transverse magnetic field.
Fig. 2. Dependence of sublattice's magnetization on the value of external transverse magnetic field.
2 D 4 4
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 D η6 þH 2 5 4 16
ð27Þ
from which it is easy to find the field of the phase transition into the FM phase: pffiffiffiffiffiffiffiffi ? H SFFM ¼ 3=2D ð28Þ ? , then the coefficients near η 2 is negative which If H oH SFFM indicates to the first order phase transition from the SF phase into the FM phase, and the line, determined by Eq. (28), is the line where SF phase becomes unstable during the phase transition into the FM phase. Fig. 1 shows the dependence of magnetization orientation in sublattices depending on the value of the transverse magnetic field. The dependence of the sublattice's magnetization on the value of the external magnetic field is shown in Fig. 2. As you can see, the phase transition from the FM phase into the SF phase occurs with hysteresis which also indicates the first order phase transition.
3. Conclusions Carried out investigations have shown that three phase states are possible in the Ising-like antiferromagnet with strong single-ion
anisotropy at varying of the external magnetic field. In case of the transverse magnetic field, these are: the non-saturated ferromagnetic (FM) phase, the spin-flop (SF) phase, and the quadrupolar (QU) phase. The supersolid (SS) magnetic phase which realizes in case of the longitudinal magnetic field cannot realize in case of the transverse magnetic field – the SF phase realizes instead. It was shown before [14] that the same phase states realize in the Heisenberg strongly anisotropic antiferromagnet; however, the Ising-like antiferromagnet, considered in the present work, exhibits some interesting peculiarities. Let us consider the results obtained with that, obtained previously in [14,15]. In case of the longitudinal magnetic field, the phase transitions the QU phase – the SS phase is of the second order in strongly anisotropic Heisenberg antiferromagnet [14]. While, as our investigations have shown [15], this phase transition is of the first order in the Ising-like antiferromagnet. At this, most likely, the SF phase realizes in the region of the QU and the SS phases co-existence. The type of the phase transition the SS phase – the FM phase is also of the first order; however, orientation of the magnetic moment of one of the sublattices remains constant in the region of the FM and the SS phases co-existence, and abruptly orients along the external magnetic field only at the phase transition line. At this, the magnetization modulus depends on the value of the field in the SS phase – it increases with the increasing external field, and reaches saturation at the phase transition into the FM phase [15]. In case of the transverse magnetic field, the SS phase does not realize in the Ising-like antiferromagnet, despite the case of Heisenberg antiferromagnet [14,25]. The SF phase realizes instead of the SS phase. Such behavior can be explained by the influence of the easyaxis exchange anisotropy in the Ising-like antiferromagnet. Both phase transitions, the SF phase – the FM phase and the SF phase – the QU phase, are of the first order. At this, as well as in the case of the longitudinal orientation of the magnetic field, the magnetization modulus depends on the magnitude of the external magnetic field in the SF and the FM phases; however, magnetization reaches saturation only at infinite magnetic field, because the operator of single-ion anisotropy does not commute with the Z-component of spin operator, and, consequently, this component of spin moment is not the integral of motion. Also, the phase transition from the FM phase into the SF phase occurs with hysteresis. To summarize the results obtained for the longitudinal and transverse orientations of the external magnetic field, let us compare the influence of the effective fields of the exchange interactions on the sublattices for different orientations of the external magnetic field. In case of the longitudinal magnetic field, the effective fields of the exchange interactions are collinear to the external magnetic field. This result in the competition between the external magnetic field and the effective exchange fields [15] – the effective fields of the exchange interactions amplify the influence of the external magnetic field in one of the sublattices, and weak it in another sublattice. This leads to different absolute values of angles of magnetizations' orientations in sublattices, and, consequently, to the realization of the SS phase. In case of the transverse external magnetic field, the effective fields of the exchange interactions act perpendicularly to the external magnetic field. At this, the absolute values of angles of magnetizations' orientations coincide for different sublattices, as they do not depend on the exchange interactions anymore (see Eq. (25)). Consequently, this leads to the realization of the SF phase. Thus, the realization of the SS phase essentially depends on the orientation of the external magnetic field, and the SS phase cannot realize in case of the transverse external magnetic field. Besides, magnons' dispersion law essentially depends on the magnetic field orientation. In case of the longitudinal magnetic field, magnon spectra are dispersionless [15], while in the transverse
Ph.N. Klevets et al. / Journal of Magnetism and Magnetic Materials 348 (2013) 68–74
magnetic field the spectra (8) and (16) depend on the wave vector k. And if such form of the magnon spectrum is common for highfrequency magnons, it requires some explanations for the lowfrequency magnons. In case of the longitudinal magnetic field, the independence of the low-frequency magnon spectra on the wave vector k in the FM and the QU phases is related with the absence of the transverse component of the spin operators (Sx and Sy) in the Hamiltonian of the system. Probably, the magnon spectra will depend on the wave vector in the intermediate SS phase, but their determination in this phase is beyond the scope of the present investigations. In case of the transverse magnetic field, the Hamiltonian of the Ising-like antiferromagnet always contains non-zero transverse components of spin operator which is exhibited in the dependence of Eqs. (8) and (16) on k. As it was mentioned in the Introduction, there are real magnetic systems in which the supersolid magnetic phase can be found experimentally. These are, for example, NiCl24SC(NH2)2 and Ba2CoGe2O7 [13,16–19]. The results, obtained in the present investigations, show that in case of the system with the Ising-like exchange interactions and strong easy-plane single ion anisotropy, the supersolid magnetic phase cannot be observed in the transverse magnetic field. The case of the longitudinal magnetic field perpendicular to the basal plane is the most favorable for experimental observation of the supersolid magnetic phase. This must be taken into account by experimentalists while trying to find experimental evidence of the supersolid magnetic phase existence.
Acknowledgments
non-diagonal Hubbard operators. These solutions allow obtaining the relationship between the material parameters of the system and the parameters of rotation. After the unitary transformations, mentioned above, the singlesite Hamiltonian takes the diagonal form: H 0 -H~ 0 ¼ UH 0 U þ ¼ ∑ En X nn ; n
where En are the energy levels of a magnetic ion. The next step is to transcribe the Hubbard operators through the pseudo-Hubbard operators which is case of S ¼1 have the following relationship with Bose operators of creation and annihilation: 11 00 þ X~ n ¼ 1anþ an bn bn ; X~ n ¼ anþ an ; 11 10 þ þ ¼ bn bn ; X~ n ¼ 1anþ an bn bn an X~ n 01 X~ n ¼ anþ 11 11 01 10 þ þ þ X~ n ¼ 1anþ an bn bn bn ; X~ n ¼ bn ; X~ n ¼ anþ bn ; X~ n ¼ bn an
Here a are the Bose-operators corresponding to the transition of a magnetic ion from the state E1 into the state E0 and vice versa, and b correspond to the transition from the state E1 into the state E 1 and vice versa. α The Lie algebra of the pseudo-Hubbard operators X~ n is the α same as the Lie algebra of the Hubbard operators X n . The direct calculations prove the following equality: h ij pq i iq pj ¼ δjp X~ n δqi X~ n : X~ n ; X~ n
The authors are grateful to Prof. Ivanov B.A. for fruitful discussion. The work was supported by the State Fund for Fundamental Research, Grant F33.2/002.
Appendix We have taken the advantage of the method of the Hubbard operator’s bosonization while obtaining the magnon' spectra. The application of this approach allows obtaining the exact expressions for the spectra of elementary excitations for a rather complex system, like, for example, that, investigated in the present work. Unfortunately, this mathematical technique is explained in details only in Russian; therefore, we have decided briefly explain its key features here. The procedure of spectra derivation is fulfilled in two steps. At first, the single-site Hamiltonian is diagonalized and re-written through the Hubbard operators describing the transition of a magnetic ion from the jM i state into the jM′i state: X M′M n
Ψ n ðM′Þ Ψ n ðM Þ [21]. Next, the Hubbard operators X α are trann α scribed through the pseudo-Hubbard operators X~ . n
Generally, the single-site Hamiltonian is not diagonal in the basis of the eigenvectors of the Sz operator. The generalized u-v transformation is required to diagonalize it [26]. At this, the singlesite Hamiltonian is rotated in the spin space according to the following rule: nn nm mm n mn U þ sin nm φ ¼ exp nmφX φ X mn¼ 1 þ cos φ 1 X þ X φ exp iμ X exp iμ X ;where φ is the parameter of rotation. At this, the law of transformation of the Hubbard operators during the unitary rotation is given by the following expression: þ X pq -U nm φ X pq U nm φ X pq φ : After these rotations, one obtains the system of equations on the parameters φi which solutions must be restricted with that satisfying the condition of vanishing of the coefficients at
73
α
However, X~ n operators are not identical to the Hubbard operators X αn . These operators cannot be equated in principle, because the Hubbard and the pseudo-Hubbard operators act in spaces of different dimensionality: the dimensionality of the physical space of X αn operators equals three (at S ¼1); while the Hilbert space of a and b operators is infinitely dimensional. Besides, there are other differences, in particular: 10 þ 11 þ pq þ 01 11 X ¼ X qp ; X~ a X~ ; X~ a X~ n
n
11 X 01 ¼ X 01 n Xn n ;
n
n
01 11 X~ n X~ n
n
n
01 þ ¼ anþ anþ bn bn a X~ n
Generally, the Hubbard operators, and, consequently, the single-site Hamiltonian, cannot be transcribed through any combinations of the Bose operators. At the same time, one can construct the Bose analog of the Hamiltonians (1) and (2), i.e., the operator acting in the infinitely dimensional Hilbert space; at this, certain part of its coefficient functions equals the coefficient functions of the Hamiltonians (1) and (2). Transcribing the Hamiltonians (1) and (2) through the Bose operators and restricting ourselves with the terms quadratic in the operators of creation and annihilation, we obtain the following general expression: þ
H ð2Þ ¼ ∑ ðE0 E1 Þakþ ak þ ∑ ðE1 E1 Þbk bk k
k
n 1 þ þ ∑ Ak akþ ak þ ∑ A~ k akþ ak þ A~ k ak ak 2 k k n 1 ~ þ þ þ þ ∑ Bk bk bk þ ∑ B k bk bk þ B~ k bk bk : 2k k
Diagonalizing this Hamiltonian with standard Bogolubov's u–v transformation, one obtains: þ
H ð2Þ ¼ ε0 þ ∑ εα ðkÞαkþ αk þ ∑ εβ ðkÞβ k βk ; k
where the coefficients at of quasi-particles.
k
α and β operators determine the spectra
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