Thermodynamics of ferromagnetic spin chains in a magnetic field: Impact of the spin–wave interaction

Physica B 442 (2014) 81–89

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Thermodynamics of ferromagnetic spin chains in a magnetic field: Impact of the spin–wave interaction Christoph P. Hofmann n Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima C.P. 28045, Mexico

art ic l e i nf o

a b s t r a c t

Article history: Received 21 December 2013 Received in revised form 22 February 2014 Accepted 25 February 2014 Available online 3 March 2014

The thermodynamic properties of ferromagnetic spin chains have been the subject of many publications. Still, the problem of how the spin–wave interaction manifest itself in these low-temperature series has been neglected. Using the method of effective Lagrangians, we explicitly evaluate the partition function of ferromagnetic spin chains at low temperatures and in the presence of a magnetic field up to three loops in the perturbative expansion where the spin–wave interaction sets in. We discuss in detail the renormalization and the numerical evaluation of a particular three-loop graph and derive the lowtemperature series for the free energy density, energy density, heat capacity, entropy density, as well as the magnetization and the susceptibility. In the low-temperature expansion for the free energy density, the spin–wave interaction starts manifesting itself at order T5/2. In the pressure, the coefficient of the T5/2-term is positive, indicating that the spin–wave interaction is repulsive. While it is straightforward to go up to three-loop order in the effective loop expansion, the analogous calculation on the basis of conventional condensed matter methods, such as spin–wave theory, appears to be beyond reach. & 2014 Elsevier B.V. All rights reserved.

Keywords: Spin chains Ferromagnet Effective Lagrangians Spin waves Finite temperature

1. Introduction In the present study we rigorously answer the question of how the spin–wave interaction manifests itself in the low-temperature properties of ferromagnetic spin chains in a weak magnetic field. Using the systematic method of effective Lagrangians, in the very recent article [1], it was argued that the spin–wave interaction only starts showing up at the three-loop level. However, the explicit evaluation of the various Feynman graphs contributing at this order to the partition function has not been addressed in that reference. This quite elaborate task is the subject of the present paper. We then provide the low-temperature series for the free energy density, energy density, heat capacity, entropy density, as well as the magnetization and the susceptibility. The effective Lagrangian method relies on the fact that the lowenergy dynamics of the system is captured by the Goldstone bosons, which result from the spontaneously broken global symmetry. In the present case, the spin rotation symmetry of the Heisenberg ferromagnet is spontaneously broken, Oð3Þ-Oð2Þ, and the spin– waves or magnons emerge as Goldstone bosons. Conceptually, it is quite remarkable that the effective Lagrangian method works in one spatial dimension. It is well-known that in a Lorentz-invariant

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http://dx.doi.org/10.1016/j.physb.2014.02.047 0921-4526 & 2014 Elsevier B.V. All rights reserved.

framework, where the Goldstone bosons (pions, kaons, η-particle) follow a linear, i.e. relativistic, dispersion relation, the method fails in one spatial dimension. However, the ferromagnet, where the spin waves obey a quadratic dispersion law, is quite peculiar: here the systematic loop expansion perfectly works as we explain below. In the low-temperature expansion of the free energy density, the spin–wave interaction generates a term of order T5/2. The general structure of this series is discussed, and the question of which contributions are due to free magnon particles and which ones are due to the spin–wave interaction is thoroughly answered. In view of the nonperturbatively generated energy gap, we also critically examine the range of validity of the effective low-temperature series, pointing out that it is not legitimate to take the limit of a zero magnetic field. The thermodynamic properties of ferromagnetic spin chains have attracted a lot of attention over the past few decades and many methods have been used to study these interesting onedimensional systems. While early investigations were based on the Bethe ansatz [2–10], the modified spin–wave theory was the method advocated in Refs. [11–13]. Further methods used to address ferromagnetic spin chains include Schwinger-boson mean-field theory [14,15], Green functions [16,18,17,19–23], variants of spin-wave theory [24], scaling methods [25,17,26–31], numerical simulations [32–38,22], and yet other approaches [39–44]. Given this abundant literature on ferromagnetic spin chains, it is really surprising that the effect of the spin–wave interaction has been

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C.P. Hofmann / Physica B 442 (2014) 81–89

largely neglected. In particular, although ferromagnetic spin chains can be solved exactly by e.g. the Bethe ansatz, the low-temperature series derived from these exact results all refer to either a tiny or a zero magnetic field, which does not cover the domain we are interested in here. We emphasize that in the problem under consideration, the effective field theory approach is more efficient than conventional condensed matter methods such as spin–wave theory, as it allows one to systematically go to higher orders in the low-temperature expansion – beyond the results provided in the literature. Above all – for the first time, to the best of our knowledge – the manifestation of the spin–wave interaction in the low-temperature behavior of ferromagnetic spin chains in a magnetic field is discussed in a systematic manner. Almost all previous theoretical studies that analyzed the structure of the low-temperature series for ferromagnetic spin chains were restricted to the idealized picture of the free magnon gas. One exception is Ref. [24] which, however, refers to a tiny magnetic field and appears to be not quite consistent, as we point out in Section 5. The rest of the paper is organized as follows. In Section 2 we provide the reader with some basic aspects of the effective Lagrangian technique. The low-temperature expansion of the partition function up to three-loop order is derived in Section 3. The nontrivial part concerns the renormalization of a particular threeloop graph which is discussed in detail in Section 4. The lowtemperature series for the free energy density, pressure, energy density, entropy density, heat capacity, as well as the magnetization and the susceptibility for ferromagnetic spin chains in a magnetic field are given in Section 5. While our conclusions are presented in Section 6, details on the numerical evaluation of a specific threeloop graph are discussed in two appendices. The model-independent and systematic effective Lagrangian method, unfortunately, is still not very well known among condensed matter physicists. We would like to convince the reader that this method indeed represents an alternative and rigorous theoretical framework to address condensed matter systems, by providing a list of articles which are also based on this method. Ferromagnets and antiferromagnets in three and two space dimensions were considered in Refs. [45–61]. Two-dimensional antiferromagnets, doped with either holes or electrons, which represent the precursors of high-temperature superconductors were analyzed in Refs. [62–71]. Moreover, it was demonstrated in Refs. [72–76] that the effective Lagrangian technique is perfectly consistent with both numerical simulations based on the loopcluster algorithm and an analytically solvable microscopic model in one spatial dimension.

2. Effective Lagrangian method In a very recent article, Ref. [1], the low-temperature expansion of partition function for the ferromagnetic spin chain in a weak magnetic field was evaluated up to two loops. Here we perform the analysis up to three-loop order, where the spin–wave interaction comes into play. Essential aspects of the effective Lagrangian method at finite temperature have been discussed in Section 2 of Ref. [1] and will not be repeated here in detail. Below, we just focus on some basic ingredients of the method. Although Section 2 of Ref. [1] is self-contained and contains all the necessary information to understand the present calculation, the interested reader may still find more details on finite-temperature effective Lagrangians in Appendix A of Ref. [48] and in the various references given therein. The systematic construction of the effective field theory is based on an inspection of the symmetries inherent in the underlying theory. In the present case, the effective Lagrangian, or more

precisely the effective action Z 2 S eff ¼ d x Leff

ð1Þ

describing the ferromagnetic spin chain, must share all the symmetries of the underlying Heisenberg model. These include the spontaneously broken O(3) spin rotation symmetry (at zero temperature), parity and time reversal. Note that here we refer to zero external field. As discussed below, the magnetic field is incorporated into the effective Lagrangian as a perturbation that explicitly breaks O(3). One also has to identify the relevant lowenergy degrees of freedom entering the effective description. In the case of the Heisenberg ferromagnet, these are the two real magnon fields – or the physical magnon particle – that arise due to the spontaneously broken spin symmetry Oð3Þ-Oð2Þ. The various terms in the effective Lagrangian are organized systematically according to the number of space and time derivatives which act on the magnon fields. At low energies or temperatures, terms which contain only a few derivatives are the dominant ones, while terms with a larger number of derivatives are suppressed [77–79]. The effective Lagrangian Leff thus amounts to a systematic derivative expansion, or, equivalently, an expansion in powers of energy and momentum. Hence the quantities of physical interest (partition function, free energy density, magnetization, etc.) derived from Leff also correspond to expansions in powers of momentum which – at finite temperature – translate into expansions in powers of temperature. The leading-order effective Lagrangian for the one-dimensional ferromagnet is of momentum order p2 and reads [50] L2eff ¼ Σ

ϵab ∂0 U a U b 1 þ U3

1 þ ΣμHU 3  F 2 ∂x1 U i ∂x1 U i : 2

ð2Þ

The fundamental object is the three-dimensional magnetization unit vector U i ¼ ðU a ; U 3 Þ, where the two real components U a ða ¼ 1; 2Þ describe the spin–wave degrees of freedom. The quantity H is the ! magnetic field which points to the third direction, H ¼ ð0; 0; HÞ with ! H ¼ j H j 4 0. While the derivative structure of the above terms is determined by the symmetries of the underlying theory, the two a priori unknown low-energy coupling constants – the spontaneous magnetization at zero temperature Σ, and the constant F – have to be fixed experimentally, in a numerical simulation or by comparison with the microscopic theory. It is important to point out that one time derivative (∂0 ) counts as two space derivatives (∂x1 ∂x1 ), i.e., two powers of momentum are on the same footing as one power of 2 energy or temperature: k p ω; T. This is characteristic of ferromagnetic systems where the magnons display a quadratic dispersion relation. The next-to-leading-order effective Lagrangian for the ferromagnetic spin chain is of order p4 and involves the two effective coupling constants l1 and l3 [1]: L4eff ¼ l1 ð∂x1 U i ∂x1 U i Þ2 þ l3 ∂x21 U i ∂x21 U i :

ð3Þ

Higher-order pieces in the effective Lagrangian are not needed for the present calculation. The systematic perturbative evaluation of the partition function relies on the suppression of loops by some power of momentum. In one spatial dimension, ferromagnetic loops are suppressed by one power of momentum [1]. The corresponding Feynman graphs for the partition function up to order p5 are depicted in Fig. 1. The leading temperature-dependent contribution stems from the oneloop graph 3 which is of order p3, as it involves a vertex from L2eff (p2) and one loop (p). The one-loop diagram 5d with an insertion from L4eff is of order p5, as it involves L4eff (p4) and one loop (p). Finally, the two-loop (three-loop) diagrams are of order p4 (p5) as they involve one (two) more loops with respect to diagram 3.

C.P. Hofmann / Physica B 442 (2014) 81–89

83

The connection between the momentum–frequency representation and the real-space imaginary-time representation is given by a Fourier transform. In one spatial dimension, the thermal Matsubara propagator takes the form Z β Z Gðk; ωn Þ ¼ dx4 ð8Þ dx1 eiωn x4  ikx1 Δðx1 ; x4 Þ; 0

where

ωn ¼ 2π n=β:

Fig. 1. Feynman diagrams referring to the low-temperature expansion of the partition function of ferromagnetic spin chains up to and including order p5 . While the number 4 attached to the vertex in diagram 5d corresponds to the next-toleading-order Lagrangian, L4eff , vertices corresponding to the leading term L2eff are denoted by a filled circle. Note that loops are suppressed by one momentum power.

ð9Þ

Moreover, it is convenient to use dimensional regularization in the effective theory, since this regularization scheme respects the symmetries of the theory. The thermal propagator, regularized in the spatial dimension ds, is then given by GðxÞ ¼

1

1

∑

ð4πγ Þ

ds =2

1

d =2 n ¼  1 xns

-2

e x

=4γ xn  μHxn

Θx n ;

ð10Þ

with Again, more details on the perturbative evaluation of the partition function can be found in Section 2 of Ref. [1].

xn  x4 þ nβ:

3. Evaluation of the partition function up to three-loop order

where Δ is the Laplace operator in the regularized spatial dimension. It should not be confused with the symbol ΔðxÞ, which represents the Euclidean propagator at zero temperature. The quantities G1 and GΔ can be decomposed into two parts: a piece that does depend on temperature, and a piece that is temperature independent:

The low-temperature expansion of the partition function for the ferromagnetic spin chain in a weak magnetic field was derived in Ref. [1] up to two-loop order. The relevant diagrams 2, 3, 4 and 5d in Fig. 1 lead to the following expression for the free energy density: z ¼  ΣμH  

1 2π 1=2 γ

T 1=2

3=2

1

 μHnβ

3l3 e T 5=2 ∑ þ Oðp5 Þ; 5=2 4π 1=2 Σγ 5=2 n¼1 n

ð4Þ

where β  1=T. The contributions of order T 3=2 and T5/2 arise from the one-loop graphs and thus describe noninteracting spin waves. While the former term only depends on the leading-order effective constants Σ and F (γ ¼ F 2 =Σ , spin stiffness), the latter also involves the next-to-leading-order effective coupling l3 from L4eff . It is quite remarkable that the spin–wave interaction does not enter at the two-loop level in the above low-temperature expansion. This is because the diagram 4 of order T2 is 0 due to parity [1]. In order to discuss the impact of the spin–wave interaction, we thus have to go up to three-loop order. Note that the effective coupling constants can be expressed in terms of microscopic quantities as follows [1]: S a

Σ¼ ;

F 2 ¼ JS2 a;

γ¼

F2

Σ

¼ JSa2 ;

l3 ¼

JS2 a3 : 24

ð5Þ

Here a is the distance between two spins and J represents the exchange integral of the Heisenberg model. It is important to point out that the exchange integral J represents the natural scale of the underlying theory. So whenever we talk of low temperature or weak magnetic field, we mean that the ratios T/J and H/J are small. For reasons that will become more evident in Section 4, it is advantageous to use the real-space imaginary-time representation for the propagators, rather than to work with the momentum– frequency representation. The thermal propagator G(x) amounts to GðxÞ ¼

1

∑

n ¼ 1

Δðx1 ; x4 þ nβÞ;

Finally we use the notation

ð6Þ

where ΔðxÞ is the Euclidean zero temperature propagator referring to ferromagnetic magnons: Z dk4 dk eikx1  ik4 x4 : ð7Þ ΔðxÞ ¼ ð2π Þ2 γ k2 ik4 þ μH

GΔ  ½ΔGðxÞx ¼ 0 ;

G1  ½GðxÞx ¼ 0 ;

G1 ¼ GT1 þ G01 ;

e  μHnβ ∑ 3=2 n¼1 n

1

ð11Þ

GΔ ¼ GTΔ þ G0Δ :

The explicit expressions for meter ds, read GT1 ¼ GTΔ ¼

1

1 ð4πγ Þ

ds =2

1

ð12Þ

∑

ð13Þ GT1

T

and GΔ , regularized in the para-

e  μHnβ

; βÞds =2  1 ds e  μHnβ  : ∑ 2γ n ¼ 1 ðnβ Þds =2 þ 1

n ¼ 1 ðn



ð4πγ Þds =2

ð14Þ

The temperature-independent pieces G01 and G0Δ do not contribute to the partition function: in dimensional regularization, these expressions vanish exactly. Hence, the only contributions which matter in our evaluation are those which are temperature dependent. These are finite if the regularization is removed, i.e., if the limit ds -1 is taken. After these technical remarks, we now address the three-loop graphs. Note that they only involve vertices from the leading-order Lagrangian L2eff . Graph 5a amounts to a product of three thermal propagators (and space derivatives thereof), which have to be evaluated at the origin: z5a ¼ 

F2

Σ3

GΔ ðG1 Þ2 :

ð15Þ

The three-loop graph 5b, remarkably, does not contribute to the partition function: z5b ¼ 0:

ð16Þ

Finally, the cateye graph 5c yields z5c ¼ 

F4 2Σ

4

Iþ

F2

Σ3

GΔ ðG1 Þ2 :

The quantity I stands for the following integral over the torus: Z d þ1 ~ s G; ~ r; s ¼ x1 ; …; xds ; I ¼ d s x ∂r G∂r G∂s G∂

ð17Þ

ð18Þ

T

displaying a product of four thermal propagators, where G ¼ GðxÞ;

G~ ¼ Gð  xÞ:

ð19Þ

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C.P. Hofmann / Physica B 442 (2014) 81–89

Since the second term in Eq. (17) cancels the contribution from graph 5a, the only relevant piece at the three-loop level is the one involving the integral I. As this contribution does not just correspond to a product of thermal propagators (or derivatives thereof) to be evaluated at x¼ 0, its renormalization and numerical evaluation are much more involved. These issues are considered in detail in the following section, as well as in Appendix A and B.

spatial dimensions (see Refs. [48,46]), contributions of class C are finite, such that these integrals can be evaluated numerically without further ado – in Appendix A we provide useful representations. Still, in order to check consistency of our method, in Appendix B we proceed along the lines of Ref. [80] and show that both variants of the method yield the same result. Gathering all terms that contribute up to order p5, and writing the integral I as

4. Cateye graph: renormalization

IðsÞ ¼ T 5=2

iðsÞ

γ

7=2

s ¼ μHβ ¼

;

μH T

;

γ¼

F2

Σ

;

ð27Þ

In order to analyze potential ultraviolet divergences in the three-loop graph 5c, it is essential that we use the real-space imaginary-time representation for the propagators. We adopt the strategy outlined in Ref. [80], where the same three-loop graph was discussed within a Lorentz-invariant context. The relevant integral Z d þ1 ~ s G; ~ I ¼ d s x ∂r G∂r G∂s G∂

the final expression for the low-temperature series of the free energy density of the ferromagnetic spin chain in a weak magnetic field is

exhibits a product of four thermal propagators. We split each one of them into two parts:

The structure of this series has been investigated previously [11,12,24] within spin–wave theory. However, except for Ref. [24], the authors restricted themselves to the case of free magnons. Although an interaction correction was given in Ref. [24], this correction, as we discuss in the next section, cannot be quite correct. We emphasize that the effective Lagrangian technique is completely systematic, unlike other approaches which are plagued with approximations or ad hoc assumptions.

z ¼  ΣμH  

T

GðxÞ ¼ GT ðxÞ þ ΔðxÞ:

ð20Þ T

While the temperature-dependent piece G ðxÞ is finite in the limit ds -1, the zero-temperature propagator ΔðxÞ may lead to ultraviolet singularities. Decomposing the integral I according to Eq. (20), one obtains nine terms that can be grouped into six different classes – for simplicity the derivatives are omitted: GT ðxÞGT ðxÞΔð  xÞGT ð  xÞ;

C : Δ ðxÞG ð xÞG ð  xÞ; G ðxÞGT ðxÞΔ ð  xÞ; 2

T

T

2

T

D : ΔðxÞG ðxÞΔð  xÞG ð  xÞ; T

T

E : Δ ðxÞΔð  xÞGT ð  xÞ; ΔðxÞGT ðxÞΔ ð  xÞ; 2

2

F : Δ ðxÞΔ ð  xÞ: 2

2

ð21Þ

Note that the product ΔðxÞΔð  xÞ of zero-temperature propagators is proportional to Θðx4 ÞΘð  x4 Þ, such that terms of the classes D; E and F do not contribute. Hence we are left with the cases A, B and C. Classes A and B do not pose any problems as the corresponding integrals over the torus Z T T T T d þ1 d s x ð∂r GT ∂r GT ∂s G~ ∂s G~ þ 4∂r Δ∂r GT ∂s G~ ∂s G~ Þ; ð22Þ

P ¼ z0  z;

ð23Þ

where we have displayed the derivatives. In the limit ds -1, the zero-temperature piece ∂x1 ΔðxÞ can be written as   x2 x1 ð24Þ ∂x1 ΔðxÞ p 3=2 exp  1 : 4γ x 4 x4 On the other hand, the first term in the Taylor expansion of ∂x1 GT ð  xÞ at the origin is linear in x1: ð25Þ

Accordingly, in the limit ds -1, the contribution in the integral I takes the form !2 Z Z x1 1 2 Ip dx1 dx4 3=2 e  x1 =2γ x4 x21 p dx4 pffiffiffiffiffi; ð26Þ x4 x4 which is not singular in the ultraviolet. Unlike in two or three

ð29Þ 5

because the system is homogeneous. Up to order p , we get P ¼ η^ 0 T 3=2 þ η^ 1 T 5=2 þ Oðp6 Þ;

ð30Þ

with coefficients η^ i given by

η^ 0 ¼ η^ 1 ¼

are finite at ds ¼1. Concerning class C, let us consider the term

∂x1 GT ð  xÞ ¼ ∂x21 GT ð  xÞjx ¼ 0 x1 þ Oðx31 Þ:

ð28Þ

We now address the thermodynamic behavior of ferromagnetic spin chains, based on the representation (28) for the free energy density. We first discuss the low-temperature series for the pressure that can be obtained from the temperature-dependent part of the free energy density

T

∂x1 ΔðxÞ∂x1 ΔðxÞ∂x1 GT ð  xÞ∂x1 GT ð  xÞ;

1 e  μHnβ 3l3 1 T 5=2 ∑  iðsÞT 5=2 þ Oðp6 Þ: 2 5=2 4π 1=2 Σγ 5=2 2Σ γ 3=2 n¼1 n

5. Ferromagnetic spin chains in a magnetic field: thermodynamics

A : GT ðxÞGT ðxÞGT ð  xÞGT ð  xÞ; B : ΔðxÞGT ðxÞGT ð  xÞGT ð  xÞ;

1 e  μHnβ 1 T 3=2 ∑ 3=2 2π 1=2 γ 1=2 n¼1 n

2π 4π

e  μHnβ ; 3=2 n¼1 n 1

1

γ

1=2 1=2

∑

1=2

Σγ

5=2

¼ η^ 1 þ η^ 1 : free

e  μHnβ 1 þ iðsÞ 2 5=2 2Σ γ 3=2 n¼1 n 1

3l3

int

∑

ð31Þ

The spin–wave interaction starts manifesting itself at order p5 p T 5=2 through the three-loop contribution proportional to the dimensionless function iðsÞ. The other contributions in the pressure stem from one-loop graphs, i.e., they refer to noninteracting magnons. The spin–wave interaction is thus governed by the function iðsÞ which we have depicted in Fig. 2. Since the function iðsÞ is positive in the whole s-range, the spin–wave interaction in the pressure always is repulsive. The smaller the ratio between magnetic field and temperature, the stronger the repulsive interaction in the pressure. However, as we discuss below, for small values of s the effective expansions are only valid if the temperature is extremely low. Although we are dealing with a three-loop effect (next-to-nextto-leading order T5/2 in the pressure), the effect of the interaction is visible. This can be appreciated in Fig. 3, where we have plotted

C.P. Hofmann / Physica B 442 (2014) 81–89

85

Using the representation (30) for the pressure, we obtain 1  sn

1 e 1 1 e  sn u ¼ 1=2 1=2 T 3=2 s ∑ 1=2 þ ∑ 3=2 2n¼1 n 2π γ n¼1 n 1  sn

3l3 e 3 1 e  sn þ 1=2 5=2 T 5=2 s ∑ 3=2 þ ∑ 5=2 2n¼1 n 4π Σγ n¼1 n

1 diðsÞ 5=2 3 iðsÞ  s þ þ Oðp6 Þ; T 2 2 ds 2Σ γ 3=2 s¼

Fig. 2. The function iðsÞ representing the three-loop spin–wave interaction contribution in the low-temperature dynamics of the ferromagnetic spin chain in a weak magnetic field. The quantity s is the dimensionless parameter s ¼ μH=T.

1  sn

e 3 1 e  sn 1=2 ∑ T s ∑ þ 1=2 2 2π 1=2 γ 1=2 n3=2 n¼1 n 1  sn n ¼ 1 1  sn

3l3 e 5 e þ 1=2 5=2 T 3=2 s ∑ 3=2 þ ∑ 5=2 2n¼1 n 4π Σγ n¼1 n

1 diðsÞ 3=2 5 ið þ Oðp4 Þ; T s Þ  s þ 2 2 ds 2Σ γ 3=2

cV ¼

1



1 e  sn 1 e  sn 3 1 e  sn 1=2 2 ∑ þ T s ∑ þ s ∑ 2π 1=2 γ 1=2 n  1=2 n1=2 4 n3=2 n ¼ 1 1  sn n ¼ 1 1  sn n ¼ 1 1  sn

3l3 e e 15 e 3=2 ∑ s2 ∑ 1=2 þ 3s ∑ 3=2 þ þ 1=2 5=2 T 4 n ¼ 1 n5=2 4π Σγ n¼1 n n¼1 n ) 2 1 diðsÞ 3=2 15 2 d iðsÞ iðsÞ  3s þ þs T ð34Þ þ Oðp4 Þ: 2 4 ds ds2 2Σ γ 3=2 1

In the above series, the contributions due to the spin–wave interaction enter at order p5 p T 5=2 (p3 p T 3=2 ) for u (s; cV ). In particular, there is no interaction term of order p4 p T 2 in u, and no interaction term of order p2 p T in s and cV. This is because the two-loop diagram 4 of Fig. 1 turns out to be 0 as a consequence of parity [1]. Finally we turn to the magnetization and the susceptibility. With the representation (28) for z, the low-temperature expansion for the magnetization Fig. 3. Interaction contribution in the term of order T 5=2 in the pressure according to Eq. (32). The curves refer to S ¼ f12; 1; 32g from top to bottom in the figure.

0

1

e  sn

s¼

11

α~ 0 ¼

;

ð32Þ

T 1



for the cases S ¼ . The higher the spin S, the smaller the impact of the spin–wave interaction in the pressure. Note that we have expressed the effective constants Σ, γ and l3 in terms of microscopic quantities according to Eq. (5). We find it quite remarkable that the interaction contribution given in Eq. (31) does not involve any higher-order effective constants, but is completely determined by the zero-temperature spontaneous magnetization Σ and the spin stiffness γ ¼ F 2 =Σ that appear in the leading-order Lagrangian L2eff . The restrictions imposed by the symmetries are thus very strong in one spatial dimension. In particular, the fact that the spin–wave interaction in the pressure is repulsive follows from symmetry considerations alone. Let us derive the low-temperature series for the energy density u, entropy density s, and heat capacity cV of the ferromagnetic spin chain: 3 2; 1; 2

s¼

∂P ; ∂T

u ¼ Ts  P;

cV ¼

∂u ∂s ¼T : ∂T ∂T

ð35Þ

amounts to

α~ 1 ¼

μH

∂z ; ∂ðμHÞ

Σ ðT; HÞ ¼ 1  α~ 0 T 1=2  α~ 1 T 3=2 þ Oðp4 Þ: Σ The coefficients α~ i depend on the ratio s ¼ μH=T and read

the ratio 2 ∑n ¼ 1 5=2 η^ int B S C n þ 1A xðsÞ ¼ free 1 int ¼ @ pffiffiffiffi iðsÞ 16 π η^ 1 þ η^ 1

Σ ðT; HÞ ¼ 

ð33Þ

ð36Þ

1 e  sn 1 ∑ ; 2π 1=2 Σγ 1=2 n ¼ 1 n1=2

1 e  sn 3l3 1 diðsÞ ∑  : 2 3 4π 1=2 Σ γ 5=2 n ¼ 1 n3=2 2Σ γ 3=2 ds

ð37Þ

On the other hand, the susceptibility of the ferromagnetic spin chain

χ ðT; HÞ ¼

∂Σ ðT; HÞ ; ∂ðμHÞ

ð38Þ

takes the form

χ ðT; HÞ ¼ κ~ 0 T  1=2 þ κ~ 1 T 1=2 þOðp2 Þ;

ð39Þ

the coefficients κ~ i given by

κ~ 0 ¼

1 e  sn 1 ∑ ; 2π 1=2 γ 1=2 n ¼ 1 n  1=2

κ~ 1 ¼

1 e  sn 3l3 1 d iðsÞ ∑ þ : 4π 1=2 Σγ 5=2 n ¼ 1 n1=2 2Σ 2 γ 3=2 ds2 2

ð40Þ

Note that these series become meaningless in the limit H-0, because the leading coefficient α~ 0 in the magnetization, and the coefficients κ~ 0 and κ~ 1 in the susceptibility then diverge. Indeed, as we have discussed in detail in Section 4 of Ref. [1], it is conceptually inconsistent to take the limit s ¼ μH=T-0 (temperature fixed), because we are then outside the domain where the

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C.P. Hofmann / Physica B 442 (2014) 81–89

effective expansion presented here is valid. The point is that an energy gap is generated nonperturbatively at finite temperatures, such that the correlation length of the magnons no longer is infinite. This means (see Section 4 of Ref. [1]) that, for a given temperature, the magnetic field cannot be arbitrarily weak. Rather, the condition   μH T2 1 Z 800 2 ð41Þ S¼ J 2 J must be satisfied. In this regime, the above low-temperature expansions are on safe grounds. The restriction (41) is equivalent to   T 1 S¼ ; ð42Þ s Z 800 J 2 implying that only at very low temperatures (T=J{1), the parameter s can take small values. So while it is true that the repulsive interaction gets stronger if we approach the limit H-0 (while keeping the temperature fixed), we have to keep in mind that the temperature has to be very low. We emphasize that almost all previous investigations on ferromagnetic spin chains neglected the effect of the spin–wave interaction. Apparently, to address the impact of the spin–wave interaction in the low-temperature series with e.g. spin–wave theory up to the order considered here is very challenging. Still, there is one exception: Ref. [24] based on spin–wave theory at constant order parameter, where the interaction is discussed. The explicit expression for the magnetization derived there is   1 rffiffiffi rffiffiffi!3 ζ mðHÞ 2 pffiffi 1 t 1 1 t ¼ 1  pffiffiffiffi t  þ S 2S v 16 2S v 2S π þ Oðt; t 3=2 v  1=2 Þ;

t¼

T H ; v¼ : JS T

ð43Þ

Up to the last term, the above expansion agrees with the leading terms of our effective result (36), if we Taylor expand in the parameter s. However, the last term – the interaction correction – has no analog in the systematic effective expansion, hinting at an inconsistency. The problem is solved by noticing that the interaction correction in Eq. (43) lies outside the parameter regime where the spin–wave picture is valid. In fact, the authors of Ref. [24] say pffiffiffiffiffiffiffi that the coefficient ð2SÞ  1 t=v is close to unity. For S ¼ 12 this means v ¼t or, equivalently,   μH T2 1 ¼2 2 : ð44Þ S¼ J 2 J So the constraint (41) is not satisfied and we are clearly outside the regime where the spin–wave picture applies. Moreover, the expansion (43) appears to be inconsistent itself for the following pffiffiffiffiffiffiffi reason: the third term in Eq. (43), i.e., ð2SÞ  1 t=v, is close to  1, which is not a small correction to the magnetization mðHÞ=S. At the end of this section, we point out an important observation regarding bound states. Although the formation of twomagnon or multi-magnon bound states becomes more important in lower space dimensions [81], unfortunately, the role of bound states in the low-temperature behavior of ferromagnetic spin chains is not well explored. In the various references cited in the present work, the existence and the effect of bound states are not really addressed. On the other hand, a simple scaling argument [82] indicates that magnon bound states in ds ¼1 at most start to show up at order T5/2 in the free energy density, i.e., at the order where the spin–wave interaction sets in. To rigorously explore the effect of bound states, it would be interesting to compare the magnitude of the T5/2-coefficient in our effective expansion for the free energy density (28) with

Bethe-ansatz results or numerical simulations. In either case one could determine whether bound states start manifesting themselves at order T5/2 and, if so, which part in the T5/2-coefficient is due to the spin–waves and which part originates from the presence of magnon bound states. Unfortunately, references based on e.g. Bethe ansatz methods, providing low-temperature expansions for the domain where our series are valid (weak but not tiny or zero magnetic field), appear to be unavailable. Alternatively, one could also incorporate magnon bound states as explicit degrees of freedom into the effective Lagrangian formalism and analyze their impact on the low-temperature expansions. Work in this direction is in progress. Since it is a fact that magnon-bound states do exist in ferromagnetic spin chains, one may be confused about the statement that the spin–wave interaction in the pressure is repulsive. The point is that one has to carefully check how “attractive” or “repulsive” is defined. In the present study, we refer to the picture of a nonideal magnon gas: if the sign of the interaction correction to the pressure is positive, we call the interaction as repulsive. This does not mean that bound states cannot form. It is also important to note that the question of bound states is already relevant at zero temperature. Here we have considered the properties of the system at finite temperature and inferred the repulsive nature of the interaction from the low-temperature behavior of the pressure. We also like to point out that the positivity of the temperature coefficient does not correspond to a self-consistency check of the assumptions underlying the effective field theory construction. While in the present case of ferromagnetic spin chains the interaction coefficient is positive in the whole parameter regime of the magnetic field (see Fig. 2), this need not necessarily be so. In fact, in the case of (pseudo-)Lorentz-invariant systems with a spontaneously broken rotation symmetry O(N) -O(N 1) that live in two spatial dimensions (N ¼3 describes the antiferromagnet, N ¼2 refers to the quantum XY model, see [48,85]), the sign of the Goldstone boson interaction correction in the pressure depends on the strength of the external field: for weak fields the sign is positive (repulsive), for stronger fields it becomes negative (attractive), provided that N 4 4 [86].

6. Conclusions Whereas a rather considerable number of articles have been devoted to the impact of the spin–wave interaction in the threedimensional ideal ferromagnet over the years – pioneered by the landmark papers by Dyson [83] and Zittartz [84] – the analogous question regarding ferromagnets in two spatial dimensions or ferromagnetic spin chains has been largely ignored. The present study, as well as the preceding Refs. [45,46,1], aimed at closing this gap in the condensed matter literature. Within the framework of effective Lagrangians, here we have derived the partition function for the ferromagnetic spin chain up to three-loop order and have discussed the low-temperature series for various thermodynamic quantities in a weak magnetic field, including the magnetization and the susceptibility. In particular, we have considered the impact of the spin–wave interaction and have shown that in the free energy density, the interaction starts manifesting itself at order T5/2. While this power of temperature immediately follows from the systematic loop counting, the derivation of the corresponding coefficient required the renormalization and numerical evaluation of a specific three-loop graph which was quite elaborate. Remarkably, the coefficient of the order-T5/2 interaction term in the pressure is positive, such that the spin–wave interaction is repulsive.

C.P. Hofmann / Physica B 442 (2014) 81–89

Although various authors have also investigated the lowtemperature behavior of ferromagnetic spin chains – using, e.g., spin–wave theory, Bethe ansatz, Schwinger-boson mean field theory and yet other methods – the impact of the spin–wave interaction in the low-temperature series has been neglected; apart from Ref. [24] which we have discussed in detail. Even though the system under consideration can in principle be treated by exact methods like the Bethe ansatz, the expansions derived from these results in the literature all refer to either a tiny or zero magnetic field. As we have outlined, this is not the domain where the effective Lagrangian formalism operates. With the present paper we close a series of articles devoted to the study of the low-temperature properties of ideal ferromagnets on the basis of the effective Lagrangian method. These studies include three- and two-dimensional ideal ferromagnets as well as ferromagnetic spin chains. In either case a complete and systematic analysis of the partition function was given up to three-loop order. By being able to systematically go to higher orders in the perturbative expansion, as compared to conventional condensed matter techniques, we hope to have convinced the reader that the effective Lagrangian method is indeed a very powerful tool to analyze ferromagnetic systems.

The author would like to thank D.V. Dmitriev for correspondence.

Appendix A. Cateye graph: numerical evaluation Here we address the numerical evaluation of the three-loop graph 5c. In the relevant expression Z T T T T 2 I ¼ d x ð∂r GT ∂r GT ∂s G~ ∂s G~ þ 4∂r Δ∂r GT ∂s G~ ∂s G~ Þ T Z T T 2 þ 2 d x ∂r Δ∂r Δ∂s G~ ∂s G~ ; ðA:1Þ T

the individual terms involve the two variables r ¼ x1 and t ¼ x4 , such that 2

d x ¼ dr dt:

ðA:2Þ

We introduce the dimensionless integration variables sffiffiffi 1 T x1 : η ¼ Tx4 ; ξ ¼ 2 γ

η and ξ: ðA:3Þ

In the first two terms of Eq. (A.1), i.e. in the integrals involving quartic and triple sums, we integrate over all space, and are thus left with one-dimensional integrals in the variable η. The expression involving the quartic sum is Z 2 d x ∂r GT ðxÞ∂r GT ðxÞ∂s GT ð  xÞ∂s GT ð  xÞ ¼

¼

Z

3 32π 3=2 γ 7=2

T 5=2

1=2  1=2

dη

1

∑

n1 …n4 ¼ 1

ððη þ n1 Þðη þn2 Þð  η þ n3 Þð  η þ n4 ÞÞ3=2

;

while for the triple sum we obtain Z 2 d x ∂r ΔðxÞ∂r GT ðxÞ∂s GT ð xÞ∂s GT ð xÞ ¼

Z

3 32π

γ

3=2 7=2

T 5=2

ðηðη þn2 Þð  η þ n3 Þð  η þ n4 ÞÞ3=2

;

ðA:5Þ

with

s¼

μH T

γ¼

;

F2

Σ

:

ðA:6Þ

The quantities Q^ ðη; n1 ; n2 ; n3 ; n4 Þ and Q^ ðη; 0; n2 ; n3 ; n4 Þ depend in a nontrivial way on the summation variables. The remaining integral of Eq. (A.1) which involves double sums is Z 2 d x ∂r ΔðxÞ∂r ΔðxÞ∂s GT ð  xÞ∂s GT ð  xÞ T

¼

1 4π 2 γ

T 5=2 7=2

1



∑

n1 ;n2 ¼ 1

Z

1=2 0

dη

Z

1 0

dξ ξ

4

e  sðn1 þ n2 Þ P^ ðξ; η; n1 ; n2 Þ;

e P^ ðξ; η; n1 ; n2 Þ ¼

 ξ ð2=η þ 1=ð  η þ n1 Þ þ 1=ð  η þ n2 ÞÞ 2

fη2 ð  η þ n1 Þð  η þ n2 Þg

3=2

:

ðA:7Þ

1=2 0

dη

1

∑

n2 …n4 ¼ 1

e  sðn2 þ n3 þ n4 Þ

In one spatial dimension, contributions of class C in the integral (18) do not contain ultraviolet singularities and the numerical integration over the torus can be performed directly as outlined in the preceding appendix. Still, we also want to follow the method described in Ref. [80], which provides us with an alternative decomposition of the cateye graph and therefore serves as a consistency check. First a sphere S of radius jSjr β =2 around the origin is cut out. The integral which involves the contributions of class C is written as Z T T d þ1 d s x ∂r Δ∂r Δ∂s G~ ∂s G~ T Z T T d þ1 ¼ d s x ∂r Δ∂r Δ∂s G~ ∂s G~ S

Z

þ T \S

d

ds þ 1

T T x ∂r Δ∂r Δ∂s G~ ∂s G~ :

ðB:1Þ

In the integral over the sphere, we then subtract the term (26) Z d þ1 d s x ∂r ΔðxÞ∂r ΔðxÞ∂s GT ð  xÞ∂s GT ð  xÞ S Z d þ1 ¼ d s x ∂r ΔðxÞ∂r ΔðxÞQ ss ðxÞ S Z d þ1 þ d s x ∂r ΔðxÞ∂r ΔðxÞ∂αs GT ð  xÞjx ¼ 0 S

 ∂βs GT ð xÞjx ¼ 0 xα xβ ;

ðB:2Þ

where the quantity Qss(x) is given by Q ss ðxÞ ¼ ∂s GT ð  xÞ∂s GT ð  xÞ

e  sðn1 þ n2 þ n3 þ n4 Þ

 Q^ ðη; n1 ; n2 ; n3 ; n4 Þ; Q^ ðη; n1 ; n2 ; n3 ; n4 Þ    5=2 1 1 1 1 þ þ þ η þ n1 η þ n2  η þ n3  η þ n4

T

 Q^ ðη; 0; n2 ; n3 ; n4 Þ; Q^ ðη; 0; n2 ; n3 ; n4 Þ    5=2 1 1 1 1 þ þ þ η η þ n2  η þ n3  η þ n4

Appendix B. Alternative decomposition of the cateye graph

Acknowledgments

T

¼

87

 ∂αs GT ð  xÞjx ¼ 0 ∂βs GT ð  xÞjx ¼ 0 xα xβ :

ðA:4Þ

ðB:3Þ

Finally we decompose the second integral on the right hand side of Eq. (B.2) according to Z d þ1 d s x ∂r ΔðxÞ∂r ΔðxÞ∂αs GT ð  xÞjx ¼ 0 ∂βs GT ð  xÞjx ¼ 0 S Z d þ1  xα xβ ¼ d s x ∂r ΔðxÞ∂r ΔðxÞ∂αs GT ð  xÞjx ¼ 0 R Z d þ1  ∂βs GT ð  xÞjx ¼ 0 xα xβ  d s x ∂r ΔðxÞ∂r ΔðxÞ R\S

 ∂αs GT ð xÞjx ¼ 0 ∂βs GT ð  xÞjx ¼ 0 xα xβ :

ðB:4Þ

88

C.P. Hofmann / Physica B 442 (2014) 81–89

the method sketched here yields the same numerical results as the one outlined in Appendix A.

The integral over all Euclidean space takes the form Z d þ1 d s x ∂r ΔðxÞ∂r ΔðxÞ∂αs GT ð  xÞjx ¼ 0 ∂βs GT ð  xÞjx ¼ 0 R

ds ðds þ2Þ T ds þ 2 ðμHÞðds  2Þ=2 23ds þ 5 π 3ds =2 γ ð3ds þ 4Þ=2 ( )2   1 e  μHnβ ds :  ∑ ðd þ 2Þ=2 Γ 1  s 2 n¼1n  xα xβ ¼

References ðB:5Þ

The above regularized expression is not divergent in the limit ds -1, and reads 1  sn 2 3 e 5=2 p ffiffiffi ffi T ∑ 3=2 : ðB:6Þ 256πγ 7=2 s n¼1 n The integral I finally amounts to Z T T T T 2 I ¼ d x ð∂r GT ∂r GT ∂s G~ ∂s G~ þ 4∂r Δ∂r GT ∂s G~ ∂s G~ Þ T Z Z T T 2 2 þ2 d x ∂r Δ∂r Δ∂s G~ ∂s G~ þ2 d x ∂r Δ∂r ΔQ ss Z

T \S

[16] [17]

S

d x ∂r Δ∂r Δ∂αs GT ð  xÞjx ¼ 0 ∂βs GT ð  xÞjx ¼ 0 xα xβ 2

2 R\S

þ



3 pffiffiffiffiT 5=2 128πγ 7=2 s

 sn 2

1

∑

n¼1

e n3=2

[18] [19]

:

ðB:7Þ

The numerical evaluation of this expression poses no problems. The following representations for the three integrals in Eq. (B.7) which involve double sums have been used in our numerical evaluation: Z 2 d x ∂r ΔðxÞ∂r ΔðxÞ∂s GT ð  xÞ∂s GT ð  xÞ T \S

¼

1 4π γ

2 7=2

Z T 5=2

S

dη

0

Z

1

dξξ pffiffiffiffiffiffiffiffiffiffiffi 2

1

4

S  η2

∑

n1 ;n2 ¼ 1

e  sðn1 þ n2 Þ

 P^ ðξ; η; n1 ; n2 Þ;  ξ ð2=η þ 1=ð  η þ n1 Þ þ 1=ð  η þ n2 ÞÞ

e P^ ðξ; η; n1 ; n2 Þ ¼  Z

2

η

2ð 

η þ n1 Þð  η þ n2 Þ

3=2 ;

ðB:8Þ

d x ∂r ΔðxÞ∂r ΔðxÞQ ss ðxÞ 2

S

¼

1 4π 2 γ 7=2

Z T

S

5=2 0

dη

2 Z pSffiffiffiffiffiffiffiffiffiffiffi  η2

0

dξξ

1

4

∑

n1 ;n2 ¼ 1

e  sðn1 þ n2 þ 2ηÞ

 Q^ ðξ; η; n1 ; n2 ; sÞ;

ðB:9Þ

with 2 Q^ ðξ; η; n1 ; n2 ; sÞ ¼ e  ξ ð2=η þ 1=ð  η þ n1 Þ þ 1=ð  η þ n2 ÞÞ

e2ηs fð  η þ n1 Þð  η þ n2 Þg



3=2



eξ

2

ð1=ð  η þ n1 Þ þ 1=ð  η þ n2 ÞÞ 3=2 3=2

n1 n2

η

3

;

ðB:10Þ

and finally, Z 2 d x ∂r ΔðxÞ∂r ΔðxÞ∂sα GT ð  xÞjx ¼ 0 xα ∂sβ GT ð xÞjx ¼ 0 xβ R\S Z 1 Z 1 1 1 4 ^ ξ; η; n1 ; n2 Þ ¼ 2 7=2 T 5=2 dη dξξ ∑ e  sðn1 þ n2 þ 2ηÞ Rð 4π γ S 0 n1 ;n2 ¼ 1 þ

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

1 T 5=2 4π 2 γ 7=2

Z 0

^ ξ; η; n1 ; n2 Þ ¼ Rð

S

Z 1 4 dη pffiffiffiffiffiffiffiffiffiffiffi dξξ 2 S η

2

2 e  2ξ = η

fη2 n1 n2 g

3=2

:

1

∑

n1 ;n2 ¼ 1

^ ξ; η; n1 ; n2 Þ; e  sðn1 þ n2 þ 2ηÞ Rð

ðB:11Þ

Note that the result for the function I must be independent of the radius of the sphere S – this provides us with a welcome consistency check on the numerics. Moreover, we have verified that

[20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68]

C.P. Hofmann, Phys. Rev. B 87 (2013) 184420. M. Takahashi, Prog. Theor. Phys. 46 (1971) 401. M. Takahashi, Prog. Theor. Phys. 50 (1973) 1519. P. Schlottmann, Phys. Rev. Lett. 54 (1985) 2131. M. Takahashi, M. Yamada, J. Phys. Soc. Jpn. 54 (1985) 2808. M. Yamada, M. Takahashi, J. Phys. Soc. Jpn. 55 (1986) 2024. P. Schlottmann, Phys. Rev. B 33 (1986) 4880. K. Lee, P. Schlottmann, Phys. Rev. B 36 (1987) 466. M. Yamada, J. Phys. Soc. Jpn. 59 (1990) 848. X.-W. Guan, M.T. Batchelor, M. Takahashi, Phys. Rev. A 76 (2007) 043617. M. Takahashi, Prog. Theor. Phys. Suppl. 87 (1986) 233. M. Takahashi, Phys. Rev. Lett. 58 (1987) 168. J. Sirker, M. Bortz, Phys. Rev. B 73 (2006) 014424. A. Auerbach, D.P. Arovas, J. Appl. Phys. 67 (1990) 5734. A. Auerbach, D.P. Arovas, in: Z. Tesanovich (Ed.), Field Theories in Condensed Matter Physics, Addison-Wesley, 1990, pp. 1–25. J. Kondo, K. Yamaji, Prog. Theor. Phys. 47 (1972) 807. L.S. Campana, A. Caramico D'Auria, U. Esposito, G. Kamieniarz, Phys. Rev. B 39 (1989) 9224. F. Suzuki, N. Shibata, C. Ishii, J. Phys. Soc. Jpn. 63 (1994) 1539. M. Hamedoun, Y. Cherriet, A. Hourmatallah, N. Benzakour, Phys. Rev. B 63 (2001) 172402. I. Junger, D. Ihle, J. Richter, A. Klümper, Phys. Rev. B 70 (2004) 104419. T.N. Antsygina, M.I. Poltavskaya, I.I. Poltavsky, K.A. Chishko, Phys. Rev. B 77 (2008) 024407. I. Juhász Junger, D. Ihle, L. Bogacz, W. Janke, Phys. Rev. B 77 (2008) 174411. M.-W. Liu, Y. Chen, C.-C. Song, Y. Wu, H.-L. Ding, Solid State Commun. 151 (2011) 503. M. Kollar, I. Spremo, P. Kopietz, Phys. Rev. B 67 (2003) 104427. P. Kopietz, Phys. Rev. B 40 (1989) 5194. H. Nakamura, M. Takahashi, J. Phys. Soc. Jpn. 63 (1994) 2563. H. Nakamura, N. Hatano, M. Takahashi, J. Phys. Soc. Jpn. 64 (1995) 1955. H. Nakamura, N. Hatano, M. Takahashi, J. Phys. Soc. Jpn. 64 (1995) 4142. N. Read, S. Sachdev, Phys. Rev. Lett. 75 (1995) 3509. M. Takahashi, H. Nakamura, S. Sachdev, Phys. Rev. B 54 (1996) R744. S. Sachdev, Quantum Phase Transitions, Cambridge University Press, New York, 2006. J.J. Cullen, D.P. Landau, Phys. Rev. B 27 (1983) 297. J.W. Lyklema, Phys. Rev. B 27 (1983) 3108. S. Kadowaki, A. Ueda, Prog. Theor. Phys. 75 (1986) 451. Y.C. Chen, H.H. Chen, F. Lee, Phys. Lett. A 130 (1988) 257. T. Delica, H. Leschke, Physica A 168 (1990) 736. A.W. Sandvik, R.R.P. Singh, D.K. Campbell, Phys. Rev. B 56 (1997) 14510. S.-J. Gu, N.M.R. Peres, Y.-Q. Li, Eur. Phys. J. B 48 (2005) 157. C. Zhou, C.P. Enz, Physica C 170 (1990) 119. V.I. Yukalov, S. Gluzman, Physica A 273 (1999) 401. A. Cuccoli, V. Tognetti, P. Verrucchi, R. Vaia, Phys. Rev. B 62 (2000) 57. A. Ceulemans, S. Cojocaru, L.F. Chibotaru, Eur. Phys. J. B 21 (2001) 511. D.V. Dmitriev, V.Ya. Krivnov, Phys. Rev. B 73 (2006) 024402. D.V. Dmitriev, V.Ya. Krivnov, Phys. Rev. B 86 (2012) 134407. C.P. Hofmann, Phys. Rev. B 86 (2012) 054409. C.P. Hofmann, Phys. Rev. B 86 (2012) 184409. C.P. Hofmann, Phys. Rev. B 65 (2002) 094430. C.P. Hofmann, Phys. Rev. B 84 (2011) 064414. C.P. Hofmann, Phys. Rev. B 60 (1999) 406. H. Leutwyler, Phys. Rev. D 49 (1994) 3033. C.P. Hofmann, Phys. Rev. B 60 (1999) 388. J.M. Román, J. Soto, Int. J. Mod. Phys. B 13 (1999) 755. J.M. Román, J. Soto, Ann. Phys. 273 (1999) 37. J.M. Román, J. Soto, Phys. Rev. B 62 (2000) 3300. C.P. Hofmann, AIP Conf. Proc. 623 (2002) 305. C.P. Hofmann, J. Phys.: Conf. Ser. 287 (2011) 012018. P. Hasenfratz, H. Leutwyler, Nucl. Phys. B 343 (1990) 241. P. Hasenfratz, F. Niedermayer, Phys. Lett. B 268 (1991) 231. P. Hasenfratz, F. Niedermayer, Z. Phys. B 92 (1993) 91. C.P. Hofmann, Phys. Rev. B 81 (2010) 014416. C.P. Hofmann, AIP Conf. Proc. 1361 (2011) 257. F. Kämpfer, M. Moser, U.-J. Wiese, Nucl. Phys. B 729 (2005) 317. C. Brügger, F. Kämpfer, M. Moser, M. Pepe, U.-J. Wiese, Phys. Rev. B 74 (2006) 224432. C. Brügger, F. Kämpfer, M. Pepe, U.-J. Wiese, Eur. Phys. J. B 53 (2006) 433. C. Brügger, C.P. Hofmann, F. Kämpfer, M. Pepe, U.-J. Wiese, Phys. Rev. B 75 (2007) 014421. C. Brügger, C.P. Hofmann, F. Kämpfer, M. Moser, M. Pepe, U.-J. Wiese, Phys. Rev. B 75 (2007) 214405. C. Brügger, C.P. Hofmann, F. Kämpfer, M. Pepe, U.-J. Wiese, Physica B 403 (2008) 1447. F.-J. Jiang, F. Kämpfer, C.P. Hofmann, U.-J. Wiese, Eur. Phys. J. B 69 (2009) 473.

C.P. Hofmann / Physica B 442 (2014) 81–89 [69] C. Brügger, C.P. Hofmann, F. Kämpfer, M. Moser, M. Pepe, U.-J. Wiese, AIP Conf. Proc. 1116 (2009) 356. [70] F. Kämpfer, B. Bessire, M. Wirz, C.P. Hofmann, F.-J. Jiang, U.-J. Wiese, Phys. Rev. B 85 (2012) 075123. [71] N.D. Vlasii, C.P. Hofmann, F.-J. Jiang, U.-J. Wiese, Phys. Rev. B 86 (2012) 155113. [72] U. Gerber, C.P. Hofmann, F. Kämpfer, U.-J. Wiese, Phys. Rev. B 81 (2010) 064414. [73] U.-J. Wiese, H.P. Ying, Z. Phys. B 93 (1994) 147. [74] U. Gerber, C.P. Hofmann, F.-J. Jiang, M. Nyfeler, U.-J. Wiese, J. Stat. Mech.: Theory Exp. (2009) P03021. [75] F.-J. Jiang, U.-J. Wiese, Phys. Rev. B 83 (2011) 155120. [76] U. Gerber, C.P. Hofmann, F.-J. Jiang, G. Palma, P. Stebler, U.-J. Wiese, J. Stat. Mech.: Theory Exp. (2011) P06002.

89

[77] S. Weinberg, Physica A 96 (1979) 327. [78] J. Gasser, H. Leutwyler, Ann. Phys. 158 (1984) 142; J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465. [79] H. Leutwyler, Ann. Phys. 235 (1994) 165. [80] P. Gerber, H. Leutwyler, Nucl. Phys. B 321 (1989) 387. [81] F. Keffer, Spin waves, in: S. Flügge, H.P.J. Wijn (Eds.), Encyclopedia of Physics— Ferromagnetism, vol. 18-2, Springer, Berlin, 1966, p. 1. [82] D.V. Dmitriev, private communication. [83] F.J. Dyson, Phys. Rev. 102 (1956) 1217; F.J. Dyson, Phys. Rev. 102 (1956) 1230. [84] J. Zittartz, Z. Phys. 184 (1965) 506. [85] C.P. Hofmann, J. Stat. Mech.: Theory Exp. (2014) P02006. [86] C.P. Hofmann, in preparation.