Magnon Energy Renormalization and Low-Temperature Thermodynamics of O(3) Heisenberg Ferromagnets
Slobodan M. Radosevic, Milan R. Pantic, Milica V. Pavkov-Hrvojevic, Darko V. Kapor
aa r X i v : . [ c ond - m a t . s t r- e l ] J un Magnon Energy Renormalization and Low-Temperature Thermodynamics of O(3)Heisenberg Ferromagnets
Slobodan M. Radoˇsevi´c, ∗ Milan R. Panti´c, Milica V. Pavkov-Hrvojevi´c, and Darko V. Kapor
Department of Physics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovi´ca 4, Novi Sad, Serbia
We present the perturbation theory for lattice magnon fields of D -dimensional O(3) Heisenbergferromagnet. The effective Hamiltonian for the lattice magnon fields is obtained starting fromthe effective Lagrangian, with two dominant contributions that describe magnon-magnon interac-tions identified as a usual gradient term for the unit vector field and a part originating in theWess-Zumino-Witten term of effective Lagrangian. Feynman diagrams for lattice scalar fields withderivative couplings are introduced, on basis of which we investigate the influence of magnon-magnoninteractions on magnon self-energy and ferromagnet free energy. We also comment appearance ofspurious terms in the low-temperature series for the free energy by examining magnon-magnoninteractions and internal symmetry of the effective Hamiltonian (Lagrangian). PACS numbers: 75.30.DS,75.10.Jm,11.10.Wx
I. INTRODUCTION
Effective field theory (EFT) is well established methodfor treating models exhibiting spontaneous symmetrybreaking [1] and is applicable to low-energy part of anysystem whose only massless excitations are Goldstonebosons [2]. Initially developed for the description oflow-energy sector of quantum chromodynamics (QCD),where it is known by the name of the chiral perturba-tion theory [2–5], EFT was also adapted for the con-densed matter problems [1, 6, 7]. In particular, an appli-cation of EFT to Heisenberg ferromagnet (HFM) has metwith considerable success [8–13]. As it is well known, theground state | i of a HFM is determined by a preferreddirection in the internal space, singled out by the totalspin S = P n S n and small fluctuations of the order pa-rameter near the ground state are described by Goldstonebosons (magnons) of spontaneously broken (spin) rota-tional symmetry. In the low dimensional ( D = 1 , D being the dimensionality of spatial lattice) isotropic fer-romagnets with short-range interactions, rotational sym-metry of the Heisenberg Hamiltonian is restored at finitetemperatures [14] and spontaneous symmetry breakingis possible only if D ≥
3. (Henceforth we will alwaysassume D ≥ n GB ), as well as the num-ber of Goldstone fields ( π a , a = 1 , , . . . n GB ), equalsthe number of broken symmetry generators ( n BS ), i.e. n GB = n BS = dim(G) − dim(H). Here G denotes spon-taneously broken (internal) symmetry group of the un-derlying system and H is the symmetry group of theground state. Even though the symmetry breaking pat-tern in HFM is O(3) → O(2), the excitation spectrumcontains only one type of magnon. This is related tothe fact that ferromagnetic magnons possess nonrelativis-tic dispersion ω ∝ k due to nonzero vacuum expecta- ∗ Electronic address: [email protected] tion values of charge densities [6, 15, 16], and a complexfield ψ ∝ π + i π describes a single particle [1, 6, 9–12, 17]. In other words, π and π represent canonicallyconjugate variables and not two distinct Goldstone fields[18]. A general theorem on SSB in Lorentz-noninvariantsystems [15] asserts that twice the number n BS − n GB equals rank of the matrix ρ , defined by its elements ρ ij = lim V →∞ ( − i /V ) h | [ Q i , Q j ] | i , where V denotes thespatial volume of the system and { Q i } is the set of bro-ken generators (integrals of charge densities). If, as usual,the spontaneous magnetization aligns in the direction ofpositive z axis, one finds ρ ∝ diag[1 , −
1] and correspond-ing single ferromagnetic magnon. This is in accordancewith standard spin-wave theory. The theorem was re-cently proved in [19] using EFT (see also related workin [15, 16, 18, 20–22]) demonstrating once again useful-ness of the effective Lagrangian method in the theorieswithout Lorentz invariance.On the other hand, the thermodynamic properties offerromagnets are usually calculated by some variant ofspin-wave theory. The predictions of linear spin-wavetheory (LSWT) are reliable almost up to T C / T C de-notes the Curie temperature), but for quantitative de-scription beyond this temperatures one needs to incor-porate the effects of magnon-magnon interactions. Asuccessful theory of the spin-wave interactions in Heisen-berg ferromagnets was put forward by Dyson [23, 24].He had shown that the kinematical interaction, aris-ing from the limitation on the maximum number (2 S )of the spin deviations on each lattice site, may safelybe ignored at temperatures not to close to T C . Dysonalso demonstrated the weakness of dynamical magnon-magnon interaction by calculating first order correctionto the free energy and spontaneous magnetization of 3DHFM, thus providing an explanation for the success ofLSWT. The weakness of magnon-magnon interactionsreflects itself through the changes in the Bloch’s law.The first correction due to magnon-magnon interactionsis only of order T , compared to the leading term pro-portional to T / . Dyson’s results were subsequentlyrederived using Holstein-Primakoff bosons [25] and thediagram technique for spin operators [26, 27]. (See [28–31] for a comprehensive reviews and list of original refer-ences.) The discovery of high-temperature superconduc-tivity (HTSC) revived interest in the Heisenberg mag-nets. Since the mid of 1980-ties, a lot of work was putin the understanding of spin-wave interactions in sys-tems of localized spins. Theoretical constructions fromthis period, dealing explicitly with the Heisenberg ferro-magnets, include the modified spin wave theory (MSWT,see [32–34]), the large N expansion of SU( N ) Heisenbergmodels and Schwinger boson mean field theory (SBMFT,see [35–37]), the self-consistent spin wave theory [38]and renormalization group (RG) methods [39, 40]. Asin the earlier works [25–27, 41], the authors of [32–38]had shown that a realistic description of the low temper-ature phase of HFM can be reached using bosonic (orcombined bosonic-fermionic [38]) representations of thespin operators within quartic approximation, or with thehelp of appropriate mean field/random phase approxi-mations (MFA/RPA), without complicated mathemat-ical constructions of Dyson. As an alternative to bo-son/fermion Hamiltonians, obtained from one of manyrepresentation of spin operators [42], several authors de-veloped the method of double time temperature Green’sfunctions (TGF). (A recent review and original referencescan be found in [43].) It is based on the equations of mo-tion for spin operators, which are turned into a solvablesystem of algebraic equations by suitable linearization.Known as the decoupling schemes in the language of thedouble time TGFs, the linearizations incorporate effectsof magnon-magnon interactions without any direct ref-erence to the nonlinear boson/fermion Hamiltonian, i.e.to the magnon-magnon interaction operator. One of themost frequently used approximations of this kind is theone by Tyablikov (TRPA, see [43]), usually described asthe one in which correlations between S z and S ± op-erators from adjacent sites are neglected. Magnon en-ergy renormalization, a consequence of Tyablikov’s ap-proximation, affects the low-temperature regime of thetheory. The low-temperature expansion of ferromagneticorder parameter for 3D lattice calculated in TRPA con-tains so called spurious term ∝ T , in disagreement withrigorous results of Dyson. Despite this, TRPA yieldsreliable predictions in accordance with Mermin-Wagnertheorem and closed system of equations for correlationfunctions is often tractable within standard numericaltools. Also, unique solution for critical temperature withself-consistently determined parameters agrees well withMonte Carlo simulations and experimental values [43].This aspect of true self-consistency comes to be impor-tant when real compounds are modeled by Heisenbergferromagnet/antiferromagnet (see e.g. [44–49] for an ap-plication of TRPA to cuprates, iron pnictides and man-ganites).The standard theories of non-linear spin waves men-tioned in the previous paragraph are based on bo-son/fermion representations of spin operators. Since thecommutation relations for spin operators and the dynam-ics of the spin system are fully satisfied only with exactboson/fermion Hamiltonian and corresponding Hilbertspace, theories of non-linear spin waves are sensitive toany form of approximation. These include, e.g, variousmean-field approximations in the Heisenberg spin Hamil-tonian [26, 27, 35, 36], or approximate boson/fermion ex- pressions for spin operators [25, 38]. Similar remark holdsfor the theories based on the equations of motion for thespin operators where the approximations are made in thecommutator for S ± operators (see [50, 51] and the sec-tion II). All these simplifications basically alter the spinnature of ( S ± , S z ) operators in a manner that may notbe obvious within a given framework. These problemsdo not arise in the EFT approach, since one works withtrue magnon operators from the beginning. All simpli-fications are directly related to the piece of Lagrangian(Hamiltonian) describing magnon-magnon interactions.This makes the influence of approximation more trans-parent.To the best of our knowledge, the perturbation theorywith lattice regularization has not yet been applied to theEFT of a ferromagnet. There are several reasons for us-ing a lattice within Hamiltonian formalism. First, unlikedimensional regularization, frequently used within EFTframework [2, 10–13], the lattice regularization preservesfull discrete symmetry of the original Heisenberg Hamil-tonian and it seems to be an appropriate method to dealwith system initially defined on a lattice. Second, wewill address to some issues inaccessible to the continuumfield theoretical methods of [9–13, 15–22], such as the in-fluence of interactions on magnon energy renormalizationover the entire Brilouin zone. Further, it is the structureof interacting Hamiltonian for ferromagnetic magnons,rather than the general form of interacting Lagrangian,that reveals certain simplifications in the diagrammaticcalculation of the magnon self-energy and free energy ofO(3) HFM. Although it lacks some of the systematiza-tion capabilities of continuum field theoretical approach,the lattice regularized theory can provide us with a use-ful information not just about spin systems but also onother standard techniques. For example, by examiningmagnon mass renormalization in sections IV and VI, wereach a clear explanation for spurious T term in Tyab-likov RPA.The section II contains brief discussion on LSWT, themagnon mass renormalization and its influence on thespontaneous magnetization. Some notation on the lat-tice theory, such as the lattice Laplacian, are likewiseintroduced there. The effective interaction Hamiltonianof lattice magnons is derived in the Section III, startingfrom the effective Lagrangian. The Feynman diagramswith colored propagators and vertices, suitable for theo-ries of lattice scalar fields with derivative couplings arealso defined in the Section III. Two-loop perturbationtheory for lattice magnon self-energy is presented in theSection IV, while three-loop analysis of the free energyis given in the Section V. Results of Section VI, basedon continuum field theoretic calculation supplement andclarify findings of two preceding Sections. An importantfeature of the effective Hamiltonian is identification ofthe two types of magnon-magnon interaction different inorigin. The careful discussion in sections III–VI offersa new answer for appearance of the spurious terms inthe low-temperature series and demonstrates the influ-ence of spin-rotation symmetry on the thermodynamicproperties of O(3) HFM. Finally, some calculation de-tails and an alternative formulation of the O(2) model ofthe Subsection IV D are collected in the Appendices. II. PRELIMINARY DISCUSSION
In this section we are motivating approach to be dis-cussed in detail latter. Also, for the clarity of presenta-tion, we find it convenient to introduce some notation onlattice fields before general perturbation theory.First, it is instructive to rewrite the Hamiltonian ofHeisenberg ferromagnet with nearest neighbor interac-tion J on a D dimensional lattice H = − J X n , λ S n · S n + λ , (1)in terms of the discrete Laplacian H = − JZ | λ | D X x S x · ∇ S x − JS ( S + 1) Z N . (2)Here [ S i x , S j y ] = i ǫ ijk S k x ∆( x − y ), S = S ( S + 1), thelattice Laplacian is (See e.g. [52]) ∇ φ ( x ) = 2 DZ | λ | X λ h φ ( x + λ ) − φ ( x ) i , (3) { λ } are the vectors that connect given site x with its Z nearest neighbors and N is the total number of lat-tice sites. As we are mainly interested in finite tem-peratures, imaginary time formalism is used through-out the paper (unless otherwise stated). Employing − ∂ τ S j ( n , τ ) = [ S j ( n , τ ) , H ] , j = 1 , ,
3, equation of mo-tion for S ( x , τ ) is found to be (imaginary time argumentsare suppressed) − ∂ τ S ( x ) = − i2 JZ | λ | D h(cid:16) ∇ S ( x ) (cid:17) × S ( x ) − S ( x ) × ∇ S ( x ) i . (4)Eq. (4) is just the lattice version of imaginary timeLandau-Lifshitz equation for operators S ( x , τ ). It canbe solved in a linear approximation. Assuming the longrange order (LRO), we may set S z ( n ) ≈ S [83]. Inthis approximation, equation of motion for S + ( x ) = S x ( x ) + i S y ( x ) takes the form of the imaginary timeequation for Schr¨odinger field on the lattice − ∂ τ S + ( x ) = − m LSW ∇ S + ( x ) , (5)where we have defined m LSW = 2 D JSZ | λ | . (6)Similar equation holds for S − ( x ) = S x ( x ) − i S y ( x ). Si-multaneously, the linearized commutation relations for S ± operators read (cid:20) S + ( x ) √ S , S − ( y ) √ S (cid:21) = ∆( x − y ) . (7) Comparing (7) to the usual form of equal-time commu-tation relations, [ ψ ( x ) , ψ † ( y )] = v − ∆( x − y ), where v denotes volume of the primitive cell, we see that in thisapproximation the Heisenberg ferromagnet is describedby the bosonic lattice Schr¨odinger fields ψ ( x , τ ) = S + ( x , τ ) √ Sv , ψ † ( x , τ ) = S − ( x , τ ) √ Sv , (8)which annihilate and create magnons at lattice site x ,respectively. Schr¨odinger field interpretation can be fur-ther justified by solving equation (5) and constructing adiagonal Hamiltonian. Finding plane-wave solutions [53]of (5) ψ ( x , τ ) = Z k a k e i k · x − ω ( k ) τ , Z k ≡ Z IBZ d D k (2 π ) D (9)and using eigenvalues of the lattice Laplacian ∇ exp[i k · x ] = − D | λ | [1 − γ D ( k )] exp[i k · x ] (10) ≡ − b k exp[i k · x ] , γ D ( k ) = Z − X { λ } exp[i k · λ ] , we find the magnon dispersion ω LSW ( k ) = b k m LSW = JZ S [1 − γ D ( k )] , (11)and diagonal magnon Hamiltonian in the linear approx-imation H = − v m LSW X x ψ † ( x ) ∇ ψ ( x ) − E (12)= V Z k ω LSW ( k ) n k − E , E = JZ N S , where V n k = a † k a k and V = (2 π ) D δ ( k − k ) = N v . a k and a † k are standard bosonic operators obeying commu-tation relations [ a k , a † q ] = (2 π ) D δ ( k − q ). Operating onthe vacuum | i , a † p creates one-magnon state a † p | i = | p i .These states are normalized as h p | q i = (2 π ) D δ ( p − q ) [6].We may now identify m LSW as (bare) mass of the latticefield quanta i. e. magnons. We shall continue to refer to m LSW as a magnon mass because of the nonrelativisticform of the dispersion relation (11), even though ferro-magnetic magnons are ”massless” from the point of viewof the Goldstone theorem.The diagonal Hamiltonian makes thermodynamicalproperties of a ferromagnet trivial to calculate. For ex-ample, at low temperatures, the spontaneous magnetiza-tion per lattice site is found to be a vacuum expectationvalue of S z x = q S ( S + 1) − [ S x x ] − [ S y x ] ≈ S − S − x S + x S = S − v ψ † ( x ) ψ ( x ) . (13)Written in terms of the thermal propagator forSchr¨odinger field D ( x − y , τ x − τ y ) = h T (cid:8) ψ ( x , τ x ) ψ † ( y , τ y ) (cid:9) i (14)= 1 β ∞ X n = −∞ Z q e i q · ( x − y ) − i ω n ( τ x − τ y ) ω ( k ) − i ω n , it is h S z i = S − v Z q h n q i = S − v D (0) . (15)Here D (0) denotes the propagator evaluated at the origin, h n q i is the free-magnon Bose distribution and we haveused the sum rule [53] β − P n [ ω ( p ) − i ω n ] − = h n p i .Results (6)-(12), which define the lattice theory of freeSchr¨odinger field, are easily seen to be those of standardlinear spin waves (LSW).The question of how to incorporate the effects ofmagnon-magnon interactions into equations like (15) haslong history and long list of answers. They are grouped inseveral categories as described in the Introduction. Theprimary goal of the present paper is to show that thermo-dynamic properties of a D dimensional O(3) Heisenbergferromagnet may be calculated within formalism of in-teracting lattice Schr¨odinger field, based on the effectiveLagrangian. We will show, e.g., that the spontaneousmagnetization of O(3) HFM to the first order in 1 /S ,can be written as S − v G (0), where G (0) is the magnonfield Green’s function calculated to the one loop. Also, inSections III–VI we develop the perturbation theory ca-pable for calculating both micro and macro properties ofO(3) HFM.Returning to the magnon dispersion, it is easily seenthat approximation S z ( n ) ≈ h S z i in (4) eventually leadsto TRPA result with the magnon energies ω RPA ( k ) = b k m RPA , m
RPA = 2 D J h S z i Z | λ | . (16)As happens in LSWT, the final result in TRPA containsno information about the short range fluctuations (SRF)of the order parameter if the operator S z ( n ) is replacedwith the site independent average h S z i . (A discussionabout the role of SRF can be found in a recent review[54].) In spite of that, TRPA incorporates certain typeof magnon-magnon interaction that renormalizes magnonmass according to (16). Since the approximation is madedirectly in the equation of motion, an explicit form ofmagnon-magnon interactions that yield (16) can’t be de-duced in TGF formalism. Tyablikov’s result (16) forHFM was subsequently re-derived by linearizing the com-mutation relations for Fourier components of S ± n opera-tors [50, 51] similarly as in this section, using the per-turbation theory for self-consistent mean field approxi-mation [26, 27], various diagram techniques for spin op-erators [28, 55], drone-fermion for S = 1 / S = 1[56] and pseudofermion representation for spin S = 1 / T term of Tyablikov. III. EFFECTIVE INTERACTION OF THELATTICE MAGNON FIELDS
We have seen in the previous section that the LSWTdescription of HFM is equivalent to a theory of free lat-tice Schr¨odinger field. The rest of the present paperwill be devoted to the influence of magnon-magnon in-teractions on microscopic and macroscopic properties ofa ferromagnet. The simplest choice of the interactionfor the Schr¨odinger field, with the Hamiltonian density ∝ [ ψ † ( x ) ψ ( x )] simply won’t work because the verticesof [ ψ † ( x ) ψ ( x )] interaction carry no momentum, so itcan not renormalize the mass of a ferromagnetic magnon.The correct form of the effective interaction is most easilyformulated in terms of the Goldstone fields π a ( x ). A. Effective Lagrangian
As noted in the Introduction, the general effective La-grangian is written in terms of Goldstone fields π a ( x ) , a =1 , . . . dim(G) − dim(H). Various terms appearing in theeffective Lagrangian are organized in the powers of mo-menta of the Goldstone fields. The leading order La-grangian collects all contributions of the order p . If thesystem is invariant under parity, which is the case withthe Heisenberg Hamiltonian (1), only terms with evenpowers of momenta are permitted. The next-to-leadingorder Lagrangian then contains all contributions of theorder p and so on. Translated into the direct space, thepowers of momenta correspond to the derivatives. Theeffective Lagrangian is constructed by adding terms withincreasing number of derivatives of the Goldstone fields,with the lowest order term containing two derivatives.It should be noted that for systems whose massless ex-citations characterize nonrelativistic dispersion ω ∝ p ,such is HFM, single time derivative counts as p , i.e.as two spatial derivatives or a single power of tempera-ture [6, 10, 11]. Expansion in the powers of momentumis always terminated at some finite order and, besideGoldstone fields and their derivatives, the effective La-grangian also includes several coupling constants whosevalues are not specified by the symmetry requirements.They can be determined by comparison of predictions ofEFT with numerical simulations, experimental results orby matching with detailed microscopic calculations [1, 2].When the effective Lagrangian is constructed, a straight-forward application of Feynmann rules enables one tocalculate the correlation functions, partition function etc.For a Heisenberg ferromagnet G = O(3) , H = O(2), andthe spontaneous symmetry breaking is accompanied bytwo real Goldstone fields π ( x ) and π ( x ). However, theferromagnetic magnons possess nonrelativistic dispersionrelation ( ω ∝ k ) and a complex field ψ ∝ π + i π de-scribes a single magnon [1, 6, 9–11, 17–19].Effective Lagrangian for HFM was introduced in [6, 8,58] (see also earlier works [59–61]), and a detailed deriva-tion of the partition function up to the three loops usingcontinuum approximation and the dimensional regular-ization, resulting with the leading corrections to Dyson’sanalysis of 3D HFM, can be found in [11] (see also [9, 10]and [12, 13] for corresponding analysis of the two di-mensional ferromagnet). In the present paper, a slightlymodified path will be followed. As one of our interestslies in the mass renormalization of the lattice magnons,we wish to preserve the full discrete symmetry of lat-tice spin Hamiltonian (1). Because of that, we find itmore convenient to work in the Hamiltonian formulationof lattice field theory [62, 63], leaving only (imaginary)time coordinate continuous. Therefore, the first task is toconstruct the effective Hamiltonian that describes inter-actions of lattice magons with nonrelativistic dispersion.Details for lattice regularization of the Lorentz-invarianteffective field theory can be found e.g. in [64, 65].The leading order real time effective Lagrangian ofO(3) ferromagnet is [6, 58, 59] L eff = Σ ∂ t U U − ∂ t U U U − F ∂ α U i ∂ α U i , (17)where two magnon fields are collected into the unit vector U i := [ U , U , U ] T ≡ [ π ( x ) , U ( x )] T , Σ = N S/V is thespontaneous magnetization per unit volume at T = 0Kand F is a constant. The first part of Lagrangian isusually denoted as Wess-Zumino-Witten (WZW) term.It gives rise to the Berry phase [66] and is respon-sible for the classical dispersion of the ferromagneticmagnons ( ω ∝ k ). The presence of WZW term makesLagrangian rotationally invariant only up to the totalderivative. As it will be shown, the inclusion of magnon-magnon interactions arising from WZW term is crucialfor correct low-temperature description of O(3) HFM.The next-to-leading order Lagrangian contains termssuch as l ( ∂ α U i ∂ α U i ) , l ( ∂ α U i ∂ β U i ) , l U i ∆ U i , or l ∂ α U i ∂ α U i with arbitrary coupling constants l , . . . l [10, 11]. These O ( p ) terms shall not be directly in-cluded in the effective Hamiltonian. Instead, higher or-der momentum contributions will appear naturally in alattice regularized theory. This regularization, however,restricts possible choices for higher order terms (See sec-tion VI). B. Transition to interaction picture
The use of perturbation theory requires clear separa-tion between the free-magnon part, which must be iden- tified with (12), and the interaction part of the Hamil-tonian [67]. To extract them from the Lagrangian (17),we may rewrite it in terms of the complex field ψ = p Σ / π + i π ] which describes the physical magnon,and follow the standard canonical prescription. However,this is not the most efficient way to construct interactionpicture. The WZW term modifies canonical momentum,from i ψ † of noninteracting theory, to 2i ψ † / [1+ U ], where U = p − (2 / Σ) ψ † ψ . Consequently, the complex fields ψ and ψ † that enter Hamiltonian are not those obeyingequal-time commutation relations. As the connection be-tween canonical momentum and ψ † is highly nonlinear itcan be solved for ψ † only iteratively. Because of that,an important part of magnon-magnon interactions is notmanifest in the Hamiltonian, since it enters the quan-tum theory through the failure of ψ and ψ † to satisfycanonical Schr¨odinger-field commutation relations. Thisis reminiscent of the situation dealt with in the spin-operator approach to Heisenberg magnets: the commu-tation relations governing the dynamics of system areneither Bose nor Fermi type and the interaction is gen-erated by expanding localized spins operators in termsof boson/fermion operators [25, 28, 30, 38, 42]. For thepresent purposes, however, it is desirable to have an ex-plicit form of the magnon-magnon interaction.A different strategy [67] makes use of the equation ofmotion, which in the present case is the Landau-Lifshitzequation [6], ∂ t U a + F Σ ε aij (∆ U i ) U j = 0 , (18)to eliminate ˙ π and ˙ π from the interaction part of theLagrangian (17). Here ∆ = ∂ α ∂ α . In this manner wefind the free-magnon Lagrangian L free = Σ2 (cid:2) ∂ t π π − ∂ t π π (cid:3) + F π · ∆ π , (19)and the interaction piece L int = F h − π − p − π i ∆ p − π − F − √ − π √ − π p − π π · ∆ π . (20)Canonical interacting quantum theory can now be easilyconstructed starting from L free and L int . To perform two-loop calculations for the self-energy ant three-loop calcu-lations for the free energy, we need to retain tjhe magnon-magnon interaction up to and including six magnon op-erators. By expanding (20) we find that terms with six π a operators precisely cancel, in contrast to a Lorentz- invariant theory [5, 64]. Remaining four-magnon termsare then collected to [84] H int = F π (cid:2) π · ∆ π − ∆ π (cid:3) . (21)Finally, by putting the free Hamiltonian and inter-action part (21) on the lattice, we obtain the effective FIG. 1: (Color online) Coordinate space representation ofa diagram contributing to the self-energy of lattice magnonfield (top) and one of its contractions in momentum space(bottom). Vertices from H (I)int and H (II)int are denoted by redand blue squares, respectively. The blue line denotes prop-agator affected by lattice Laplacians of H (II)int . The doublecolored line represents propagator acted on by Laplacians ofboth H (I)int and H (II)int . Hamiltonian for lattice magnon fields H eff = H + H int , (22) H = − m v X x ψ † ( x ) ∇ ψ ( x ) , m = Σ2 F H int = F v X x π ( x ) (cid:2) π ( x ) · ∇ π ( x ) − ∇ π ( x ) (cid:3) ≡ H (I)int + H (II)int , (23)where ∇ denotes the lattice Laplacian and the latticeSchr¨odinger field is (Σ = S/v in a lattice theory) ψ ( x ) = r S v (cid:2) π ( x ) + i π ( x ) (cid:3) . (24)In what follows, ψ will always be written to the right inexpressions like π · π . H is basically LSWT Hamilto-nian (12) and with this choice for L free and L int (i.e. H and H int ), ψ and ψ † do satisfy canonical commutationrelations for Schr¨odinger field. Unlike in its continuouscounterpart, the discrete symmetry of the original Hamil-tonian (1) that modifies magnon dispersion in higher or-ders of momentum is fully preserved in (22) and (23).Hence, all higher order terms in momentum, i.e. in spa-tial derivatives, that resolve the lattice structure at the same time describing the free magnons are collected in H . This fact simplifies further calculations. Magnon-magnon interactions in accord with lattice structure andinternal symmetries, to the order considered here, arecollected in H int . One can choose constant F to beΣ / (2 m LSW ) = JS Z | λ | / (2 Dv ), so that the energy offree lattice magnons is measured in units of J [85], asin LSWT (see (11) and (12)). Of course, this is unneces-sary, since the value of F can be deduced from the exper-imental data on magnon dispersion. In terms of the unitvector U , spontaneous magnetization can be calculatedas h S z i = S h U i ≈ S − ( S/ h π i , which coincides with(15). Hamiltonian (22) resembles the Hamiltonian firstobtained by Dyson, starting from nonorthogonal multispin-wave states [23]. It was subsequently rederived us-ing boson representations for the spin operators [25, 68].However, (22) is expressed in terms of true magnon fieldoperators, with no direct connection to the localized spinsof (1). Also, it is seen from the derivation of (22) that H (II)int has the form of the usual gradient contribution inthe Hamiltonian of a unit vector field, while H (I)int de-scribes magnon-magnon interactions originating in theWZW term (A part of interactions form WZW-term ofthe form π ∇ π are present in H (II)int too). athe magnon-magnon interactions collected in H (I)int are therefore essen-tial for preserving the spin characteristics of bosonic field U ( x ).After the free and interaction parts of the Hamiltonianhave been constructed, the perturbation theory may beapplied to calculate the Green’s function G ( x − y , τ x − τ y ) = h T (cid:8) ψ ( x , τ x ) ψ † ( y , τ y ) U ( β ) (cid:9) i h U ( β ) i (25)with U ( β ) = T exp n − R β d τ H int ( τ ) o , and the free en-ergy, thereby determining the influence of interaction onmagnon energies and thermodynamic properties of thesystem. By expanding the exponential in definition of U ( β ), we arrive at the Feynman rules for interacting lat-tice magnon fields in O(3) HFM. We shall now introducea convenient variant of Feynman diagrams. C. Diagrammar
To define graphical calculations suitable for particularinteraction in (22), i.e. in (23), consider the two-loopdiagram for self energy depicted at FIG 1, and a typicalcontraction proportional to (we abbreviate ψ ( x , τ x ) as ψ x , etc.) v X a , b Z β d τ a Z β d τ b h | T { ψ x ψ † a ψ a ∇ a (cid:0) ψ † a ψ a (cid:1) ψ † b ψ b ψ † b ∇ b ψ b ψ † y }| i . (26)Two lattice Laplacians, acting upon magnon propagators[which are explicitly given in (14)], appear in the up- per integral. The expressions containing discrete Lapla- FIG. 2: (Color online) All possible momentum-space (colored) contractions corresponding to the upper diagram of FIG 1. Eachcolored line denotes the lattice Laplacian acting on a magnon propagator. The line carrying two colors represent propagatoraffected by two Laplacians (See the text). cians could be rewritten as the difference between valueof lattice fields on a given site and on all of its nearestneighbors (see (3)), which would seemingly simplify ex-pressions like (26). However, the eigenvalues of ∇ areproportional to the free magnon energies and the physicalinterpretation favors the use of lattice Laplacian. There-fore, we will stick to the form explicitly containing latticeLaplacian. To distinguish between two or more Lapla-cians in diagrams, we introduce colored lines and vertices.Each vertex carries single color (red or blue in our exam-ple), representing a single Laplacian contained in it. Thelines could be single- or multi-color valued, depending onweather one or more Laplacians acts upon them. Therest of the lines are simply black. All lines are labeled by D +1 momentum k = [ k , ω n ] T and colored ones also carryeigenvalue − b k . The standard momentum-conservationrules at vertices equally apply for black and colored lines.Note that the Laplacian of H (I)int always acts on a singlepropagator. Thus, only single colored line, with the colorof the propagator being the same as that of vertex, canend in . However, it can be single or multi color-valued, depending on if it is affected by a Laplacian ofanother vertex. In contrast, a single colored line of thesame color as that of the vertex is passing through ,and it carries eigenvalue − b k of the algebraic sum of theincoming and outgoing momenta. The other three linesattached to , as well as the remaining two ofcould be colored differently than the vertex or be black.For example, the momentum-space representation of in-tegral (26) is given at the bottom of FIG 1 and the cor-responding integrand is proportional to \ k − q b p .The full consistency of Feynman diagrams with col-ored propagators is achieved by supplementing the rulesof preceding paragraph with additional conventions con-cerning loops closed around a single vertex. These ap-pear, for example, in the one-loop corrections to themagnon propagator as well as in the perturbative cor-rections to the free energy. Consider first the one-loopdiagrams. If the Laplacian of H (I)int acts on a single prop-agator, the loop will be drawn half-colored, so that onlyone colored part of the loop ends at . If the samesituation occurs with , the line is in full color. Thus we have ∝ β ∞ X n = −∞ Z p D ( p , ω n ) b p (27)but, also ∝ β ∞ X n = −∞ Z p D ( p , ω n ) \ p − p = 0 (28)with D ( p , ω n ) denoting the Fourier components of latticemagnon propagator (14). These rules also apply to thediagrams with two loops attached to . Also, they holdif the Laplacian of H (II)int acts on propagators belongingto the same loop, as in (28). If, hoverer, the Laplacianof H (II)int affects propagators from different loops, they areto be drawn half-colored. For example ∝ β ∞ X n,m = −∞ Z p , q D ( p ) D ( q ) \ p − q (29)with p = [ p , ω n ] T , q = [ q , ω m ] T . Of course, it is of noimportance if the upper or the lower part of diagram in(29) is colored. Hoverer, both of these are not to becounted, since they represent the same contraction.As an example, in FIG 2 we give the full set of coloredmomentum space diagrams corresponding to the upperdiagram of FIG 1. Each of these diagrams is to be multi-plied by factor 2, due to two identical sets of contractionsgenerated by two ψ † b operators of H (II)int . We shall refer tothe diagrams with colored lines and vertices as coloredcontractions.In a final remark, we note that extension of multi-colorline formalism to higher order interactions is straightfor-ward. A glance on (20) reveals that all vertices of effectiveinteraction, regardless on the number of π a fields, carrysingle discrete Laplacian. Also, the method of multi-colorFeynman diagrams is, with minimal interventions, appli-cable to various theories of scalar fields with derivativecouplings. In particular, we shall find them very usefulin the Section VI. FIG. 3: One-loop corrections to the lattice propagator (14).The numbers associated with the vertices refer to H (I)int and H (II)int of Eq. (23). IV. MAGNON SELF-ENERGY AT TWO-LOOPA. One-loop correction to the magnon self-energy
The graphs occurring at the one-loop approximationare given in FIG 3. The explicit form of the correctionarising from the first vertex is easily found using Feynmanrules defined above:Σ (1)I ( k , ω n ) = . (30)It is understood that external legs, black and colored, areto be amputated. Further,Σ (1)II ( k , ω n ) =+ . (31)Since the first two diagrams of (31) vanish, by performingsummation over the Matsubara frequencies, we obtainΣ (1) ( k ) = Σ (1)I ( k ) + Σ (1)II ( k ) (32)= b k m S | λ | D v Z q h n q i b q where we have exploited cubic symmetry of the lat-tice, and h n q i represents the Bose distribution for freemagnons. According to (32), magnons acquire mass[ m (1)r ] − = m − (cid:20) − S | λ | D v Z q h n q i b q (cid:21) ≡ m − [1 − A ( T )] (33)This result can be made self-consistent bu further sum-mation, i. e. by replacing the propagators with full Green’s functions in (30) and (31)[ m (1)R ] − = m − (cid:20) − S | λ | D v Z q h n q i b q (cid:21) , (34)with h n q i denoting the Bose distribution for magnonswith energies b q / (2 m (1)R ). If constant F is chosen sothat H of (22) fully coincides with (12), i.e. F = Σ2 m = JS Z | λ | Dv (35)then b k / (2 m (1)R ) is precisely renormalized spin-wave en-ergy, obtained for the first time in [41] by minimization ofthe free energy of a ferromagnet, where the spin Hamil-tonian (1) is written in terms of Dyson-Maleev (DM)bosons with only diagonal part of the interaction be-ing retained. It was also obtained by the bubble dia-gram summation [69], again using DM representation.However, only the derivation of (34) using effective La-grangian clearly shows that the effects of two distincttypes of magnon-magnon interactions are accounted forin (34). B. One-loop approximations for the spontaneousmagnetization
The one-loop corrections to the LSWT result for spon-taneous magnetization are found by substituting magnonpropagator with Green’s function calculated to the oneloop in (15). This is easily obtained by keeping the ex-ternal legs in (30) and (31). The result is h S z i = S − v Z p h n p i + δ h S z i ,δ h S z i = − v Z p Σ (1) ( p ) h n p i [ h n p i + 1] T . (36)There is an obvious virtue in writing the spontaneousmagnetization as in Eq. (36). The term S − v R q h n q i de-scribes the reduction of spontaneous magnetization dueto free magnons. Its low-temperature expansion for 3DHFM contains well known contributions proportional to T / (Bloch’s law), T / , T / and so on. On the otherhand, the corrections arising from the magnon-magnoninteractions are entirely collected in the integral propor-tional to 1 /T . More generally, the low-temperature seriesfor spontaneous magnetization of a D − dimensional sim-ple cubic HFM in the one-loop approximation consists oftwo parts h S z i = S + δ h S z i free + δ h S z i int , (37)where δ h S z i free = α T D/ + α T ( D +2) / + α T ( D +4) / + α T ( D +6) / + O (cid:16) T ( D +8) / (cid:17) (38) FIG. 4: (Color online) Spontaneous magnetization of S =1 / J = 10K 3D HFM calculated using LSWT (Eq.(15)), Tyablikov RPA, one-loop approximation (36), renor-malized magnons (RM) of (33) and self-consistent renormal-ized magnons (SCRM) given in (34). and the free-magnon coefficients α i are given by α = − (cid:18) √ π (cid:19) D √ π Γ( D/ ζ (cid:18) D (cid:19) (cid:20) Σ a F (cid:21) D/ ,α = − (cid:18) √ π (cid:19) D D ζ (cid:18) D + 22 (cid:19) (cid:20) Σ a F (cid:21) D +22 ,α = − (cid:18) √ π (cid:19) D D [ D + 8]512 ζ (cid:18) D + 42 (cid:19) (cid:20) Σ a F (cid:21) D +42 ,α = − (cid:18) √ π (cid:19) D D ζ (cid:18) D + 62 (cid:19) (cid:20) Σ a F (cid:21) D +62 × (cid:20)
25 + 3 D + D (cid:21) . (39)The temperature expansion of one-loop correction toLSWT results is δ h S z i int = β T D +1 + β T D +2 + O (cid:0) T D +3 (cid:1) , (40)with β = − S (cid:18) π (cid:19) D Dπ D (cid:20) Σ a F (cid:21) D +1 × ζ (cid:18) D (cid:19) ζ (cid:18) D + 22 (cid:19) ,β = − S (cid:18) π (cid:19) D D [ D + 2] π D (cid:20) Σ a F (cid:21) D +2 × ((cid:20) ζ (cid:18) D + 22 (cid:19)(cid:21) + ζ (cid:18) D (cid:19) ζ (cid:18) D + 42 (cid:19)) . (41)In the formulae above, ζ ( x ) denotes the Riemann zetafunction and D ≥ D = 3, the lowestorder correction from magnon-magnon interaction comesto be ∝ T , in agreement with Dyson [24]. We note thatthe correct form of leading order contribution is foundeasily, evaluating only a single type of diagram indicatedat FIG 3. This should be compared with continuumfield-theoretical calculations [10, 11], where number of FIG. 5: Two-loop graphs for magnon self-energy that con-tribute solely to the magnon mass.FIG. 6: Two-loop diagrams for magnon self-energy that causefinite magnon lifetime. diagrams to be evaluated becomes greater with increas-ing dimensionality of the lattice (see also the Section VI).Also, the lattice regularized theory allows for a compar-ison with LSWT and other methods, such as TyablikovRPA, even at not too low temperatures.The plot of spontaneous magnetization of spin S = 1 / J = 10K calculated by LSWT (Eq.(15)), Tyablikov RPA, the one-loop approximation of Eq.(36) and dressed magnons of (33) and (34) is presentedat FIG. 4. We have set F = JS Σ a in (36) to workwith a common energy scale. The TRPA result for T C is ≈ .
065 K (For precise calculation of the critical tem-perature in TRPA, see [49, 70] and references therein).
C. Two-loop corrections to the self-energy
The self-energy graphs with two loops, involving ver-tices and can be classified in two groups. Graphspresented at FIG 5 contribute purely to the magnonmass, i.e. they have no imaginary parts. The dia-grams from FIG 6, however, produce the finite magnonlifetime. (The influence of finite magnon lifetime shallnot be discussed further in the present paper. For de-tails and references, see [29, 71, 72].) All these diagramscan be evaluated using formalism of colored propagators,0as explained in III C. An example for decomposition ofcompact two-loop diagram of FIG 6 into its momentum-space contractions is given at FIG 2. Note that thereare only two distinct Matsubara summations at the two- loop. The first one is common to all graphs from FIG5 and the second one appears in all graphs form FIG 6,since the lattice Laplacian leaves n -index untouched.The diagrams from FIG 5 then evaluate toΣ FIG 5 ( k ) = 1 S b k m D | λ | α ( T )2 m T (cid:20) | λ | D (cid:21) v Z p h n p i [ h n p i + 1] T (cid:2) b p (cid:3) ≡ b k m A ( T ) B ( T ) (42)The contribution from diagrams presented at FIG 6 consist of two parts, both of which change the geometry ofmagnon dispersion: The first part is proportional to b k Σ ( a )FIG 6 ( k ) = 12 S b k [2 m ] (cid:20) | λ | D (cid:21) v Z p , q F kp , q (i ω n ) b q b p h b q − \ p − q i (43)and the other one to b k Σ ( b )FIG 6 ( k ) = 12 S " b k m (cid:20) | λ | D (cid:21) v Z p , q F kp , q (i ω n ) b q h b q − \ p − q i . (44)We have introduced here a shorthand notation for the vertex function, obtained by double Matsubara-index summation F kp , q (i ω n ) = h n p i [1 + h n q i + h n k + p − q i ] − h n q i h n k + p − q i ω ( k + p − q ) − ω ( p ) + ω ( q ) − i ω n . From (32), (42), (43) and (44), we find the magnon energies at two loop ω ( k ) = ω ( k ) − δω ( k ) , δω ( k ) = lim δ → ReΣ( k , ω ( k ) + i δ ) , (45)Σ( k , ω ( k ) + i δ ) = Σ (1) ( k ) + Σ FIG 5 ( k ) + Σ ( a )FIG 6 ( k , ω ( k ) + i δ ) + Σ ( b )FIG 6 ( k , ω ( k ) + i δ )= b k m [ A ( T ) + A ( T ) B ( T )]+ b k m S (cid:20) | λ | D (cid:21) v m Z p , q F kp , q ( ω ( k ) + i δ ) b q (cid:16) b p + b k (cid:17) (cid:16) b q − \ p − q (cid:17) . (46)It is seen from (46) that magnons remain gapless at twoloop ( ω ( k ) → | k | →
0) just as do pions inLorentz-invariant models [64, 65, 73].On FIG 7 we plot the free-magnon dispersion ω ( k x , k y ,
0) [Equation (11)] for J = 10K, S = 1 / T = 1K, along with δω ( k x , k y , A = 2 . × − and B = 7 . × − . D. TRPA as an effective field theory
Now that the picture of HFM as a interacting magnonfield is complete, we can make some observation onTRPA result for spontaneous magnetization and freeenergy. They may not be apparent, or even accessi-ble within conventional TGF methodology or any otherapproach that relies on boson/fermion representationof spin operators. Present derivation of TRPA dis-persion relation for magnons, and latter discussion onspurious T term, clearly isolates the influence of re- tained magnon-magnon interactions from the neglectionof short-ranger fluctuations in mean number of magnonsper lattice site.Consider a system of magnons for which the free La-grangian is (19) and interaction is described by e L int ob-tained from (20) by neglecting the second term propor-tional to π · ∆ π and keeping only √ − π in the squarebracket. Corresponding Hamiltonian that includes up tofour magnon operators, e H = H + e H int , is easily con-structed. Since e H int = − H (II)int , the one-loop self energyis found from (31) e Σ( k ) = 1 S v m Z p h n p i \ p − k . (47)We shall now assume that the mean number of excitedmagnons is the same at each lattice site. This simplifica-tion mimics TRPA replacement of the operator S z n withthe site-independent average h S z i . Then v Z p h n p i γ ( p ) = v Z p h n p i (48)1 FIG. 7: (Color online) Reduced free magnon energies ¯ ω ( k ) = ω ( k )[ m | λ | /D ] (left axis, purple curve) and 2-loop cor-rection δ ¯ ω ( k ) = δω ( k ) [ m | λ | /D ] (right axis, bluecurve). ω ( k ) is defined in (11), and δω ( k ) is given in(45) and (46). S = 1 / , D = 3 J = 10K and T = 1K for bothcurves equals the mean number of magnons on each lattice site, h n x i . The magnon energies may now be written as e ω ( k ) = ω ( k ) − e Σ( k ) = b k m S − h n x i S , (49)which we may, for low temperatures, identify with TRPAenergies (16). In other words, the effective Hamiltonianof Tyablikov RPA, written in terms of lattice magnonfields is e H ≡ H RPAeff = H + F v X x π ( x ) ∇ π ( x ) (50)with H defined in (22). Reversing the arguments thatlead to the H RPAeff , and also to the correct interactingHamiltonian of lattice magnons (23), we see that TRPAresults are generated starting from the leading order ef-fective Lagrangian L RPAeff = Σ2 (cid:0) ∂ t U U − ∂ t U U (cid:1) − F ∂ α U i ∂ α U i − F π ( x )∆ π ( x ) . (51)This Lagrangian manifestly violates spin-rotational in-variance of the original Heisenberg Hamiltonian (1).Various explanations for the spurious T term inTRPA expansion of the ferromagnetic order parameterand the error caused by the Tyablikov decoupling at lowtemperatures have been offered by many authors. Forexample, in the Tyablikov’s monograph, it is attributedto the ”approximate character” (of the decoupling ap-proximation) and the ”neglection of the fluctuation oforder parameter” [74]. In the context of the spin-diagramtechnique, authors of [26, 27] state that T arises since”in the decoupling methods terms after r − are takeninto the account incorrectly”. (Here r − represents theformal expansion parameter in the spin-diagram tech-nique, namely the reciprocal interaction volume.) In[55], the main feature of TRPA is recognized as being”uncontrolled expansion to all orders in 1 /Z ”. On theother hand, the authors of [51] conclude that erroneous T term in TRPA ”comes from taking expectation val-ues in the equation of motion too soon”. Finally, in[54], Tyablikov RPA is described as an approximation”in which contributions of static fluctuations of spinsare neglected”. However, it is also noted in this ref-erence that T term will appear in any approximationthat incorrectly treats spectral density entering the cor-relation function h S − S + i k . All arguments quoted aboverely directly on the localized spin operators [54, 55, 74]that define Heisenberg Hamiltonian (1) or on their bo-son/fermion representations [26, 27, 51]. The deriva-tion of Tyablikov RPA in terms of lattice magnon fields,as given in the present paper, provides a simple andstraightforward answer based on the internal symmetriesof the Heisenberg model. It is seen from the equations(50)-(51) that Tyablikov RPA incorrectly describes O(3)HFM at low temperatures since it eventually results fromthe effective Lagrangian (51) that does not preserve spin-rotational symmetry of the Heisenberg ferromagnet (1).Explicitly, interactions of the form π π ·∇ π arising fromthe WZW term H (1)int are omitted in TRPA. It may also besaid that due this reduction of magnon-magnon interac-tions in the effective Lagrangian, localized spins of HFMare inadequately described by the unit vector U ( x ).We note that essential error in TRPA is made whenmagnon-magnon interactions arising from the WZWterm are omitted. Neglection of the SRF of the orderparameter, i.e. neglection of the fluctuations in the meannumber of magnons at adjacent sites (the replacement of v R q h n q i γ ( q ) with v R q h n q i = h n x i ) merely modifiescoefficient of the T term. To show this, we find the firstorder correction to the spontaneous magnetization basedon Eq. (31) δ h S z i = − S F T Σ v Z p J ( p ) h n p i [ h n p i + 1] ,J ( p ) = v Z q h n q i ( \ p − q ) . (52)The leading order term in temperature expansion of Eq.(52) for D dimensional simple cubic lattice is − S F Dπ D (cid:16) a π (cid:17) D (cid:20) Σ F (cid:21) D +1 (53) × ((cid:20) ζ (cid:18) D (cid:19)(cid:21) + ζ (cid:18) D (cid:19) ζ (cid:18) D − (cid:19)) T D . and for D = 3, it corresponds to spurious T term. If anadditional assumption on the absence of SRF is included,the term ∝ ζ ( D/ ζ ( D/ −
1) is missing from (53).The Tyablikov’s T term [74, 75], − S (cid:18) πJS (cid:19) (cid:20) ζ (cid:18) (cid:19)(cid:21) T (54)is found by setting F = JS Σ a (See (35)). An alter-native formulation of O(2) model (50) is given in theAppendix B.2 V. FREE ENERGY AT THREE-LOOP
For the subsequent analysis of the low temperaturethermodynamics, we shall include weak external mag-netic field directed along the 3-axis. It opens the gap inmagnon spectrum ω ( p ) −→ ω ( p , H ) = ω ( p ) + µH, (55)and it is included in the effective Lagangian by standardZeeman term [6]. L H = Σ µHU . (56)Note that the interaction Hamiltonian does not containterms proportional to the external field H . This is anexact result to all orders in π , and it follows from theequation of motion. A. Two-loop correction to the free energy
The first-order correction to the free energy (see e.g.[76]) of lattice magnons, involving the two-loop graphs,is given by δF = − T − T , (57)with the notation introduced in previous sections.According to the Feynman rules defined in III C, thesediagrams decompose to= + , (58)and = + , (59)so that the first correction to the free energy per latticesite is given by δf = δF N = − S F Σ | λ | D (cid:20) v Z p h n p i b p (cid:21) , (60)The temperature expansion of (60) starts with T D +2 term. Specifically, for D − dimensional cubic lattices, it is δf = − S Dπ D a D (cid:18) π (cid:19) D (cid:18) Σ a F (cid:19) D +1 T D +2 × " ∞ X n =1 e − µHn/T n ( D +2) / + O ( T D +3 ) . (61)If D = 3, this gives leading-order part of Dyson’s T term[24]. It receives contribution from higher-loop diagramsin the lattice regularized theory (see the subsection V C).At this point we also justify one-loop calculationof spontaneous magnetization from previous section byshowing that corrections to LSWT from (36) can beobtained from the first-order correction to the free en-ergy of lattice magnons. The first correction to spon-taneous magnetization (per lattice site) is δ h S z i = − ∂ ( δf ) /∂ ( µH ) | H =0 . Differentiating (60) with re-spect to µH and setting H = 0 we readily recover equa-tion (36). B. Three-loop corrections to the free energy
Three-loop contribution to the magnon free energy isrepresented by diagrams from FIG 8. They can be clas-sified into two categories, distinguished by a and b super-scripts in the following equations.Each of the three upper diagrams from FIG 8, to beclassified as a -type, consists of a number of differentcolored contractions. For the graph containing verticessolely from H (I)int , each of distinct colored contractions re-peats four times, so that δf ( a )I,I = − S (cid:20) m (cid:21) v Z k , p , q h n q i [ h n q i + 1] T × h n p i h n k i h b p b q + b k b q + b p b k + b q b q i . (62)Further, for the graph with two vertices from H (II)int , wefind δf ( a )II,II = − S (cid:20) m (cid:21) v Z k , p , q h n q i [ h n q i + 1] T × h n p i h n k i (cid:20) \ p − q \ k − q (cid:21) . (63)This term consists of two different colored contractions,each appearing twice. The total contribution of thegraphs that involve vertices of both H (I)int and H (II)int is δf ( a )I,II = 1 S (cid:20) m (cid:21) v Z k , p , q h n q i [ h n q i + 1] T × h n p i h n k i (cid:20) b p \ k − q + b q \ k − q (cid:21) . (64)In this case, one of the colored contractions appears eighttimes and the other two four times each.3 FIG. 8: Six distinct three-loop diagrams contributing to the free energy.
Finally, by putting contributions (62)-(64) together, we find the first part of three-loop contribution to the magnonfree-energy per lattice site δf ( a )3loop = − S (cid:20) m (cid:21) v Z q h n q i [ h n q i + 1] T (cid:20) v Z p h n p i (cid:16) \ p − q − b p − b q (cid:17)(cid:21) . (65)The calculation resumes in a similar manner for remaining diagrams of b -type, i.e. for lower three diagrams of FIG8. They add up to δf ( b )3loop = − S (cid:20) m (cid:21) v Z k , p , q G qk . p (cid:20) ( \ k + p − q ) + b q − \ p − q − \ k − q (cid:21) (cid:20) b p + b k − \ p − q − \ k − q (cid:21) (66)with three-index Matsubara sum G qk , p = h n k i h n p i [1 + h n q i + h n k + p − q i ] − h n q i h n k + p − q i [ h n k i + h n p i + 1] ω ( k + p − q ) + ω ( q ) − ω ( p ) − ω ( k ) . Now we consider limit a → C. Analysis of the three-loop integrals
First, it is easily seen that δf ( a )3loop vanishes in contin-uum limit, thus contributing nothing when the latticeanisotropies are neglected. This was pointed out in [11]for D = 3. Its lowest order contribution is proportionalto T [3 D +6] / . For D dimensional simple cubic lattice, thisterm is − v Σ (cid:18) Σ F (cid:19) (3 D +4) / (cid:18) a D (cid:19) D [ D + 2]32 × (cid:18) √ π (cid:19) D (cid:20) ζ (cid:18) D + 22 (cid:19)(cid:21) T [3 D +6] / . (67)It turns out, however, that this is not the leading ordercorrection to the Dyson’s T D +2 term.The leading order correction to T D +2 term in thelow-temperature expansion of free energy originates in δf ( b )3loop of (66), but the continuum limit of these diagramsshould be handled with care. In the direct continuumlimit ( a → D ≥
3. Instead of subtractingthis infinite contribution, we may fully exploit the lat-tice regularization to remove the divergent term. If thewave vectors, whose energies does not appear in Bosefactors of (66), are kept within Brilouin zone, we findthat first part of δf ( b )3loop renormalizes (60), i.e. (61): δf → δf [1 + Q ( D ) /S ]. For the D dimensionalsimple cubic lattice, the renormalizing factor is Q ( D ) = 1 D Z x cos x − γ D ( x ) . (68)Here x denotes dimensionless wave vector within Brilouinzone, − π ≤ x α ≤ π, α = 1 , , · · · D .Now we calculate the finite part of δf ( b )3loop in the con-tinuum limit, the correction to (61) of order T (3 D +2) / due to magnon-magnon interactions. Introducing dimen-sionless wavevectors x = k / √ m T , y = p / √ m T , and z = q / √ m T , we obtain the leading order term in δf ( b )3loop : δf ( b )3loop ( T, h ) ≈ − v Σ (cid:18) Σ F (cid:19) D/ I ( D, h ) T (3 D +2) / (69)4 TABLE I: Numerical values of integrals I ( D ) and J ( D ) D I ( D ) J ( D )3 5 . × − − . × − . × − − . × − . × − − . × − . × − − . × − where I ( D, h ) = Z x , y , z h n x ( h ) ih n y ( h ) ih n z ( h ) i ( x · y ) z + x · y − z · ( x + y ) (70)and h n x ( h ) i = (cid:2) exp( x + h ) − (cid:3) − , h = µH/T. (71)In the absence of external field H , (69) reduces to δf ( b )3loop ( T ) ≈ − v Σ (cid:18) Σ F (cid:19) D/ I ( D ) T (3 D +2) / (72)with I ( D ) ≡ I ( D, h = 0). The numerical values of inte-grals I ( D ) are listed in the Table I for D = 3 , , T / termagrees perfectly with the calculations of [11] based on thedimensional regularization. For more details concerningnumerical evaluation of the integrals I ( D ), we refer tothe Appendix A.The result (72) enable us to calculate correspond-ing correction to the spontaneous magnetization in-duced by magnon-magnon interaction. Since δ h S z i = − ∂ ( δf ) /∂ ( µH ) | H =0 , we find δ h S z i ( T ) ≈ − v Σ (cid:18) Σ F (cid:19) D/ J ( D ) T D/ (73)where J ( D ) = (74) − Z x , y , z h n x ih n y ih n z i [3 + h n x i + h n y i + h n z i ]( x · y ) z + x · y − z · ( x + y )and h n x i = h n x ( h = 0) i . The values of integrals J ( D )are also listed in Table I. Again, for D = 3, we findexcellent agreement with [11].At this point we may compare results from three-loopcalculations for the free energy with those of section IV.The two leading order terms in low temperature series forthe spontaneous magnetization due to magnon-magnoninteractions were shown there to carry D + 1 and D + 2powers of temperature, respectively. Results of three-loop analysis for the free energy reveal term with 3 D/ D = 3, the T D/ term in-deed represents the first correction to Dyson’s T D +1 re-sult. However, already for D = 4, contribution from T D +2 and T D/ terms are of the same order. For D ≥ T D +1 term is pro-portional to T D +2 , and is given in (40) and (41). Furtherunderstanding of relationship between T D +2 and T D/ terms may be reached by directly examining diagram-matic series for low-temperature expansion of free energy(see VI B). VI. SYMMETRY OF THE EFFECTIVELAGRANGIAN AND SPURIOUS TERMS INLOW-TEMPERATURE EXPANSIONS
To gain further insight into the thermodynamics of D dimensional O(3) Heisenberg ferromagnet − in particularto see how various terms in the low-temperature expan-sion of free energy are generated by magnon-magnon in-teractions − we shall now obtain the most importantresults of Section V with the help of power countingscheme for magnon fields. The power counting schemesand structure of diagrams for low-temperature expansionof the free energy in cases D = 2 and D = 3 are given in[11, 12]. We shall now generalize these results for arbi-trary D ≥
3. In this section, the focus will be on the freeenergy per unit volume ( f = F/V ), in contrast to theSection V, devoted to the calculation of the free energyper lattice site.
A. Effective Lagrangian and power counting
Following e.g [77], we start from the lattice Hamilto-nian (22), and obtain continuous Hamiltonian density upto | p | , organized in the powers of magnon momenta. Wealso include weak external magnetic field H = H e z H = F ∂ α π · ∂ α π − Σ µH (cid:0) − π / (cid:1) , H (2) = F (cid:2) π π · ∆ π − π ∆ π (cid:3) ≡ H (2)I + H (2)II , H (4) = − l ∂ α π · ∂ α π (75)+ l (cid:2) ∂ α (cid:0) π π (cid:1) · ∂ α π − ∂ α π ∂ α π (cid:3) , H (6) = c ∂ α π · ∂ α π + c (cid:2) − ∂ α (cid:0) π π (cid:1) · ∂ α π + ∂ α π ∂ α π (cid:3) . H describes free magnons with rotationally invariantclassical dispersion E ( k ) = F Σ k + µH, (76)and the rest of terms in (75) are treated as a perturbation.While first terms in H (4) and H (6) modify dispersion dueto the lattice anisotropies, the other contributions includemagnon-magnon interactions. The formal values for cou-pling constants are l = F a / , c = F a / L = L (2) + L (4) + L (6) (77)where L (2) is given in (17) L (2) = Σ ∂ t U U − ∂ t U U U − F ∂ α U · ∂ α U + Σ µHU , (78)and L (4) = l ∂ α U · ∂ α U , L (6) = c U · ∂ α ∂ α U . (79)5We see that, unlike spin-rotation symmetry, space-rotation symmetry is lost starting from p terms. Thecrucial point in systematic EFT calculation of the freeenergy density comes from the observation that loop di-agrams carry additional powers of momentum and thusyield terms with increasing powers of temperature. Forthe model of current interest, each loop is suppressed by D powers of momentum, as can bee seen from (14) [Seealso [10–12]]. This, along with the fact that p countsas T , allows for systematic organization of diagrams con-tributing free energy density [5].Before discussing the low-temperature expansion, wenote that the single vertex three-lop diagrams , (80)with rectangle denoting vertices from H (2) , H (4) or H (6) (i.e. n = 1 , , U andconsistent with symmetries of the Heisenberg model. Tosee how this happens, observe that WZW term alwaysproduce the six-magnon vertex with opposite sign thanthe one arising from terms with spatial derivatives only,as can be shown by employing the equation of motion.Thus, for model (77), cancellation of six-magnon verticesis exact to all orders in | p | . B. Magnon magnon interactions andlow-temperature series for the free energy density
Basically, two types of diagrams contribute to free en-ergy. In the first of them, only vertices with two magnonfields appear. Their general form is (81)where vertex with 2 n i powers of momentum appears k i times ( i = 1 , , · · · R ). As loops are suppressed by | p | D for Lagrangian (77), it is easy to see that dia-grams of equation (81) contribute to low-temperatureexpansion of free energy with terms whose power of T is D/ P Ri =1 k i ( n i − α i coefficients of the low-temperature expansion of sponta-neous magnetization, given in (38). Since all two-magnonvertices carry an even power of momentum, contributions from lattice anisotropies produce the terms with powersof temperature equal to D/ , D/ H (2) .By dimensional arguments, it contributes with a termproportional to T D in low-temperature expansion of freeenergy. However, this term vanishes due to the space-rotation symmetry [10, 11]. Moreover, we shall now showtat even general class of diagrams, containing four vertexof H (2) and two-magnon vertices of H (2 n ) always vanish,i.e. they do not contribute to the free energy of O(3)HFM. To this end, consider the diagram (82)where we have, following standard conventions [5, 10, 11],denoted the vertex from H (2) by a dot. The rectanglesdenote two-magnon vertices carrying 2 n and 2 m pow-ers of p , respectively. First, split the diagram (82) intotwo pieces, one including four-magnon vertex of H (2)I andthe other one with vertex of H (2)II (i.e. =+ ). and bellow refer to H (2)I and H (2)II of (75). Now= ++ + , (83)= + (84)where first two diagrams in (83) and first one in (84) areto be multiplied by a factor of 2 and the rest of them by afactor of four. (The last two diagrams in (83), as well asthe second one in (84) possess additional symmetry dueto permutation of blue and green vertices.) There arefour more colored contractions in equation (84) which wehave not displayed, since they are of the form (28) andthus vanish. The diagrams from (83) and (84) are to6be evaluated using diagrammatic rules from the SectionIII, modified to compensate the replacement of latticemagnon dispersion with (76). Finally, we see that dia-gram (82) is proportional to Z p , q G ( p , q , T ) p · q (cid:2) n p m + p n q m (cid:3) = 0 (85)with G ( p , q , T ) = h n p i [ h n p i + 1] h n q i [ h n q i + 1] T , p n = ( − ) n D X α =1 p nα . A special cases of vanishing diagram (82) are the oneswith single two-magnon vertex or with four-magnon ver-tex only.In a similar manner, one can also demonstrate the van-ishing of three-loop diagram (86)(see also [11] and discussion just before (67)).Thus, a list of different diagrams arising from magnon-magnon interactions that actually contribute to the freeenergy of O(3) HFM is quite limited. The first term inthe low-temperature expansion of the free energy density,carrying D + 2 powers of temperature (the Dyson term),is generated by the two-loop diagram . (87)Its value is δf D +2 = − l Dπ D (cid:18) π (cid:19) D (cid:18) Σ F (cid:19) D +2 T D +2 × " ∞ X n =1 e − µHn/T n ( D +2) / . (88)Out of the two-loop graphs, now comes the T D +3 term.It is given by + , (89)so that δf D +3 = − Dπ D (cid:18) π (cid:19) D (cid:18) Σ F (cid:19) D +3 T D +3 × (cid:20) l [ D + 4] F − c (cid:21) × ∞ X n,m =1 e − µH [ n + m ] /T n ( D +2) / m ( D +4) / . (90) Next enter three-loop diagrams. The lowest non-zero di-agram is . (91)This diagram contains an infinite contribution (See theSection V). Its finite part is given by δf D/ = − (cid:18) Σ F (cid:19) D/ I ( D, h ) T (3 D +2) / , (92)where numbers I ( D, h ) are defined in (70). If the exter-nal magnetic field is switched off, equations (88), (90)and (92) give three lowest contributions to free energydensity due to magnon-magnon interactions for D = 3 , T D/ term of (92) represents the leading correctionto the Dyson’s T D +2 term only for D = 3. This is un-derstood quite naturally in the language of EFT: The T D/ -term comes from three-loop graph (91), andwith increasing D it is being pushed up in the temper-ature expansion compared to two-loop graphs (89). At D = 6, one also needs to include four-magnon vertices oforder | p | to calculate T D +4 term from two-loop graphs,which is of the same order as T D/ term of (91). C. Spurious terms
Now we examine in detail what is, in the section IV D,shown to be the effective field theory for TRPA. TheHamiltonian density obtained from (50), up to the termsof order | p | reads H (2)RPA = F π ∆ π , H (4)RPA = − l ∂ α π · ∂ α π + l ∂ α π ∂ α π , (93) H (6)RPA = c ∂ α π · ∂ α π + c ∂ α π ∂ α π . H and the coefficients l and c are the same as ones in(75). The corresponding Lagrangian up to | p | consistsof three parts, L RPA = L (2)RPA + L (4)RPA + L (6)RPA , where eachterm contains contributions that explicitly break internalsymmetry of the Heisenberg Hamiltonian (1) L (2)RPA = Σ2 (cid:0) ∂ t U U − ∂ t U U (cid:1) − F ∂ α U · ∂ α U + Σ µHU − F π ∆ π , (94) L (4)RPA = l ∂ α U · ∂ α U − l π ∂ α ∂ α π , L (6)RPA = l ∂ α U · ∂ α U − l π ∂ α ∂ α π . It is easily seen from (93) that TRPA contains thesame one-loop diagrams (81) as correct effective theory7for HFM defined by (75). The most important differencein low temperature expansion concerns the two-loop di-agram of four-magnon vertex H (2)RPA . It is now given by= + (95)so that δf RPA D +1 = − Dπ D (cid:18) π (cid:19) D (cid:18) Σ F (cid:19) D T D +1 × ∞ X n,m =1 e − µH [ n + m ] /T n ( D +2) / m D/ . (96)The dot in (95) represents four-magnon vertex of H (2)RPA from (93). Clearly, (96) is responsible for the spurious T D +1 term (See the equations (53) and (54)). In thismanner, we have reached the same conclusion as in latticeregularized theory: the main reason for spurious T D +1 term in low-temperature series for free energy is the ex-plicit violation of internal symmetry of HFM by discard-ing magnon-magnon interactions from the WZW term.The effective action for TRPA Lagrangian is no longerO(3) invariant. Even though it was shown that space-rotation symmetry of leading order Lagrangian guar-anties absence of T D +1 term (See [10, 11] for D = 3),we see that it only does so if the full spin-rotational sym-metry of effective action is preserved (order by order, inthe perturbative expansion).The importance of WZW term in effective Lagrangianfor ferromagnet was stressed in [6, 61] where it was shownthat it is crucial for obtaining correct qualitative magnonspectrum in linear approximation. The present paperputs conclusions of [6, 61] a step forward in the sensethat it is demonstrated how the thermodynamical prop-erties of a ferromagnet depend on the magnon-magnoninteractions generated by WZW term. VII. SUMARRY
The method of effective Lagrangians is a powerful toolfor analyzing the low-energy (low temperature) domain ofmodels with strongly interacting constituents, providedspontaneous symmetry breaking occurs. Employed firstfor the description of low-energy sector of QCD, it soonfound its way to the strongly correlated systems of con-densed matter. A notable example for the latter class ofsystems is the isotropic Heisenberg ferromagnet, exhibit-ing O(3) −→ O(2) spontaneous breakdown of internalspin-rotation symmetry. Due to a symmetry breakdown,massless particles (Goldstone bosons) appear in the spec-trum. In the case of Heisenberg magnets, Goldstonebosons are called magnons and for O(3) HFM, there aretwo broken generators and a single magnon. The EFTline of reasoning then constitutes in writting the most general Lagrangian (i.e. action) consistent with internalsymmetries of the Heisenberg Hamiltonian, organized inthe powers of momenta of Goldstone fields. By employ-ing appropriate Feynman rules for time-ordered Green’sfunctions of the effective Lagrangian, one can calculatefree energy and other physical quantities of interest.EFT is usually formulated in the functional integralframework using dimensional regularization. Lattice reg-ularization, however, enables the effective Hamiltonian toinherit the full discrete symmetry of the original Heisen-berg ferromagnet which modifies the free magnon dis-persion from k / (2 m ) to b k / (2 m ). In the spirit of chi-ral perturbation theory, where original quark and gluonsof QCD are replaced with Goldstone bosons of sponta-neously broken chiral symmetry, no reference is made tothe original degrees of freedom, i. e. to the localizedspins. The theory includes bosonic (magnon) fields fromthe beginning. Because of that, calculations in latticeEFT are free of certain types of approximations unavoid-able in approaches relaying on equations of motion forspin operators or on their bosonic/fermionic representa-tions. A vital part of our analysis is careful inclusion ofthe magnon-magnon interactions arising from the WZWterm in the effective Lagrangian.Calculations in EFT reduce to evaluation of variousFeynman diagrams involving propagators of Goldstonefields. As Goldstone fields are derivatively coupled, thelattice regularization necessary induces diagrams con-taining at least one lattice Laplacian acting on propaga-tors. The number of lattice Laplacians increases with theorder of diagram, i.e. with number of loops. The struc-ture of diagrams becomes even more involved if interac-tion part of Hamiltonian consists of several terms differ-ent in derivative structure since more than one latticeLaplacian may act on given propagator, as in the presentpaper. To get around these complications as much aspossible, we have devised a variant of Feynman diagramssuitable for scalar fields with derivative couplings. Theyare applicable for lattice as well for continuum field theo-ries. Corresponding Feynman rules are based on coloredlines and vertices and are exposed in Section III C. Allcalculations in the present paper rely on this version ofFeynman diagrams.Once the effective Hamiltonian of lattice magnon fieldsis found, it can be used to study magnon-magnon inter-actions and their influence on magnon self energy, spon-taneous magnetization and free energy of D ≥ T D +2 , T D +3 andso on, while the three-loop corrections start with termproportional to 3 D/ T D/ term in case of 3,4,5 and 6-dimensional latticealong with general expression valid for all D ≥
3. Whileresults for D = 3 are found to be in excellent agree-ment with recent literature, those for D = 4 , | p | .Also, using colored diagrams, we have shown that cer-tain two and three-loop diagrams for free energy vanish.At the end of Section VI we show that spurious T D termin the low-temperature expansion of spontaneous magne-tization characterizes theories inconsistent with internalO(3) symmetry of D dimensional HFM. For example,explicit breakdown of O(3) symmetry in Tyablikov RPAis caused by neglection of magnon-magnon interactionsof the form π π · ∇ π generated solely by the WZWterm, thus making TRPA a theory of pure bosonic latticeSchr¨odinger field. This becomes especially clear whenTRPA effective theory is put in the form given in theAppendix B.The Heisenberg model and the spin waves on hypercu-bic lattice have been studied in the past. These papers,however, focus on general properties of magnetic systemson hypercubic lattices, such as existence of the Landau-Lifshitz equation [78]. Also, their analysis relies on MFapproximations [35], coupled-cluster [79] and 1 /Z [55]expansion or RG methods [39, 80]. In contrast, we in-vestigate the low-energy magnon-magnon interactions indetail. Explicit expressions describing their influence onthe magnon self energy, the ferromagnet free energy andthe spontaneous magnetization are presented in the pa-per. We also discuss subtleties concerning the influenceof spatial dimensionality of the lattice and the numberof loops entering the low-temperature expansion of fer-romagnet free energy, which can not be found in earlierworks.To conclude, we may say that the lattice regulariza-tion offers new perspectives on EFT approach to Heisen-berg ferromagnet. By keeping full magnon dispersion it extends EFT range of validity at not too low temper-atures. It also makes comparison between EFT resultsand those of nonlinear spin waves/TRPA direct and pre-cise. Finally, although the lattice regularization for effec-tive Lagrangians of more complex models may not be asstraightforward as for HFM, there seems to be no princi-pal obstacle in extending this method to other systems. Acknowledgement
This work was supported by the Serbian Ministry ofEducation and Science, Project OI 171009. The authorsacknowledge the use of the Computer Cluster of the Cen-ter for Meteorology and Environmental Predictions of theDepartment of Physics, Faculty of Sciences, University ofNovi Sad, Novi Sad, Serbia.
Appendix A: Integrals I ( D ) and J ( D ) In this Appendix we give some details concerning eval-uation of integrals I ( D ) and J ( D ) that appear as numer-ical prefactors of T (3 D +2) / term in the low-temperatureseries for free energy and T D/ term of low-temperatureexpansion of spontaneous magnetization, respectively.The integrals in question are defined in (70) for h = 0and in (74). The 3 D -fold integration may be reduced to4-fold one using D − dimensional spherical coordinates x = x sin θ D − sin θ D − · · · sin θ sin θ ,x = x sin θ D − sin θ D − · · · sin θ cos θ ,x = x sin θ D − sin θ D − · · · sin θ cos θ ,x = x sin θ D − sin θ D − · · · sin θ cos θ , ... x D = x cos θ D − (A1)with x = | x | , 0 ≤ θ ≤ π and all other angles rangingfrom 0 to π . By appropriate change of variables, the θ D − integral of z vector may be reduced to the integralrepresentation of hypergeometric function [81] F (cid:18) , D −
12 ; D −
1; 2 z | x + y | z + x · y + z | x + y | (cid:19) . (A2)Out of remaining 3 D − D − I ( D ), for D = 3 , , D = 6, are listedbelow I (3) = 2 (cid:18) π (cid:19) Z ∞ d x Z ∞ d y Z ∞ d z x y z h n x ih n y ih n z i Z π d θ sin θ cos θλ ( x, y, θ ) ln (cid:12)(cid:12)(cid:12)(cid:12) A ( x, y, z, θ ) B ( x, y, z, θ ) (cid:12)(cid:12)(cid:12)(cid:12) , (A3) I (4) = 12 (cid:18) π (cid:19) Z ∞ d x Z ∞ d y Z ∞ d z x y z h n x ih n y ih n z i× Z π d θ sin θ B ( x, y, z, θ ) cos θλ ( x, y, θ ) (cid:20) − exp (cid:18)
12 ln (cid:12)(cid:12)(cid:12)(cid:12) A ( x, y, z, θ ) B ( x, y, z, θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19)(cid:21) , (A4)9 I (5) = 16 (cid:18) π (cid:19) Z ∞ d x Z ∞ d y Z ∞ d z x y z h n x ih n y ih n z i× Z π d θ sin θ cos θλ ( x, y, θ ) (cid:20) zλ ( x, y, θ ) (cid:0) z + xy cos θ (cid:1) − A ( x, y, z, θ ) B ( x, y, z, θ ) ln (cid:12)(cid:12)(cid:12)(cid:12) A ( x, y, z, θ ) B ( x, y, z, θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (A5) I (6) = 136 (cid:18) π (cid:19) Z ∞ d x Z ∞ d y Z ∞ d z x y z h n x ih n y ih n z i Z π d θ sin θ B ( x, y, z, θ ) cos θλ ( x, y, θ ) × " exp (cid:18)
32 ln (cid:12)(cid:12)(cid:12)(cid:12) A ( x, y, z, θ ) B ( x, y, z, θ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) − zλ ( x, y, θ ) B ( x, y, z, θ ) ( − zλ ( x, y, θ ) B ( x, y, z, θ ) − (cid:18) zλ ( x, y, θ ) B ( x, y, z, θ ) (cid:19) ) , (A6)where we have introduced shorthand notations commonto all integrands h n x i = 1e x − , etc λ ( x, y, θ ) = p x + y + 2 xy cos θA ( x, y, z, θ ) = z + xy cos θ − zλ ( x, y, θ ) B ( x, y, z, θ ) = z + xy cos θ + zλ ( x, y, θ ) . (A7)Remaining integrals are rapidly convergent and may beevaluated, e.g., by Gaussian quadrature (See [82] andreferences therein). The numerical values for I (3), I (4), I (5) and I (6) are given in the main text (Table I).Integrals J ( D ), appearing as numerical coefficients oflow-temperature expansion of spontaneous magnetiza-tion are obtained by replacing h n x ih n y ih n z i with −h n x ih n y ih n z i [3 + h n x i + h n y i + h n z i ] (A8)in (A3)-(A6). Their numerical values are listed in theTable I. Appendix B: TRPA as the two-body Schr¨odingertheory
In this Appendix, we reformulate results of the Subsec-tion IV D in the standard language of diagrammatic per- turbation theory for nonrelativistic bosons, i.e. withoutderivative coupling of magnon fields and Feynman rulesof III C. By simple manipulation we cast e H int = − H (II)int in the usual form of a two-body interaction for theSchr¨odinger field [66] e H int = v X x , y ψ † ( x ) ψ † ( y ) V ( x − y ) ψ ( y ) ψ ( x ) , (B1)where V ( x − y ) = 2 D [ F/ Σ] Z | λ | v X λ h ∆( y = x + λ ) − ∆( y = x ) i . The one-loop magnon self energy for this model can bewritten as e Σ( k , ω ) = . (B2)The dashed line here represents the (minus of)Fourier transform of the two-body potential, V ( k ) = − DF / ( | λ | Σ )[1 − γ ( k )]. All results of the Subsec-tion IV D could be deduced equally well starting from(B1)-(B2). [1] C.P. Burgess, Phys. Rep. , 193 (2000)[2] H. Leutwyler, Ann. Phys. , 165 (1994)[3] S. Weinberg, Physica A , 327 (1979)[4] J. Gasser, H. H. Leutwyler, Ann. Phys. , 142 (1984)[5] P. Gerber, H. H. Leutwyler, Nucl. Phys. B , 387(1989)[6] H. Leutwyler, Phys. Rev. D , 3033 (1994)[7] T. Brauner, Symmetry , 609 (2010)[8] J. M. Roman, J. Soto, Int. J. Mod. Phys B
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