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Noise-driven nonlinear sigma model
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Physica A 370 (2006) 601–612 www.elsevier.com/locate/physa
Noise-driven nonlinear sigma model M. Malard Sales, A.S.T. Pires, Ronald Dickman, M.C. Nemes Departamento de Fı´sica, Universidade Federal de Minas Gerais, 30123-970 Belo Horizonte, Brazil Received 20 January 2005; received in revised form 22 September 2005 Available online 17 April 2006
Abstract We present a noise-driven model for obtaining the gap and line-width as functions of the temperature in the nonlinear sigma model. The method is phenomenological and rests on the following physical idea: a classical external stochastic field is introduced representing the coupling of the sigma field with a noise source. Moreover, we assume that the inelastic scattering length is much longer than the elastic one, justifying the neglect of dissipation for temperatures such that the nonlinear sigma model is a good approximation for antiferromagnetic spin chains. This phenomenological approach is justified by comparison with other theoretical predictions and with experiment. r 2006 Elsevier B.V. All rights reserved. Keywords: Nonlinear sigma model; Integer-spin chains; White noise; Temperature-dependent gap; Temperature-dependent line width
1. Introduction The nonlinear sigma model (NlSM) describes physical systems of many types. In particle physics, the twodimensional model is a prototype for nonabelian gauge theories in four dimensions [1]. In condensed matter physics, the NlSM is an approximation, through the Haldane mapping, to the continuum limit of the antiferromagnetic Heisenberg model. The quantum Hall effect can be analyzed via the NlSM as well [2]. Finally, the model is also useful in understanding superconductivity [3]. Due to its broad range of application, the NlSM has been a subject of intensive study for many years. There are many theoretical works concerning the temperature dependence of the gap in the NlSM. Comparison with experimental data for S ¼ 1 one-dimensional Heisenberg antiferromagnets shows that the NlSM describes such systems well at low temperatures [4]. In particular, the generation of a finite gap at zero temperature explains, in light of Goldstone’s theorem [5], the disordered ground state of these antiferromagnets. This property was not understood until 1983, when Haldane proposed his conjecture [6,7]. Based on topological arguments, Haldane proved that, at zero temperature, integer-spin antiferromagnetic Heisenberg chains have a finite gap, and that half-integer chains are gapless. The traditional way of introducing temperature in the NlSM is through the well known Matsubara technique [8]. It predicts discrete frequencies, in contrast to the model at zero temperature. Basically, what is done is an association between the Wick-rotated time variable in the path integral and the temperature of the Corresponding author.
E-mail address: mmalard@fisica.ufmg.br (M. Malard Sales). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.03.012
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statistical mechanical partition function. This method is the theoretical sound and well tested Boltzmann prescription for the inclusion of the temperature in the problem. In the present paper, we propose a different approach for including the influence of temperature on the model. What we suggest is to couple the system to an external white noise, resembling the Langevin approach for studying the effect of fluctuations in macroscopic systems. Fluctuations are introduced by including a stochastic external field or noise in the equations of motion. We call this phenomenological approach ‘‘noisedriven’’. The general scenario of the NlSM in contact with an external field is presented in Section 2. In Section 3, we determine the gap as a function of temperature, and find good agreement with experimental data. We believe that the calculations involved are simpler than those involved in the Boltzmann prescription. The latter requires some extra algebraic effort involving series summations, modified Bessel functions and various approximations to reach a final expression for the gap as a function of the temperature. Our results are obtained using the saddle-point approximation. This leads to a complex expression, the real part representing the gap as function of temperature. The imaginary part is interpreted as representing lifetime effects due to the coupling of the sigma field with an external field. A more detailed discussion of this subject is presented in Section 4 where the calculation of the line width is performed. Again, good agreement is achieved with experimental data. The curves for the gap and the line width versus temperature are explicit evaluated for the case S ¼ 1 in order to fit them with the available experimental data. Nevertheless, our method applies for any integer-spin antiferromagnetic chain other than the more frequently studied spin-1 chain. Finally, in Section 5 we present our conclusions concerning the validity of the noise-driven method for the inclusion of temperature in the NlSM. It also includes a discussion on the difference between the physical pictures addressed by the usual Boltzmann prescription and the noise-driven model. 2. Nonlinear sigma model in the presence of an external field ~ is given by The Lagrangian density of the NlSM in the presence of an external field AðxÞ ~ ~ ~ ¼ 1 ðqm ji Þðqm ji Þ þ lðxÞð~ L½qm ~ j2 1Þ þ AðxÞ j, j; ~ j; l; A 2g
(1)
~ðxÞ, lðxÞ is the Lagrange multiplier scalar field needed to where ji is the ith component of the vector field j impose the constraint j2 ¼ 1 on the modulus of the field ~ jðxÞ and g is an arbitrary constant. As usual, Greek indices run over the space–time vector x ¼ ðt; ~ xÞ where ~ x is M-dimensional. The Latin indices stand for the N ~ components of the vectorial fields ~ jðxÞ and AðxÞ. ~ The goal of this work is to consider the external field AðxÞ as a stochastic variable, that is, as an external noise acting on the system in a way to reproduce the effects of temperature fluctuations in the model. Posed in these terms, the argument rests on the same reasoning as the Langevin approach of finding the effect of fluctuations in macroscopic systems. In the following paragraphs we attempt to provide a justification of the validity of our Langevin-like method for introducing temperature in the NlSM. In the Boltzmann prescription the introduction of temperature is done through the equivalence between the statistical partition function and the path integral functional, which arises from the introduction of the Wickrotated imaginary time ðt ¼ itÞ. The time parameter t is then associated with temperature through the equation t ¼ 1=ðkB TÞ. At zero temperature, the integration variable t runs from zero to infinity and a Fourier transform introduces a continuous and integrated frequency o. Instead, for finite temperature, a Fourier series is needed in place of the Fourier transform leading to the discrete Matsubara frequencies. These, having a linear dependence on T, generate the expected temperature dependence of any physical quantity. Our reasoning is to follow the standard calculation procedure for zero temperature and hope that the introduction of a temperature dependent noise can do the job of reproducing the thermal behavior of the model. At low enough temperatures, it is reasonable to think of thermal fluctuations as a perturbation on the model at zero temperature, thus making our prior assumption very plausible. Furthermore, our method is supported by the stochastic description of fluctuations on macroscopic systems by the master equation [9]. The master equation, when not solved exactly, is treated in a systematic power series expansion in a parameter O defined in terms of a size parameter of the system. The systematic O-expansion is the basis for the existence of
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a macroscopic deterministic description of systems that are intrinsically stochastic. It justifies as a first approximation the standard treatment in terms of a deterministic equation with noise added, as in the Langevin approach. Moreover, in the lowest approximation, it happens that the noise is Gaussian, definitely justifying its use to treat thermal fluctuations. Finally, the good agreement found between experimental data and our results for the mass and line width as functions of temperature supports the validity of the method. ~ is therefore treated as a white noise with the following statistical properties [9]: The stochastic term AðxÞ ~ hAðxÞi ¼ 0,
(2)
hAi ðxÞAj ðx0 Þi ¼ 2GkB Tdðx x0 Þdij ,
(3)
where kB is Boltzmann’s constant and T is temperature. The factor 2GkB T is a measure of the noise intensity so that, for low temperatures, it is reasonable to think of it as a perturbation on the system at zero temperature. In the usual Langevin approach, the fluctuation–dissipation theorem [9] implies that G is the damping constant. The physical picture is that random kicks, caused by thermal fluctuations, drive the system away from the state of minimum energy while damping tends to restore it to this state. At this point a fundamental difference between our approach and the usual Langevin description becomes evident: we have not included a dissipation term in the equation of motion for ~ j. Indeed, to do so would require introducing a velocity-dependent potential in the Lagrangian density, Eq. (1). This would turn the problem into a different and more complicated one. The very purpose of our approach is to investigate whether the simple Lagrangian density, Eq. (1), suffices to represent thermal effects in the NlSM. Thus, here the constant G in Eq. (3) is a mere parameter, not connected to a dissipation force as required by the fluctuation–dissipation theorem. From elementary considerations, in particular the fact that the motion of an undamped harmonic oscillator subjected to white noise is unbounded, we expect the mean energy of the system defined by Eqs. (1)–(3) to grow without limit with time, rather than attain thermal equilibrium. The saddle-point approximation used here to analyze the model seems to be blind to this fact, and, for low temperatures, we recover physically reasonable results even in the absence of damping. A suitable choice of parameter G leads to results in good accord with experiment. The absence of damping can be understood with the physical supposition that the disorder in the system is caused mainly by elastic collisions with impurities. Therefore, we assume that the inelastic scattering length is much larger than the elastic one. In this context, it makes sense to think of a purely diffusive perturbation for low enough temperatures such that inelastic effects which could led to dissipation can be neglected. It is worth mentioning here that it will suffice for our objectives to consider only the stochastic properties (2) ~ and (3) without worrying about the specific probability distribution of AðxÞ. Actually, we do not make explicit use of the fact that thermal fluctuations, being independent stochastic variables, have Gaussian probability distribution as assured by the central limit theorem [9]. The fact that the distribution is Gaussian is needed solely in the legitimation of the method of introducing temperature via an external stochastic field, as argued before in the context of the master equation. In any case, the method as it stands is completely general with ~ respect to the probability distribution of the field AðxÞ. Therefore, it applies for any stochastic external field having finite variance. ~ fields, following the same arguments of Eq. (1) assumes that the coupling is linear in both the ~ j and A Lee and Boyanovsky in their analysis of the dynamics of phase transitions induced by a heat bath [10]. The linear coupling restriction may be relaxed at the expense of additional complications. Moreover, the linear coupling applied in the context of the noise-driven NlSM is further justified once its results agree with experience. In order to proceed, it is necessary to impose the following asymptotic conditions on the field ~ j and its derivatives: jxj!1
~ jðxÞ ! 0, jxj!1
qm ~ jðxÞ ! 0
(4) 8m ¼ 0; 1; 2; . . . ; M.
(5)
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The behavior of the model can be obtained from the Lagrangian density, Eq. (1), through the path integral functional defined as Z Z i ~ ~ ¼ j; l; A, (6) Z½A D~ jDl exp S½~ _ where ~ ¼ S½~ j; l; A
Z
~ dx L½qm ~ j; ~ j; l; A
is the action of the model. We now define the field: Z ~ 0Þ ~ ~ jðxÞ ¼~ jðxÞ dx0 Gðx; x0 ÞAðx
(7)
(8)
where Gðx; x0 Þ ¼ gðq2 þ 2glðxÞÞ1 dðx x0 Þ
(9)
~ ¼ 0 and q2 stands for qm qm ¼ q2 r ~2 , the covariant Laplacian. The is the propagator for the system with A t 1 inverse of the propagator is the function G ðx; x0 Þ, which satisfies Z dx00 Gðx; x00 ÞG 1 ðx00 ; x0 Þ ¼ dðx x0 Þ. (10) Then, the inverse of the propagator can be written as G1 ðx; x0 Þ ¼
1 ðq2 þ 2glðxÞÞdðx x0 Þ. g
(11)
Using the asymptotic conditions (4) and (5) in performing integrations by parts and observing that ~ ~ ~~ j ¼ 1 is the Jacobian of the transformation (8), the path integral ~ ¼ ½qj=q~ ~ jD~ Dj j ¼ D~ j, where ½qj=q~ functional may be written as Z Z Z lðxÞ i 1 1 0~ 0 ~ 0 0 ~ þ dx AðxÞ:Gðx; x ÞAðx Þ þ Tr½lnðG ðx; x ÞÞ , Z½A ¼ Dl exp N dx (12) r 2r 2 where, for future convenience, we defined the parameter r ¼ _N. To reach Eq. (12) we have Wick rotated to ~ Accordingly, the covariant imaginary time (t ¼ it) in order to obtain a Gaussian exponential in the field ~ j. ~2 . The quadratic vector field ~ Laplacian q2 is replaced by the Euclidean one q2 ¼ q2t þ r j~ was integrated out through a limiting process going from the continuous space–time to a hyper-cubic lattice of m sites, with lattice spacing e, and letting m ! 1, e ! 0 with me ¼ constant, at the end of the calculation. The convergence of this limiting process is ensured by the introduction of a measure constant M m that eliminates the infinities of the procedure [1]. Since G1 ðx; x0 Þ is related to a Hermitian operator through Eq. (11), its matrix representation in ~ ~ becomes a product of Nm uncoupled Gaussian the discrete limit G 1 x;x0 is diagonalizable. The integral over j ~ integrals generating the above result for the functional Z½A. We are left, in Eq. (12), with an integral in l that cannot be solved analytically as it is not quadratic in that field. What is done is to expand the exponent around its saddle point retaining terms only up to second order. This is justified when one takes small deviations from the saddle-point solution which, in turn, is required for convergence when the limit N ! 1 is taken. In addition, in that limit we take r to remain constant when we simultaneously set _ ! 0. In other words, N, the overall multiplicative factor in the exponent, will work as an expansion parameter and the limit N ! 1 will lead to a Gaussian integral in the field l. This will lead us to the calculation of the gap.
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3. The gap Let us define the effective action functional: Z Z lðxÞ i ~ 0 Þ þ 1 Tr½lnðG 1 ðx; x0 ÞÞ. ~ þ dx0 AðxÞ:Gðx; f ½l ¼ dx x0 ÞAðx r 2r 2 One may expand f ½l around a solution lðxÞ ¼ lc such that df ½l ¼ 0. dlðxÞl¼lc
(13)
(14)
It will suffice for our purpose of obtaining the gap to explore the saddle-point condition, Eq. (14), without worrying about the full expansion of f ½l itself. From Eq. (9) we obtain dGðx; x0 Þ ¼ 2Gðx; x00 ÞGðx; x0 Þ, dlðx00 Þ
(15)
which must be substituted into the derivative of Eq. (13) with respect to l. Further, in order to obtain physically meaningful results we must average Eq. (14) such that the stochastic ~ is taken into account. The averaged saddle-point condition together with the properties (2) character of AðxÞ ~ lead to and (3) of the field AðxÞ Z 1 ð2GkB TÞ 00 ¼ G lc ð0Þ 1 i dxG lc ðx x Þ , (16) r r where the Euclidean version of Eq. (9) for Gðx; x0 Þ evaluated at the saddle point leads to Z 0 dp eiðp=_Þðxx Þ 2 0 0 G lc ðx; x Þ ¼ G lc ðx x Þ ¼ gðc_Þ , ð2p_Þd c2 p2 þ D2
(17)
where d stands for the dimension of the space–time and p ¼ ðp0 ; ~ pÞ is the four-momentum vector in the reciprocal space. Also, D2 ¼ 2glc ðc_Þ2 stands for the gap in the proper units of energy where c is the spinwave velocity. Substitution of Eq. (17) into Eq. (16) leads to 2 2 4p c L þ D2 gc2 _ð2GkB TÞ ¼ ln 1 i , (18) Ng_ D2 ND2 where the constant L jpjmax is a cut-off parameter introduced for convergence of the double integrals ðd ¼ 2Þ in the momentum–frequency plane. The cut-off is quite natural when applying the model in solid state physics, since in this case there exists a finite separation a between the lattice sites, so that the momentum is restricted to the first Brillouin zone of the crystal. The frequency will also be limited since it is related to the momentum via the dispersion relation. In the limit L ! 1, Eq. (18) gives the following result for DðT ¼ 0Þ D0 : 2p (19) D0 ¼ cL exp Ng_ reproducing a well known result for the NlSM in the limit L ! 1 [11]. As is the case for the NlSM, explicit assignment of values to the constants involved in Eq. (19) cannot provide a quantitative prediction for D0 in accordance with the experimental value or with numerical simulations for one-dimensional antiferromagnets. The latter predicts D0 ¼ 0:41J for a spin-one chain, in units of the antiferromagnetic (AFM) intrachain exchange coupling J [12]. In particular, the constant L in Eq. (19) carries the effect of passing from a manifestly continuous theory to a discrete one and cannot be assigned an accurate value by direct estimation. Nevertheless, as the NlSM mapping for _ ¼ 1 predicts g ¼ 2=S [8], the model is able to reproduce the expected qualitative behavior [12], D0 / epS . Actually, up to the present moment, neither the NlSM nor any other analytical approach is able to predict quantitatively D0 as a function
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of microscopic parameters. What is usually obtained is an expression for finite temperature involving ratios such as DðTÞ=D0 . In this spirit, let us define what we will call a reduced temperature and a reduced gap, respectively, as t¼
T , D0
dðtÞ ¼
(20)
DðtÞ , D0
(21)
where, in units such that kB ¼ 1, D0 can be given in kelvins making the reduced temperature t dimensionless. The reduced gap is, by definition, also dimensionless. In terms of these new variables and applying the limit L ! 1, Eq. (18) can be rewritten to first order in T as t 2 d ðtÞ ¼ exp ig 2 , (22) d ðtÞ where 2g_c2 G c2 L 2 ln g¼ ND0 D20
! (23)
must be a constant to be adjusted in order that Eq. (22) fits the experimental data. In principle, from the obtained value of g we can obtain the noise parameter G if we are able to deal with all the constants involved in Eq. (23). Fortunately, we can skip this quite troublesome task for, in Section 4, a further fit with experiment will render G an adjustment parameter. In this way, it will be possible to assign it a reasonable value without explicit considerations concerning the constants of the model. Inspection of Eq. (22) shows that dðtÞ must be a complex quantity. Its real part gives the mass or gap, mðtÞ, of the related quasi-particles—magnons for application of the model to AFM spin chains. The imaginary part is related to the magnon finite lifetime generated by the coupling between the model and noise. This will be discussed in detail in the next section. Returning to Eq. (22), we see that it is transcendental in d2 ðtÞ and, therefore, cannot be solved analytically for mðtÞ. Nevertheless, we can plot the corresponding curves for a judicious choice of the free parameter g. Fig. 1 shows the reduced mass m as a function of reduced temperature t, for S ¼ 1, obtained from the real part of Eq. (22). The reduced mass remains almost constant, equal to unity, until around t ¼ 0:1, and then begins to increase approximately linearly with reduced temperature. This is indeed what is expected for the NlSM for S ¼ 1 [13]. The experimental data reported by Sakaguchi were obtained via inelastic neutron scattering on Y2 BaNiO5 , considered the best candidate to realize a spin-1 AFM chain [4]. The best fit between theory and experiment is achieved when the free parameter g equals 1:36. The asymptotic ðt ! 0Þ expression for the NlSM reduced mass in the Boltzmann prescription is [4] mðtÞ 1 þ ð2ptÞ1=2 expð1=tÞ. The dashed line in Fig. 1 shows the Boltzmann prediction for mðtÞ. This curve begins to increase faster than the experimental results around t ¼ 0:4, in contrast with the noise-driven model. 4. The line width At this point there arises a fundamental difference between the noise-driven method and the Boltzmann prescription for the introduction of temperature in the model. As it was shown in the previous section, Eq. (22) has an imaginary part. The saddle-point method applied to the Boltzmann prescription does not lead to a corresponding imaginary term in the gap for, in that approximation, the system behaves like an ensemble of free magnons. Neglect of interactions leads to an infinite magnon lifetime. In contrast, the noise-driven method provides a finite lifetime even in the saddle-point approximation. The NlSM treated via Boltzmann prescription includes information on finite lifetime effects in the lðxÞj2 term of the Lagrangian density, Eq. (1), which endeavor the interactions between magnons. The saddle-point applied in this context is blind to these effects because in this approximation lðxÞ is made constant through the
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Boltzmann prescription
1.6
Noise-driven method Experimental data
Reduced Mass
1.4
1.2
1
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Reduced Temperature Fig. 1. Reduced mass versus reduced temperature for a S ¼ 1 AFM Heisenberg chain. The curve shows near coincidence with the experimental data for Y2 BaNiO5 [4] when g ¼ 1:36. The dashed line is the Boltzmann prediction for the NlSM reduced mass [4].
assumption that there is a solution lc that satisfies Eq. (14). Now, temperature enters the noise-driven method ~ through the external stochastic field AðxÞ introducing another decaying channel in the system in addition to the magnon–magnon interaction. In this context, the saddle-point approximation is blind to lifetime effects coming from interactions between magnons but cannot dismiss the effects of the explicitly introduced thermal bath. In fact, one should expect that magnon decay is caused not only by the interactions between the magnons themselves, but also by the thermal fluctuations that a system at finite temperature is subjected to. In order to consistently analyze the problem of magnon decay in the noise-driven NlSM, the following section is devoted to the calculation of the line width, the inverse of the magnon lifetime. The use of the total propagator in this task amounts to count at once the joint contributions of both thermal fluctuations and magnon–magnon interactions to lifetime effects. ~ The standard method for obtaining the propagator is via the introduction of a fictitious field, say JðxÞ, ~ ~ ~ ~ which couples linearly with the field ~ jðxÞ, just as AðxÞ does. Then, Z½A becomes Z½A; J and we obtain the propagator via the formula ~ J ~ i_ d2 Z½A; 0 G N ðx; x Þ ¼ (24) , ~ J ~ ~ 0 Þ ~ dJðxÞd Jðx Z½A; ~ J¼0
where we set the fictitious field J~ ¼ 0 at the end of the calculation. As the coupling is assumed to be linear, ~ is a trivial task. The only change in Eq. (12) for the functional introducing the field J~ in the expression for Z½A 0 ~ 0 ~ ~ þ JðxÞÞ:Gðx; ~ 0 Þ þ Jðx ~ ~ 0 ÞÞ. Together with the noise will be that, instead of AðxÞ:Gðx; x ÞAðx Þ, one has ðAðxÞ x0 ÞðAðx
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property (3), the calculation leads to GN ¼ G i
A 2 G , r
(25)
where for simplicity we set A ¼ 2GkB T
(26)
and Eq. (25) is written in matrix form. Making the correspondence between the propagator (multiplied by i) and the correlation between the values of the field ~ j at two different space–time points we see from Eq. (25) that thermal effects reduce the correlation. Since the fluctuations tend to disorder the system, it is reasonable that correlations are reduced under thermal fluctuations. Making use of Eq. (25), we define the saddle-point noise-driven propagator G 0N obtained from G 0 , the saddle-point T ¼ 0 propagator of the NlSM as G 0N ¼ G 0 i
A 2 G . r 0
(27)
It is now evident that the occurrence of a ‘‘thermal line width’’ even in the saddle-point approximation comes naturally from the A dependent imaginary part in the propagator, Eq. (27), as also required by the complexity of dðtÞ in Eq. (22). The explicit outcome of the ‘‘thermal lifetime’’ in our noise-driven method is an inevitable consequence of considering the thermal bath as a noise field coupled classically to the system, i.e., a source of decaying. The self-energy Sðx; x0 Þ is defined through the Dyson equation for the propagator which reads G ¼ G0 þ G 0 SG,
(28)
where G 0 is the highest order known propagator for the model in question. For the definition of the noise-driven NlSM self-energy SN one may write GN ¼ G 0N þ G0N SN GN ,
(29)
which iteratively expanded in terms of G 0N rewritten via Eq. (27) leads to the following expansion in powers of G 0 : GN ¼ G0 i
A 2 A A G þ G 0 SN G 0 i G0 SN G 20 i G20 SN G0 þ G0 SN G0 SN G0 þ OðG 40 Þ. r 0 r r
(30)
Now, using Eq. (25) for GN and then expanding G in powers of G0 following Eq. (28) we find GN ¼ G0 i
A 2 A A G 0 þ G 0 SG 0 i G 0 SG20 i G20 SG 0 þ G 0 SG 0 SG 0 þ OðG 40 Þ. r r r
(31)
Comparing Eqs. (30) and (31) we have SN ¼ S,
(32)
i.e., for G N calculated up to third order in G 0 , the noise-driven self-energy is the same as in the model with ~ ¼ 0. From Eq. (31), it is clear that G N ¼ G for T ¼ 0, and up to order G3 , temperature affects the A 0 propagator through the A linearly dependent terms. The Dyson equation (29) can be written for the noise-driven propagator in momentum space as GN ðpÞ ¼
1 , SðpÞ
G 1 0N ðpÞ
(33)
where p ¼ ðo; kÞ is the two-dimensional frequency–momentum vector and SN ðpÞ was already set equal to SðpÞ, the known self-energy of the NlSM given at order 1=N by Z 2 0 d p ½G 0 ðp þ p0 Þ G 0 ðp0 Þ , (34) SðpÞ ¼ Pðp0 Þ ð2pÞ2
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where we have absorbed the term Sð0Þ into the expression for the self-energy in order to assure its convergence. The propagator Pðp0 Þ is associated with the Lagrange multiplier field lðxÞ and for c ¼ _ ¼ 1 is given by Z Z d2 p d2 p g2 0 0 Pðp Þ ¼ . (35) G ðp þ p ÞG ðpÞ ¼ 0 0 ð2pÞ2 ð2pÞ2 ½ðp þ p0 Þ2 þ D20 ½p2 þ D20 Lifetime effects occur through the imaginary part of the denominator in Eq. (33). With G 0N given by Eq. (27), the total line width can be given after some algebra by " #1 " # Z N A g2 1 1 1 lwðpÞ ¼ d 2 p0 A N ðp2 þ D20 Þ2 2g ðp þ p0 Þ2 þ D20 p0 2 þ D20 " #1 ! 1=2 2 0 2 1=2 ð1 þ 4D =p Þ 1 2 2 2 0 p0 1 þ 4D20 =p0 ln , ð36Þ þ p2 ð1 þ 4D20 =p0 2 Þ1=2 þ 1 where we used the Feynman relation Z 1 1 1 ¼ dx . ab ½ða bÞx þ b2 0
(37)
In the limit of validity of Eq. (32), i.e., for G N up to third order in G 0 , Eq. (36) shows that the total line width separates in a thermal contribution (coming from G1 0N ) plus a magnon interaction contribution (coming from S). If higher order terms were considered in the expansion equations (30) and (31) the noise-driven selfenergy SN would be a nontrivial function of S and A so that the total line width would contain mixed contribution from thermal and magnon interaction effects. From Eqs. (20) and (26) we will rewrite A as A ¼ 2GD0 t
(38)
under the condition kB ¼ 1 for a dimensionless reduced temperature t. The constant G will be our free parameter valued as to fit the experimental data. Sakaguchi et al. performed constant k ¼ p=2 scans corresponding to the scattering vector k ¼ 0 in the ð1 þ 1Þ NlSM. Since o2 ¼ k2 þ M 2 , we evaluate Eq. (36) at p ¼ ðM; 0Þ ¼ ðD0 mðtÞ; 0Þ, obtaining " #1 3 1 8G t lwðtÞ ¼ 2GD0 t 3D30 ðm2 ðtÞ þ 1Þ2 " # Z 1 þ1 1 1 dk do 2 4 1 k þ o2 þ 2D0 mðtÞo þ D20 ðm2 ðtÞ þ 1Þ k2 þ o2 þ D20 #1 ! 1=2 " 2 2 2 1=2 4D20 1 2 2 2 ð1 þ 4D0 =ðk þ o ÞÞ 2 Ln , ð39Þ ðk þ o Þ 1 þ 2 þp k þ o2 ð1 þ 4D20 =ðk2 þ o2 ÞÞ1=2 þ 1 where we set N ¼ 3 and g ¼ 2 since for _ ¼ 1 the NlSM mapping predicts g ¼ 2=S. For D0 we use Sakaguchi’s experimental value D0 ¼ 9:5 meV for the compound Y2 BaNiO5 . After numerically integrating Eq. (39), we plot the line width in meV versus reduced temperature, as shown in Fig. 2. The best fit with experimental data was achieved for G ¼ 3:84 102 . Remember that G was defined in Eq. (3) as a parameter related to the noise intensity. It is known that the NlSM represents Heisenberg AFM chains only in the low-temperature regime where the effects of thermal fluctuations are expected to be weak. The above small value of G is in accord with this expectation. Reasonable agreement with experimental data is achieved in the low-temperature regime, up to around t ¼ 0:6. Above this temperature, the experimental values lie above the theoretical curve. A more accurate theoretical prediction requires evaluation of an extended perturbation series for the self-energy. In any case, the one-loop correction presented here is sufficient to show that the noise-driven model permits calculation of the NlSM line width as well as the mass.
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5
Noise-driven Line width Experimental data
Line width (meV)
4
3
2
1
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Reduced Temperature Fig. 2. Line width versus reduced temperature for a S ¼ 1 AFM Heisenberg chain for G ¼ 3:84 102 . The experimental data are for Y2 BaNiO5 [4].
One could expect that the thermal contribution to Eq. (39) equals the imaginary part of Eq. (22). However, comparison between the two curves shows that they do not agree perfectly. The reason is that they were obtained by quite different methods. The imaginary part of Eq. (22) is the result of an integration over the first Brillouin zone of all the modes p ¼ ðo; kÞ allowed for the system. Differently, the thermal contribution to Eq. (39) is calculated for p ¼ ðM; 0Þ ¼ ðD0 mðtÞ; 0Þ where Sakaguchi’s experiment was performed. In any case, they share the same qualitative behavior: both vanish at t ¼ 0, increase with temperature and are of the same order in magnitude. The outcome of Eq. (39), the total line width of the model, is plotted together with the imaginary part of Eq. (22) in Fig. 3. The thermal contribution is zero at t ¼ 0, as expected. In the vicinity of zero temperature (up to t ¼ 0:2) the difference between the curves is about 87%. As the temperature increases, the thermal effects become more significant but, up to the temperature for which experimental data are available (t ¼ 0:8), the maximum thermal contribution does not exceed 18% of the total result. We see that, at least inside the range of temperature where the NlSM is expected to reasonably approximate AFM integer spin chains, interactions between magnons are more significant to lifetime effects than their interaction with the thermal bath. The thermal line width may be interpreted as being due to loss of phase coherence of the excitations under the influence of fluctuations. 5. Conclusions The present work analyzes the temperature dependence of important quantities in the NlSM. The approach consists of understanding thermal effects as a noise coupled to the system and investigating the response of the gap, the model’s propagator and line width to this coupling.
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5.5 5
Thermal Line width Total Line width
4.5
Line width (meV)
4 3.5 3 2.5 2 1.5 1 0.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Reduced Temperature Fig. 3. Thermal and total line widths versus reduced temperature for a S ¼ 1 AFM Heisenberg chain. Near zero temperature, the influence of thermal fluctuations over lifetime effects is small compared to the contribution coming from the interactions between magnons.
The gap of the NlSM is modified by the noise. We find that the mass as a function of temperature generated by the noise-driven model agrees with experiment. The noise-driven propagator G N differs from the T ¼ 0 NlSM propagator G by a complex term proportional to G 2 . The total line width is calculated from the imaginary part of the inverted noise-driven propagator. The influence of temperature appears explicit in the imaginary part of the zeroth order noise-driven propagator and in the function mðtÞ, obtained via the saddlepoint approximation. Again, good agreement is found with experiment, even in the one-loop approximation for the self-energy. Furthermore, the noise-driven model, even in the saddle-point approximation, yields a line width for the model excitation spectrum. This is found to be smaller than the magnon–magnon interaction line width. Therefore, for the range of temperature where the NlSM can be applied, one can say that thermal fluctuations are secondary for lifetime effects when compared to the magnon–magnon interaction contribution. The outcome of a thermal line width is a special feature of the noise-driven method of introducing temperature in the model, that is, through an external noise coupled to the system. The noise field accounts for the thermal fluctuations that, in addition to the interaction between elementary excitations, may generate magnons with finite lifetime. Now a word about the physical pictures addressed by the present model and the Boltzmann prescription is in order. While the ‘‘Boltzmann picture’’ shows a system in thermal equilibrium with a bath, the ‘‘noise-driven picture’’ goes for a system coupled with a stochastic external field. This apparently subtle distinction is consequential in what concerns the physical situation under analysis even if the very purpose of the external field is to mimic the thermal behavior of the system. In other words, the noise-driven model and the Boltzmann prescription are not equivalent in physical grounds. The reason is twofold: first, in the absence of
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damping, the system will not attain thermal equilibrium when acted by a noise. In Section 2 we argued that, for low enough temperatures, the equilibrium situation can be reconciled for an accurate choice of the free parameter G. Also, as discussed in Section 4, the external field introduces extra decaying channels to the system which are related to the thermal fluctuations. The decaying process in the Boltzmann formalism results solely from the magnon–magnon interactions, no consideration is made regarding thermal effects. The noisedriven model intrinsically considers this possibility once treating the thermal bath as a filed coupled to the system, i.e., a source of decaying. These by no means forbid us to regard the results provided by the noisedriven model as adequate for the NLSM thermal behavior. In particular, the temperature dependence for the gap and line-width (due to magnon–magnon interactions summed to the thermal contribution) are in good accord with experimental data. The central motivation of this work is to suggest a recourse for introduction of temperature in quantum field theory models other than the usual Boltzmann prescription. Usually, temperature enters the problem via the association between the Wick-rotated time variable in the path integral and the temperature of the statistical partition function through Matsubara frequencies. In contrast to this procedure, we assume that thermal effects can be treated as a white noise that couples linearly with the system, and consequently we conceive the noise-driven model. We do not, however, include the dissipation term required for the system to attain thermal equilibrium with the bath. This is equivalent to the assumption that the thermal perturbation is purely diffusive in the temperature regime we are interested in. We implemented this method for a useful model for many physical systems—namely the NlSM. Here, the noise-driven model proved efficient to recover the temperature dependence of the gap, providing, in addition, information about eventual lifetime effects due to thermal fluctuations. Since the noise-driven approach succeeds when applied to the NlSM, we believe it may be useful to attack other problems in condensed matter and particle physics, in which the treatment of thermal fluctuations may be difficult by the available methods. Acknowledgment This work was partially supported by CNPq, Brazil. References [1] R. Rajaraman, Solitons and Instantons, An Introduction to Solitons and Instantons in Quantum Field Theory, North-Holland, Amsterdam, New York, Oxford, 1982 (and references cited therein). [2] R.E. Prange, S.M. Girvin, The Quantum Hall Effect, Springer, Berlin, 1987. [3] E. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, Reading, MA, 1991 (The Advanced Book Program, and references cited there in). [4] T. Sakaguchi, K. Kakurai, T. Yokoo, J. Akimitsu, J. Phys. Soc. Japan 65 (9) (1996) 3025. [5] D. Lurie, Particles and Fields, Interscience Publishers, a division of John Wiley and sons, New York, London, Sydney, 1968. [6] F.D.M. Haldane, Phys. Lett. A 93 (1983) 464. [7] F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153. [8] E. Ercolessi, G. Morandi, P. Pieri, M. Roncaglia, Phys. Rev. B 62 (22) (2000) 14860. [9] N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1992. [10] D.S. Lee, D. Boyanovsky, Nucl. Phys. B 406 (1993) 631. [11] D. Senechal, Phys. Rev. B 47 (13) (1993) 8353. [12] A.S.T. Pires, M.E. Gouvea, J. Magn. Magn. Mater. 241 (2002) 315. [13] Th. Jolicoeur, O. Golinelli, Phys. Rev. B 50 (13) (1994) 9265.
Physica A 370 (2006) 601–612 www.elsevier.com/locate/physa
Noise-driven nonlinear sigma model M. Malard Sales, A.S.T. Pires, Ronald Dickman, M.C. Nemes Departamento de Fı´sica, Universidade Federal de Minas Gerais, 30123-970 Belo Horizonte, Brazil Received 20 January 2005; received in revised form 22 September 2005 Available online 17 April 2006
Abstract We present a noise-driven model for obtaining the gap and line-width as functions of the temperature in the nonlinear sigma model. The method is phenomenological and rests on the following physical idea: a classical external stochastic field is introduced representing the coupling of the sigma field with a noise source. Moreover, we assume that the inelastic scattering length is much longer than the elastic one, justifying the neglect of dissipation for temperatures such that the nonlinear sigma model is a good approximation for antiferromagnetic spin chains. This phenomenological approach is justified by comparison with other theoretical predictions and with experiment. r 2006 Elsevier B.V. All rights reserved. Keywords: Nonlinear sigma model; Integer-spin chains; White noise; Temperature-dependent gap; Temperature-dependent line width
1. Introduction The nonlinear sigma model (NlSM) describes physical systems of many types. In particle physics, the twodimensional model is a prototype for nonabelian gauge theories in four dimensions [1]. In condensed matter physics, the NlSM is an approximation, through the Haldane mapping, to the continuum limit of the antiferromagnetic Heisenberg model. The quantum Hall effect can be analyzed via the NlSM as well [2]. Finally, the model is also useful in understanding superconductivity [3]. Due to its broad range of application, the NlSM has been a subject of intensive study for many years. There are many theoretical works concerning the temperature dependence of the gap in the NlSM. Comparison with experimental data for S ¼ 1 one-dimensional Heisenberg antiferromagnets shows that the NlSM describes such systems well at low temperatures [4]. In particular, the generation of a finite gap at zero temperature explains, in light of Goldstone’s theorem [5], the disordered ground state of these antiferromagnets. This property was not understood until 1983, when Haldane proposed his conjecture [6,7]. Based on topological arguments, Haldane proved that, at zero temperature, integer-spin antiferromagnetic Heisenberg chains have a finite gap, and that half-integer chains are gapless. The traditional way of introducing temperature in the NlSM is through the well known Matsubara technique [8]. It predicts discrete frequencies, in contrast to the model at zero temperature. Basically, what is done is an association between the Wick-rotated time variable in the path integral and the temperature of the Corresponding author.
E-mail address: mmalard@fisica.ufmg.br (M. Malard Sales). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.03.012
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statistical mechanical partition function. This method is the theoretical sound and well tested Boltzmann prescription for the inclusion of the temperature in the problem. In the present paper, we propose a different approach for including the influence of temperature on the model. What we suggest is to couple the system to an external white noise, resembling the Langevin approach for studying the effect of fluctuations in macroscopic systems. Fluctuations are introduced by including a stochastic external field or noise in the equations of motion. We call this phenomenological approach ‘‘noisedriven’’. The general scenario of the NlSM in contact with an external field is presented in Section 2. In Section 3, we determine the gap as a function of temperature, and find good agreement with experimental data. We believe that the calculations involved are simpler than those involved in the Boltzmann prescription. The latter requires some extra algebraic effort involving series summations, modified Bessel functions and various approximations to reach a final expression for the gap as a function of the temperature. Our results are obtained using the saddle-point approximation. This leads to a complex expression, the real part representing the gap as function of temperature. The imaginary part is interpreted as representing lifetime effects due to the coupling of the sigma field with an external field. A more detailed discussion of this subject is presented in Section 4 where the calculation of the line width is performed. Again, good agreement is achieved with experimental data. The curves for the gap and the line width versus temperature are explicit evaluated for the case S ¼ 1 in order to fit them with the available experimental data. Nevertheless, our method applies for any integer-spin antiferromagnetic chain other than the more frequently studied spin-1 chain. Finally, in Section 5 we present our conclusions concerning the validity of the noise-driven method for the inclusion of temperature in the NlSM. It also includes a discussion on the difference between the physical pictures addressed by the usual Boltzmann prescription and the noise-driven model. 2. Nonlinear sigma model in the presence of an external field ~ is given by The Lagrangian density of the NlSM in the presence of an external field AðxÞ ~ ~ ~ ¼ 1 ðqm ji Þðqm ji Þ þ lðxÞð~ L½qm ~ j2 1Þ þ AðxÞ j, j; ~ j; l; A 2g
(1)
~ðxÞ, lðxÞ is the Lagrange multiplier scalar field needed to where ji is the ith component of the vector field j impose the constraint j2 ¼ 1 on the modulus of the field ~ jðxÞ and g is an arbitrary constant. As usual, Greek indices run over the space–time vector x ¼ ðt; ~ xÞ where ~ x is M-dimensional. The Latin indices stand for the N ~ components of the vectorial fields ~ jðxÞ and AðxÞ. ~ The goal of this work is to consider the external field AðxÞ as a stochastic variable, that is, as an external noise acting on the system in a way to reproduce the effects of temperature fluctuations in the model. Posed in these terms, the argument rests on the same reasoning as the Langevin approach of finding the effect of fluctuations in macroscopic systems. In the following paragraphs we attempt to provide a justification of the validity of our Langevin-like method for introducing temperature in the NlSM. In the Boltzmann prescription the introduction of temperature is done through the equivalence between the statistical partition function and the path integral functional, which arises from the introduction of the Wickrotated imaginary time ðt ¼ itÞ. The time parameter t is then associated with temperature through the equation t ¼ 1=ðkB TÞ. At zero temperature, the integration variable t runs from zero to infinity and a Fourier transform introduces a continuous and integrated frequency o. Instead, for finite temperature, a Fourier series is needed in place of the Fourier transform leading to the discrete Matsubara frequencies. These, having a linear dependence on T, generate the expected temperature dependence of any physical quantity. Our reasoning is to follow the standard calculation procedure for zero temperature and hope that the introduction of a temperature dependent noise can do the job of reproducing the thermal behavior of the model. At low enough temperatures, it is reasonable to think of thermal fluctuations as a perturbation on the model at zero temperature, thus making our prior assumption very plausible. Furthermore, our method is supported by the stochastic description of fluctuations on macroscopic systems by the master equation [9]. The master equation, when not solved exactly, is treated in a systematic power series expansion in a parameter O defined in terms of a size parameter of the system. The systematic O-expansion is the basis for the existence of
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a macroscopic deterministic description of systems that are intrinsically stochastic. It justifies as a first approximation the standard treatment in terms of a deterministic equation with noise added, as in the Langevin approach. Moreover, in the lowest approximation, it happens that the noise is Gaussian, definitely justifying its use to treat thermal fluctuations. Finally, the good agreement found between experimental data and our results for the mass and line width as functions of temperature supports the validity of the method. ~ is therefore treated as a white noise with the following statistical properties [9]: The stochastic term AðxÞ ~ hAðxÞi ¼ 0,
(2)
hAi ðxÞAj ðx0 Þi ¼ 2GkB Tdðx x0 Þdij ,
(3)
where kB is Boltzmann’s constant and T is temperature. The factor 2GkB T is a measure of the noise intensity so that, for low temperatures, it is reasonable to think of it as a perturbation on the system at zero temperature. In the usual Langevin approach, the fluctuation–dissipation theorem [9] implies that G is the damping constant. The physical picture is that random kicks, caused by thermal fluctuations, drive the system away from the state of minimum energy while damping tends to restore it to this state. At this point a fundamental difference between our approach and the usual Langevin description becomes evident: we have not included a dissipation term in the equation of motion for ~ j. Indeed, to do so would require introducing a velocity-dependent potential in the Lagrangian density, Eq. (1). This would turn the problem into a different and more complicated one. The very purpose of our approach is to investigate whether the simple Lagrangian density, Eq. (1), suffices to represent thermal effects in the NlSM. Thus, here the constant G in Eq. (3) is a mere parameter, not connected to a dissipation force as required by the fluctuation–dissipation theorem. From elementary considerations, in particular the fact that the motion of an undamped harmonic oscillator subjected to white noise is unbounded, we expect the mean energy of the system defined by Eqs. (1)–(3) to grow without limit with time, rather than attain thermal equilibrium. The saddle-point approximation used here to analyze the model seems to be blind to this fact, and, for low temperatures, we recover physically reasonable results even in the absence of damping. A suitable choice of parameter G leads to results in good accord with experiment. The absence of damping can be understood with the physical supposition that the disorder in the system is caused mainly by elastic collisions with impurities. Therefore, we assume that the inelastic scattering length is much larger than the elastic one. In this context, it makes sense to think of a purely diffusive perturbation for low enough temperatures such that inelastic effects which could led to dissipation can be neglected. It is worth mentioning here that it will suffice for our objectives to consider only the stochastic properties (2) ~ and (3) without worrying about the specific probability distribution of AðxÞ. Actually, we do not make explicit use of the fact that thermal fluctuations, being independent stochastic variables, have Gaussian probability distribution as assured by the central limit theorem [9]. The fact that the distribution is Gaussian is needed solely in the legitimation of the method of introducing temperature via an external stochastic field, as argued before in the context of the master equation. In any case, the method as it stands is completely general with ~ respect to the probability distribution of the field AðxÞ. Therefore, it applies for any stochastic external field having finite variance. ~ fields, following the same arguments of Eq. (1) assumes that the coupling is linear in both the ~ j and A Lee and Boyanovsky in their analysis of the dynamics of phase transitions induced by a heat bath [10]. The linear coupling restriction may be relaxed at the expense of additional complications. Moreover, the linear coupling applied in the context of the noise-driven NlSM is further justified once its results agree with experience. In order to proceed, it is necessary to impose the following asymptotic conditions on the field ~ j and its derivatives: jxj!1
~ jðxÞ ! 0, jxj!1
qm ~ jðxÞ ! 0
(4) 8m ¼ 0; 1; 2; . . . ; M.
(5)
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The behavior of the model can be obtained from the Lagrangian density, Eq. (1), through the path integral functional defined as Z Z i ~ ~ ¼ j; l; A, (6) Z½A D~ jDl exp S½~ _ where ~ ¼ S½~ j; l; A
Z
~ dx L½qm ~ j; ~ j; l; A
is the action of the model. We now define the field: Z ~ 0Þ ~ ~ jðxÞ ¼~ jðxÞ dx0 Gðx; x0 ÞAðx
(7)
(8)
where Gðx; x0 Þ ¼ gðq2 þ 2glðxÞÞ1 dðx x0 Þ
(9)
~ ¼ 0 and q2 stands for qm qm ¼ q2 r ~2 , the covariant Laplacian. The is the propagator for the system with A t 1 inverse of the propagator is the function G ðx; x0 Þ, which satisfies Z dx00 Gðx; x00 ÞG 1 ðx00 ; x0 Þ ¼ dðx x0 Þ. (10) Then, the inverse of the propagator can be written as G1 ðx; x0 Þ ¼
1 ðq2 þ 2glðxÞÞdðx x0 Þ. g
(11)
Using the asymptotic conditions (4) and (5) in performing integrations by parts and observing that ~ ~ ~~ j ¼ 1 is the Jacobian of the transformation (8), the path integral ~ ¼ ½qj=q~ ~ jD~ Dj j ¼ D~ j, where ½qj=q~ functional may be written as Z Z Z lðxÞ i 1 1 0~ 0 ~ 0 0 ~ þ dx AðxÞ:Gðx; x ÞAðx Þ þ Tr½lnðG ðx; x ÞÞ , Z½A ¼ Dl exp N dx (12) r 2r 2 where, for future convenience, we defined the parameter r ¼ _N. To reach Eq. (12) we have Wick rotated to ~ Accordingly, the covariant imaginary time (t ¼ it) in order to obtain a Gaussian exponential in the field ~ j. ~2 . The quadratic vector field ~ Laplacian q2 is replaced by the Euclidean one q2 ¼ q2t þ r j~ was integrated out through a limiting process going from the continuous space–time to a hyper-cubic lattice of m sites, with lattice spacing e, and letting m ! 1, e ! 0 with me ¼ constant, at the end of the calculation. The convergence of this limiting process is ensured by the introduction of a measure constant M m that eliminates the infinities of the procedure [1]. Since G1 ðx; x0 Þ is related to a Hermitian operator through Eq. (11), its matrix representation in ~ ~ becomes a product of Nm uncoupled Gaussian the discrete limit G 1 x;x0 is diagonalizable. The integral over j ~ integrals generating the above result for the functional Z½A. We are left, in Eq. (12), with an integral in l that cannot be solved analytically as it is not quadratic in that field. What is done is to expand the exponent around its saddle point retaining terms only up to second order. This is justified when one takes small deviations from the saddle-point solution which, in turn, is required for convergence when the limit N ! 1 is taken. In addition, in that limit we take r to remain constant when we simultaneously set _ ! 0. In other words, N, the overall multiplicative factor in the exponent, will work as an expansion parameter and the limit N ! 1 will lead to a Gaussian integral in the field l. This will lead us to the calculation of the gap.
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3. The gap Let us define the effective action functional: Z Z lðxÞ i ~ 0 Þ þ 1 Tr½lnðG 1 ðx; x0 ÞÞ. ~ þ dx0 AðxÞ:Gðx; f ½l ¼ dx x0 ÞAðx r 2r 2 One may expand f ½l around a solution lðxÞ ¼ lc such that df ½l ¼ 0. dlðxÞl¼lc
(13)
(14)
It will suffice for our purpose of obtaining the gap to explore the saddle-point condition, Eq. (14), without worrying about the full expansion of f ½l itself. From Eq. (9) we obtain dGðx; x0 Þ ¼ 2Gðx; x00 ÞGðx; x0 Þ, dlðx00 Þ
(15)
which must be substituted into the derivative of Eq. (13) with respect to l. Further, in order to obtain physically meaningful results we must average Eq. (14) such that the stochastic ~ is taken into account. The averaged saddle-point condition together with the properties (2) character of AðxÞ ~ lead to and (3) of the field AðxÞ Z 1 ð2GkB TÞ 00 ¼ G lc ð0Þ 1 i dxG lc ðx x Þ , (16) r r where the Euclidean version of Eq. (9) for Gðx; x0 Þ evaluated at the saddle point leads to Z 0 dp eiðp=_Þðxx Þ 2 0 0 G lc ðx; x Þ ¼ G lc ðx x Þ ¼ gðc_Þ , ð2p_Þd c2 p2 þ D2
(17)
where d stands for the dimension of the space–time and p ¼ ðp0 ; ~ pÞ is the four-momentum vector in the reciprocal space. Also, D2 ¼ 2glc ðc_Þ2 stands for the gap in the proper units of energy where c is the spinwave velocity. Substitution of Eq. (17) into Eq. (16) leads to 2 2 4p c L þ D2 gc2 _ð2GkB TÞ ¼ ln 1 i , (18) Ng_ D2 ND2 where the constant L jpjmax is a cut-off parameter introduced for convergence of the double integrals ðd ¼ 2Þ in the momentum–frequency plane. The cut-off is quite natural when applying the model in solid state physics, since in this case there exists a finite separation a between the lattice sites, so that the momentum is restricted to the first Brillouin zone of the crystal. The frequency will also be limited since it is related to the momentum via the dispersion relation. In the limit L ! 1, Eq. (18) gives the following result for DðT ¼ 0Þ D0 : 2p (19) D0 ¼ cL exp Ng_ reproducing a well known result for the NlSM in the limit L ! 1 [11]. As is the case for the NlSM, explicit assignment of values to the constants involved in Eq. (19) cannot provide a quantitative prediction for D0 in accordance with the experimental value or with numerical simulations for one-dimensional antiferromagnets. The latter predicts D0 ¼ 0:41J for a spin-one chain, in units of the antiferromagnetic (AFM) intrachain exchange coupling J [12]. In particular, the constant L in Eq. (19) carries the effect of passing from a manifestly continuous theory to a discrete one and cannot be assigned an accurate value by direct estimation. Nevertheless, as the NlSM mapping for _ ¼ 1 predicts g ¼ 2=S [8], the model is able to reproduce the expected qualitative behavior [12], D0 / epS . Actually, up to the present moment, neither the NlSM nor any other analytical approach is able to predict quantitatively D0 as a function
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of microscopic parameters. What is usually obtained is an expression for finite temperature involving ratios such as DðTÞ=D0 . In this spirit, let us define what we will call a reduced temperature and a reduced gap, respectively, as t¼
T , D0
dðtÞ ¼
(20)
DðtÞ , D0
(21)
where, in units such that kB ¼ 1, D0 can be given in kelvins making the reduced temperature t dimensionless. The reduced gap is, by definition, also dimensionless. In terms of these new variables and applying the limit L ! 1, Eq. (18) can be rewritten to first order in T as t 2 d ðtÞ ¼ exp ig 2 , (22) d ðtÞ where 2g_c2 G c2 L 2 ln g¼ ND0 D20
! (23)
must be a constant to be adjusted in order that Eq. (22) fits the experimental data. In principle, from the obtained value of g we can obtain the noise parameter G if we are able to deal with all the constants involved in Eq. (23). Fortunately, we can skip this quite troublesome task for, in Section 4, a further fit with experiment will render G an adjustment parameter. In this way, it will be possible to assign it a reasonable value without explicit considerations concerning the constants of the model. Inspection of Eq. (22) shows that dðtÞ must be a complex quantity. Its real part gives the mass or gap, mðtÞ, of the related quasi-particles—magnons for application of the model to AFM spin chains. The imaginary part is related to the magnon finite lifetime generated by the coupling between the model and noise. This will be discussed in detail in the next section. Returning to Eq. (22), we see that it is transcendental in d2 ðtÞ and, therefore, cannot be solved analytically for mðtÞ. Nevertheless, we can plot the corresponding curves for a judicious choice of the free parameter g. Fig. 1 shows the reduced mass m as a function of reduced temperature t, for S ¼ 1, obtained from the real part of Eq. (22). The reduced mass remains almost constant, equal to unity, until around t ¼ 0:1, and then begins to increase approximately linearly with reduced temperature. This is indeed what is expected for the NlSM for S ¼ 1 [13]. The experimental data reported by Sakaguchi were obtained via inelastic neutron scattering on Y2 BaNiO5 , considered the best candidate to realize a spin-1 AFM chain [4]. The best fit between theory and experiment is achieved when the free parameter g equals 1:36. The asymptotic ðt ! 0Þ expression for the NlSM reduced mass in the Boltzmann prescription is [4] mðtÞ 1 þ ð2ptÞ1=2 expð1=tÞ. The dashed line in Fig. 1 shows the Boltzmann prediction for mðtÞ. This curve begins to increase faster than the experimental results around t ¼ 0:4, in contrast with the noise-driven model. 4. The line width At this point there arises a fundamental difference between the noise-driven method and the Boltzmann prescription for the introduction of temperature in the model. As it was shown in the previous section, Eq. (22) has an imaginary part. The saddle-point method applied to the Boltzmann prescription does not lead to a corresponding imaginary term in the gap for, in that approximation, the system behaves like an ensemble of free magnons. Neglect of interactions leads to an infinite magnon lifetime. In contrast, the noise-driven method provides a finite lifetime even in the saddle-point approximation. The NlSM treated via Boltzmann prescription includes information on finite lifetime effects in the lðxÞj2 term of the Lagrangian density, Eq. (1), which endeavor the interactions between magnons. The saddle-point applied in this context is blind to these effects because in this approximation lðxÞ is made constant through the
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Boltzmann prescription
1.6
Noise-driven method Experimental data
Reduced Mass
1.4
1.2
1
0.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Reduced Temperature Fig. 1. Reduced mass versus reduced temperature for a S ¼ 1 AFM Heisenberg chain. The curve shows near coincidence with the experimental data for Y2 BaNiO5 [4] when g ¼ 1:36. The dashed line is the Boltzmann prediction for the NlSM reduced mass [4].
assumption that there is a solution lc that satisfies Eq. (14). Now, temperature enters the noise-driven method ~ through the external stochastic field AðxÞ introducing another decaying channel in the system in addition to the magnon–magnon interaction. In this context, the saddle-point approximation is blind to lifetime effects coming from interactions between magnons but cannot dismiss the effects of the explicitly introduced thermal bath. In fact, one should expect that magnon decay is caused not only by the interactions between the magnons themselves, but also by the thermal fluctuations that a system at finite temperature is subjected to. In order to consistently analyze the problem of magnon decay in the noise-driven NlSM, the following section is devoted to the calculation of the line width, the inverse of the magnon lifetime. The use of the total propagator in this task amounts to count at once the joint contributions of both thermal fluctuations and magnon–magnon interactions to lifetime effects. ~ The standard method for obtaining the propagator is via the introduction of a fictitious field, say JðxÞ, ~ ~ ~ ~ which couples linearly with the field ~ jðxÞ, just as AðxÞ does. Then, Z½A becomes Z½A; J and we obtain the propagator via the formula ~ J ~ i_ d2 Z½A; 0 G N ðx; x Þ ¼ (24) , ~ J ~ ~ 0 Þ ~ dJðxÞd Jðx Z½A; ~ J¼0
where we set the fictitious field J~ ¼ 0 at the end of the calculation. As the coupling is assumed to be linear, ~ is a trivial task. The only change in Eq. (12) for the functional introducing the field J~ in the expression for Z½A 0 ~ 0 ~ ~ þ JðxÞÞ:Gðx; ~ 0 Þ þ Jðx ~ ~ 0 ÞÞ. Together with the noise will be that, instead of AðxÞ:Gðx; x ÞAðx Þ, one has ðAðxÞ x0 ÞðAðx
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property (3), the calculation leads to GN ¼ G i
A 2 G , r
(25)
where for simplicity we set A ¼ 2GkB T
(26)
and Eq. (25) is written in matrix form. Making the correspondence between the propagator (multiplied by i) and the correlation between the values of the field ~ j at two different space–time points we see from Eq. (25) that thermal effects reduce the correlation. Since the fluctuations tend to disorder the system, it is reasonable that correlations are reduced under thermal fluctuations. Making use of Eq. (25), we define the saddle-point noise-driven propagator G 0N obtained from G 0 , the saddle-point T ¼ 0 propagator of the NlSM as G 0N ¼ G 0 i
A 2 G . r 0
(27)
It is now evident that the occurrence of a ‘‘thermal line width’’ even in the saddle-point approximation comes naturally from the A dependent imaginary part in the propagator, Eq. (27), as also required by the complexity of dðtÞ in Eq. (22). The explicit outcome of the ‘‘thermal lifetime’’ in our noise-driven method is an inevitable consequence of considering the thermal bath as a noise field coupled classically to the system, i.e., a source of decaying. The self-energy Sðx; x0 Þ is defined through the Dyson equation for the propagator which reads G ¼ G0 þ G 0 SG,
(28)
where G 0 is the highest order known propagator for the model in question. For the definition of the noise-driven NlSM self-energy SN one may write GN ¼ G 0N þ G0N SN GN ,
(29)
which iteratively expanded in terms of G 0N rewritten via Eq. (27) leads to the following expansion in powers of G 0 : GN ¼ G0 i
A 2 A A G þ G 0 SN G 0 i G0 SN G 20 i G20 SN G0 þ G0 SN G0 SN G0 þ OðG 40 Þ. r 0 r r
(30)
Now, using Eq. (25) for GN and then expanding G in powers of G0 following Eq. (28) we find GN ¼ G0 i
A 2 A A G 0 þ G 0 SG 0 i G 0 SG20 i G20 SG 0 þ G 0 SG 0 SG 0 þ OðG 40 Þ. r r r
(31)
Comparing Eqs. (30) and (31) we have SN ¼ S,
(32)
i.e., for G N calculated up to third order in G 0 , the noise-driven self-energy is the same as in the model with ~ ¼ 0. From Eq. (31), it is clear that G N ¼ G for T ¼ 0, and up to order G3 , temperature affects the A 0 propagator through the A linearly dependent terms. The Dyson equation (29) can be written for the noise-driven propagator in momentum space as GN ðpÞ ¼
1 , SðpÞ
G 1 0N ðpÞ
(33)
where p ¼ ðo; kÞ is the two-dimensional frequency–momentum vector and SN ðpÞ was already set equal to SðpÞ, the known self-energy of the NlSM given at order 1=N by Z 2 0 d p ½G 0 ðp þ p0 Þ G 0 ðp0 Þ , (34) SðpÞ ¼ Pðp0 Þ ð2pÞ2
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where we have absorbed the term Sð0Þ into the expression for the self-energy in order to assure its convergence. The propagator Pðp0 Þ is associated with the Lagrange multiplier field lðxÞ and for c ¼ _ ¼ 1 is given by Z Z d2 p d2 p g2 0 0 Pðp Þ ¼ . (35) G ðp þ p ÞG ðpÞ ¼ 0 0 ð2pÞ2 ð2pÞ2 ½ðp þ p0 Þ2 þ D20 ½p2 þ D20 Lifetime effects occur through the imaginary part of the denominator in Eq. (33). With G 0N given by Eq. (27), the total line width can be given after some algebra by " #1 " # Z N A g2 1 1 1 lwðpÞ ¼ d 2 p0 A N ðp2 þ D20 Þ2 2g ðp þ p0 Þ2 þ D20 p0 2 þ D20 " #1 ! 1=2 2 0 2 1=2 ð1 þ 4D =p Þ 1 2 2 2 0 p0 1 þ 4D20 =p0 ln , ð36Þ þ p2 ð1 þ 4D20 =p0 2 Þ1=2 þ 1 where we used the Feynman relation Z 1 1 1 ¼ dx . ab ½ða bÞx þ b2 0
(37)
In the limit of validity of Eq. (32), i.e., for G N up to third order in G 0 , Eq. (36) shows that the total line width separates in a thermal contribution (coming from G1 0N ) plus a magnon interaction contribution (coming from S). If higher order terms were considered in the expansion equations (30) and (31) the noise-driven selfenergy SN would be a nontrivial function of S and A so that the total line width would contain mixed contribution from thermal and magnon interaction effects. From Eqs. (20) and (26) we will rewrite A as A ¼ 2GD0 t
(38)
under the condition kB ¼ 1 for a dimensionless reduced temperature t. The constant G will be our free parameter valued as to fit the experimental data. Sakaguchi et al. performed constant k ¼ p=2 scans corresponding to the scattering vector k ¼ 0 in the ð1 þ 1Þ NlSM. Since o2 ¼ k2 þ M 2 , we evaluate Eq. (36) at p ¼ ðM; 0Þ ¼ ðD0 mðtÞ; 0Þ, obtaining " #1 3 1 8G t lwðtÞ ¼ 2GD0 t 3D30 ðm2 ðtÞ þ 1Þ2 " # Z 1 þ1 1 1 dk do 2 4 1 k þ o2 þ 2D0 mðtÞo þ D20 ðm2 ðtÞ þ 1Þ k2 þ o2 þ D20 #1 ! 1=2 " 2 2 2 1=2 4D20 1 2 2 2 ð1 þ 4D0 =ðk þ o ÞÞ 2 Ln , ð39Þ ðk þ o Þ 1 þ 2 þp k þ o2 ð1 þ 4D20 =ðk2 þ o2 ÞÞ1=2 þ 1 where we set N ¼ 3 and g ¼ 2 since for _ ¼ 1 the NlSM mapping predicts g ¼ 2=S. For D0 we use Sakaguchi’s experimental value D0 ¼ 9:5 meV for the compound Y2 BaNiO5 . After numerically integrating Eq. (39), we plot the line width in meV versus reduced temperature, as shown in Fig. 2. The best fit with experimental data was achieved for G ¼ 3:84 102 . Remember that G was defined in Eq. (3) as a parameter related to the noise intensity. It is known that the NlSM represents Heisenberg AFM chains only in the low-temperature regime where the effects of thermal fluctuations are expected to be weak. The above small value of G is in accord with this expectation. Reasonable agreement with experimental data is achieved in the low-temperature regime, up to around t ¼ 0:6. Above this temperature, the experimental values lie above the theoretical curve. A more accurate theoretical prediction requires evaluation of an extended perturbation series for the self-energy. In any case, the one-loop correction presented here is sufficient to show that the noise-driven model permits calculation of the NlSM line width as well as the mass.
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Noise-driven Line width Experimental data
Line width (meV)
4
3
2
1
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Reduced Temperature Fig. 2. Line width versus reduced temperature for a S ¼ 1 AFM Heisenberg chain for G ¼ 3:84 102 . The experimental data are for Y2 BaNiO5 [4].
One could expect that the thermal contribution to Eq. (39) equals the imaginary part of Eq. (22). However, comparison between the two curves shows that they do not agree perfectly. The reason is that they were obtained by quite different methods. The imaginary part of Eq. (22) is the result of an integration over the first Brillouin zone of all the modes p ¼ ðo; kÞ allowed for the system. Differently, the thermal contribution to Eq. (39) is calculated for p ¼ ðM; 0Þ ¼ ðD0 mðtÞ; 0Þ where Sakaguchi’s experiment was performed. In any case, they share the same qualitative behavior: both vanish at t ¼ 0, increase with temperature and are of the same order in magnitude. The outcome of Eq. (39), the total line width of the model, is plotted together with the imaginary part of Eq. (22) in Fig. 3. The thermal contribution is zero at t ¼ 0, as expected. In the vicinity of zero temperature (up to t ¼ 0:2) the difference between the curves is about 87%. As the temperature increases, the thermal effects become more significant but, up to the temperature for which experimental data are available (t ¼ 0:8), the maximum thermal contribution does not exceed 18% of the total result. We see that, at least inside the range of temperature where the NlSM is expected to reasonably approximate AFM integer spin chains, interactions between magnons are more significant to lifetime effects than their interaction with the thermal bath. The thermal line width may be interpreted as being due to loss of phase coherence of the excitations under the influence of fluctuations. 5. Conclusions The present work analyzes the temperature dependence of important quantities in the NlSM. The approach consists of understanding thermal effects as a noise coupled to the system and investigating the response of the gap, the model’s propagator and line width to this coupling.
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5.5 5
Thermal Line width Total Line width
4.5
Line width (meV)
4 3.5 3 2.5 2 1.5 1 0.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Reduced Temperature Fig. 3. Thermal and total line widths versus reduced temperature for a S ¼ 1 AFM Heisenberg chain. Near zero temperature, the influence of thermal fluctuations over lifetime effects is small compared to the contribution coming from the interactions between magnons.
The gap of the NlSM is modified by the noise. We find that the mass as a function of temperature generated by the noise-driven model agrees with experiment. The noise-driven propagator G N differs from the T ¼ 0 NlSM propagator G by a complex term proportional to G 2 . The total line width is calculated from the imaginary part of the inverted noise-driven propagator. The influence of temperature appears explicit in the imaginary part of the zeroth order noise-driven propagator and in the function mðtÞ, obtained via the saddlepoint approximation. Again, good agreement is found with experiment, even in the one-loop approximation for the self-energy. Furthermore, the noise-driven model, even in the saddle-point approximation, yields a line width for the model excitation spectrum. This is found to be smaller than the magnon–magnon interaction line width. Therefore, for the range of temperature where the NlSM can be applied, one can say that thermal fluctuations are secondary for lifetime effects when compared to the magnon–magnon interaction contribution. The outcome of a thermal line width is a special feature of the noise-driven method of introducing temperature in the model, that is, through an external noise coupled to the system. The noise field accounts for the thermal fluctuations that, in addition to the interaction between elementary excitations, may generate magnons with finite lifetime. Now a word about the physical pictures addressed by the present model and the Boltzmann prescription is in order. While the ‘‘Boltzmann picture’’ shows a system in thermal equilibrium with a bath, the ‘‘noise-driven picture’’ goes for a system coupled with a stochastic external field. This apparently subtle distinction is consequential in what concerns the physical situation under analysis even if the very purpose of the external field is to mimic the thermal behavior of the system. In other words, the noise-driven model and the Boltzmann prescription are not equivalent in physical grounds. The reason is twofold: first, in the absence of
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damping, the system will not attain thermal equilibrium when acted by a noise. In Section 2 we argued that, for low enough temperatures, the equilibrium situation can be reconciled for an accurate choice of the free parameter G. Also, as discussed in Section 4, the external field introduces extra decaying channels to the system which are related to the thermal fluctuations. The decaying process in the Boltzmann formalism results solely from the magnon–magnon interactions, no consideration is made regarding thermal effects. The noisedriven model intrinsically considers this possibility once treating the thermal bath as a filed coupled to the system, i.e., a source of decaying. These by no means forbid us to regard the results provided by the noisedriven model as adequate for the NLSM thermal behavior. In particular, the temperature dependence for the gap and line-width (due to magnon–magnon interactions summed to the thermal contribution) are in good accord with experimental data. The central motivation of this work is to suggest a recourse for introduction of temperature in quantum field theory models other than the usual Boltzmann prescription. Usually, temperature enters the problem via the association between the Wick-rotated time variable in the path integral and the temperature of the statistical partition function through Matsubara frequencies. In contrast to this procedure, we assume that thermal effects can be treated as a white noise that couples linearly with the system, and consequently we conceive the noise-driven model. We do not, however, include the dissipation term required for the system to attain thermal equilibrium with the bath. This is equivalent to the assumption that the thermal perturbation is purely diffusive in the temperature regime we are interested in. We implemented this method for a useful model for many physical systems—namely the NlSM. Here, the noise-driven model proved efficient to recover the temperature dependence of the gap, providing, in addition, information about eventual lifetime effects due to thermal fluctuations. Since the noise-driven approach succeeds when applied to the NlSM, we believe it may be useful to attack other problems in condensed matter and particle physics, in which the treatment of thermal fluctuations may be difficult by the available methods. Acknowledgment This work was partially supported by CNPq, Brazil. References [1] R. Rajaraman, Solitons and Instantons, An Introduction to Solitons and Instantons in Quantum Field Theory, North-Holland, Amsterdam, New York, Oxford, 1982 (and references cited therein). [2] R.E. Prange, S.M. Girvin, The Quantum Hall Effect, Springer, Berlin, 1987. [3] E. Fradkin, Field Theories of Condensed Matter Systems, Addison-Wesley, Reading, MA, 1991 (The Advanced Book Program, and references cited there in). [4] T. Sakaguchi, K. Kakurai, T. Yokoo, J. Akimitsu, J. Phys. Soc. Japan 65 (9) (1996) 3025. [5] D. Lurie, Particles and Fields, Interscience Publishers, a division of John Wiley and sons, New York, London, Sydney, 1968. [6] F.D.M. Haldane, Phys. Lett. A 93 (1983) 464. [7] F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153. [8] E. Ercolessi, G. Morandi, P. Pieri, M. Roncaglia, Phys. Rev. B 62 (22) (2000) 14860. [9] N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1992. [10] D.S. Lee, D. Boyanovsky, Nucl. Phys. B 406 (1993) 631. [11] D. Senechal, Phys. Rev. B 47 (13) (1993) 8353. [12] A.S.T. Pires, M.E. Gouvea, J. Magn. Magn. Mater. 241 (2002) 315. [13] Th. Jolicoeur, O. Golinelli, Phys. Rev. B 50 (13) (1994) 9265.