The Wilson Renormalization Group Approach of the Principal Chiral Model Around Two Dimensions

The Wilson Renormalization Group Approach of the Principal Chiral Model around Two Dimensions.

B. Delamotte 1, D. Mouhanna 1, P. Lecheminant 2 1 Laboratoire de Physique Th´eorique et Hautes Energies. Universit´es Paris VI Pierre et Marie Curie - Paris VII Denis Diderot, 2 Place Jussieu, 75252 Paris C´edex 05, France. Laboratoire associ´e au CNRS UMR 7589 2 Laboratoire de Physique Th´eorique et Mod´elisation, Universit´e de Cergy-Pontoise, Site de Saint Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France (Received:)

We study the Principal Chiral Ginzburg-Landau-Wilson model around two dimensions within the Local Potential Approximation of an Exact Renormalization Group equation. This model, relevant for the long distance physics of classical frustrated spin systems, exhibits a fixed point of the same universality class that the corresponding Non-Linear Sigma model. This allows to shed light on the long-standing discrepancy between the different perturbative approaches of frustrated spin systems. PACS No: 75.10.Hk, 11.10.Hi, 64.60.-i, 64.60.Ak

preprint PAR-LPTHE 98-26 N H = J ei .eej = J Tr t R i R j . (1) − α α − α=1 There is now a general agreement about the field theo- X X X retical treatment of the SO(N) spin system. The pertur- The Hamiltonian (1) is invariant under the SO(N) bative approaches performed around four dimensions on SO(N) group of left U SO(N) and right V SO(N⊗) the Ginzburg-Landau-Wilson (GLW) model, around two global transformations: ∈Ri URiV. Since, in∈ the low dimensions on the Non-Linear Sigma (NLσ)modeland temperature phase, the residual→ symmetry group con- in a 1/N expansion give a consistent picture of the criti- sists in a (diagonal) SO(N), Eq. (1) is indeed a lat- cal physics of this system everywhere between D =2and tice version of the PC model. Whereas the microscopic D=4[1]. This picture has also been confirmed by non- derivation for frustrated spin systems leads in general to perturbative methods based on truncations of Wilson Ex- anisotropic interactions between the eα’s, i.e. J is α- act Renormalization Group (RG) equations [2–8].Amaz- dependent, we consider here the isotropic case where all ingly, there is no such agreement for many systems whose the Jα’s are equal. It has been shown for a large class symmetry breaking pattern is not given by SO(N) of frustrated spin systems, among which the AFT model, SO(N 1) among which are superfluid 3He [9,10],frus-→ that the anisotropy is anyway irrelevant, at least near two trated− antiferromagnets [11–14], superconductors [15,16], dimensions, for the critical properties we are interested electroweak phase transition [17,18], etc. Generically per- in [13,14]. turbation theories predict that these systems undergo a Let us first sketch out the experimental and numerical first order phase transition near D = 4 and a second or- situation for frustrated spin systems which, in D =3,is der one around D =2[19,17,13,14,18]. The origin of this far from being clear. Indeed, the behaviour of systems discrepancy is not yet understood and calls for a non- that are supposed to be described by the PC model like perturbative approach. AFT (CsVCl3, RbNiCl3) and Helimagnets (Ho, Dy, Tb) In this paper we study, by means of the Wilson Renor- are affected by the presence of disorder localized near malization Group approach, the Principal Chiral (PC) the sample surface and, possibly, by topological defects. model, corresponding to the symmetry breaking scheme As a consequence, the critical exponents strongly vary SO(N) SO(N) SO(N), which is the simplest one from one compound to another [21,22]. Numerically, the exhibiting⊗ the non→ trivial features previously quoted. situation is also confused since simulations performed on The PC model is the low energy effective field theory the PC model and directly on the AFT model lead re- of a whole class of systems among which frustrated an- spectively to first order and second order behaviour with tiferromagnets. A particularly important example is exponents of an unknown universality class [23]. the Heisenberg antiferromagnet on the triangular lattice Beyond this apparent lack of universality at the ex- (AFT). Due to the frustration, the order parameter is a perimental and numerical level, the theoretical situation triad of orthonormal vectors, i.e. a SO(3) rotation ma- already exhibits the puzzling features previously men- [20,14] trix R =(e1,ee2,ee3) . We consider, in the following, tioned. Around D = 2, the critical physics is obtained by the generalization to N orthonormal vectors eα’s with means of a low temperature expansion performed on the N components, i.e. SO(N) matrices. The long distance PC NLσ model. A fixed point is found for any N>2in physics of this generalized AFT model is thus equivalent D=2+dimensions so that a second order phase tran- to that of orthonormal frames interacting ferromagneti- sition is expected [24,25]. On the other hand, the weak cally: coupling expansion performed in D =4 on the PC GLW model suggests a first order phase transition− since

1 no fixed point is found for any N>2[11]. As such, proach has been questionned [27]. Clearly, the answer to the situation is not paradoxical since perturbation theo- these questions escapes a perturbative treatment. In gen- ries are only trustable in the immediate vicinity of their eral, the 1/N expansion provides a powerful tool to link respective critical dimension. However if, as usual, we up different perturbative methods. In the case of matrix extrapolate the perturbative results to D =3,thetwo models such an analysis is however plagued by technical results come into conflict. It is thus of first importance to difficulties. Some progress have been recently obtained clarify this theoretical situation before hoping to describe but are confined to the leading order [28,29]. The Wil- real materials. son’s scheme, which has been successfully used in many From the theoretical point of view, the crucial fact is topics [30–34,39,8], turns out to be the most efficient ap- that the calculation of the β functions in the two different proach. In this paper, we study the PC GLW model near perturbative approaches relies on qualitatively different D = 2 by means of the Wilson - Polchinski Exact Renor- grounds. Indeed, the β function of a NLσ model built on malization Group within the Local Potential Approxima- a manifold G/H only depends on the symmetry breaking tion (LPA). We show that the GLW and NLσ approaches scheme G H [25] – i.e. on Goldstone modes – whereas can be reconciled in the vicinity of two dimensions. More that of the→ GLW near D = 4 is sensitive to the repre- precisely we show by a RG analysis that the two models sentation of G spanned by the order parameter chosen to belong to the same universality class near two dimensions realize the symmetry breaking scheme. This feature can since the GLW model exhibits a non trivial fixed point be fully appreciated in the N = 3 PC model. Indeed, identical to that of the NLσ model. since SO(3) SO(3) is isomorphic to SO(4) the symme- The partition function of the PC GLW model is ob- try breaking⊗ pattern is that of the usual four component tained by writing the most general SO(N) SO(N)in- spin system: SO(4) SO(3). The perturbative β func- variant potential, at most quartic in N N ⊗real matrices tion of the N =3PCNL→ σmodel in D =2+is thus M, that favours orthogonal matrices for× the field M: identical to that of the N = 4 vector model, although 1 the symmetry breaking scheme is realized with a rotation Z = DM exp dDx Tr ( t M. M) matrix which is a SO(4) tensor and not with a four com- − 2 ∇ ∇ Z  Z ponent vector [13,14]. If this perturbative result remained (2) r true beyond D =2+, as it is believed in the SO(N) + Tr t MM +µTr (t MM)2 +λ(Tr tMM)2 . vectorial case, we could expect the critical behaviour of 2 the PC model to be determined by the same fixed point  as the N = 4 vector model everywhere between two and The domain of parameters of interest for us is given by four dimensions. This is however not the case, at least λ>0 since, in this case, the minimum of the poten- perturbatively in the vicinity of D = 4, since there is no tial in the broken phase is given by M(x)=R0 where R SO(N). In this phase, the model displays a SO(N) fixed point in the GLW approach. 0 ∈ The origin of the discrepancy between the two ap- symmetry, so that the symmetry breaking scheme is SO(N) SO(N) SO(N) and thus corresponds to the proaches can be ultimately traced back to the (non per- ⊗ → turbative) spectrum of both models. Whereas it is very GLW version of the PC model. likely that in the SO(N) case with a vectorial order pa- Our aim being to make contact with the NLσ model, rameter the NLσ and GLW models share the same low let us show how the orthogonality of the lattice order pa- energy degrees of freedom everywhere between two and rameter of (1) can be recovered from (2). Let r and µ four dimensions, it is no longer the case for models with go to infinity, the ratio r/4µ being fixed. In this limit, more general order parameters and symmetries. For ex- one recovers the partition function of the PC NLσ model ample, for the N = 3 PC model, the spectrum of the where a delta function enforces the orthogonality con- D =2NLσmodel consists in four massive particles [26] straint on M at each point: whereas the spectrum of the D = 4 GLW in the high 1 temperature phase involves nine massive particles. Ide- Z = DM exp dDx Tr ( t M. M) −2 ∇ ∇ ally, we should understand at a non-perturbative level Z  how these two field contents are linked together in D =3 R (3) r 2 and how they are related to the degrees of freedom of the +2µTr t MM + +2λ(Tr tMM)2 underlying microscopical system. This is a formidable 4µ    task that will not be tackled here. The question we address here is the possibility of a 1 1 matching between the NLσ and GLW approaches when DM δ tMM exp dDx Tr ( t M. M) → − g2 −2 ∇ ∇ varying the dimension. This allows, at the same time, to Z  0  Z test the validity of the NLσ model for frustrated systems, (4) at least around D = 2. Indeed, due to the discrepancy 2 between the two perturbative approaches and the ab- up to an overall constant. The quantity 1/g0 = r/4µ sence of experimental and numerical evidence of an O(4) which corresponds to the minimum of (3) (when λ− µ) critical behaviour, the reliability of the NLσ model ap- identifies with the inverse temperature of the NLσ model.

2 Of course, since the preceding limit is made on the bare u(χ, λ (t) )= λ (t) { p1,q1;...;pn,qn } p1,q1;...;pi,qi action, it does not allow to conclude how both models i pk,qk are related under RG transformations. We shall show X{X} q q that, around two dimensions, the GLW and NLσ models [Tr (χ 1)p1 ] 1 ... [Tr (χ 1)pi ] i . actually converge to the same renormalized trajectory in − − the continuum limit. (8) To realize this program we now study the evolution of The Wilson-Polchinski equation (7) generates the flow of the PC GLW model under RG transformations within all the λp ,q ;...;p ,q (t)’s. When combined with (6) we the LPA. This approximation consists in truncating the 1 1 n n also get the evolution of gt: effective Wilsonian action to its potential part V (M)= D 2 4 d xv(M(x)). Note that the LPA thus misses the field dgt 2 1 gt = (D 2)gt + renormalization. The Wilson-Polchinski equation for the dt − − 4π 2λ2,1(t)+2Nλ1,2(t) potentialR density v(M)isgivenby[2,4]: 2 ∂v D 2 1 (12N + 12)λ3,1(t)+24Nλ1,3(t)+4(N +N+4)λ2,1;1,1(t) = Dv − M v0 + v00 v0 v0 (5) ∂t − 2 ij ij 4π ij,ij − ij ij 

2 where vij0 = δv/δMij and t = ln Λ, Λ being the dimen- +4(2N +1)λ2,1(t)+4(N +2)λ1,2(t) . sionless running scale.  There are two different ways to exploit Eq. (5). The (9) first one is to search for an exact solution in any dimen- sion, having recourse to numerical integration. This pro- The flow analysis shows that all the λp1 ,q1;...;pn,qn (t)’s vides a powerful way to obtain precise values for critical are irrelevant coupling constants: after an exponentially quantities [8]. The second one is to solve Eq. (5) in a rapid transient regime, their scale dependence is entirely low temperature expansion. We follow this route since controlled by that of gt: we are interested in qualitative features of the RG flow ¯(o) ¯(1) 2 4 λp ,q ;...;p ,q (t) λ + λ g + O(g ) . and we want to make contact with the standard pertur- 1 1 n n → p1,q1;...;pn,qn p1,q1;...;pn,qn t t bative analysis of the NLσ model. Mitter and Ramadas (10) used the same techniques in the SO(N) case for a proof of perturbative renormalizability of the NLσ model [35]. Therefore, for any initial conditions, the flow is Let us parametrize the potential density v by v(M )= driven towards a one-dimensional renormalized trajec- 2 2 t tory parametrized by gt whose evolution, obtained from u(χ)/gt with χ = gt MM. In a perturbative approach the running potential has always a minimum as it is the (9) and (10), is given at leading orders by: case for the initial potential in (3) for tMM =1/g2.The 2 0 dgt N 1 running temperature g is thus defined via: = (D 2)g2 + − g4 + O(g6) . (11) t dt − − t 4π t t ∂u This β function identifies with that of the temperature =0. (6) ∂χ of the PC NLσ model calculated perturbatively [24].It χ =1 however differs from the standard expression where the We now write the Wilson-Polchinski equation for the po- coefficient N 1 is replaced by N 2. The origin of this − − tential density u within the LPA: difference is that, within the LPA, the field renormaliza- tion is set equal to one. If the field renormalization had ∂u been taken into account, which is the case in the next or- = Du (D 2)χ u0 ∂t − − ij ji ders in the derivative expansion, we would have obtained the correct coefficient. This difference is irrelevant for our purpose. ujl0 ujk0 + ujl0 ukj0 + ulj0 ujk0 + ulj0 ukj0 χlk − Let us indicate how, in two dimensions, our previous   g2 results allow to recover, in the continuum limit, the hard + t u + u + u + u χ +2Nu constraint of the NLσ model (4). After the transient 4π js,jp00 sj,jp00 js,pj00 sj,pj00 sp ii0 regime - i.e. Eq. (10) - u(χ, λp ,q ;...;p ,q (t) )hascon- 
 { 1 1 n n } verged towardsu ¯(χ, gt)whichcanbeexpandedinpowers 2 d 1 +g χij u0 u . of g2: t dt g2 ji − t  t   2 k (k) (7) u¯(χ, gt)= (gt) u¯ (χ). (12) k 0 Under the iterations of the RG, all SO(N) SO(N)in- X≥ variant terms are generated so that the evolved⊗ potential We have found the exact form ofu ¯(0)(χ) so that the dom- writes: inant part of the potential density at low temperature writes:

3 2 1 1 [5] A. Hasenfratz and P. Hasenfratz, Nucl. Phys. B270, 687 v(M) u¯(0)(χ)= Tr g 2 t MM 1 . (13) 2 2 t (1986). ∼ gt 2gt − q  [6] C. Bagnuls and C. Bervillier, J. Phys. Stud. 1, 366 Suppose now that, after blocking, the model having con- (1997). verged to the one-dimensional renormalized trajectory, [7] G. Zumbach, Nucl. Phys. B413, 754 (1994). the effective running temperature has reached the value [8] T.R. Morris, in New Developpements in Quantum Field Theory (NATO ASI series, Plenum Press) hep- gµ,atscaleµ. Reversing the flow on this trajectory, towards the ultraviolet, Eq. (11) gives the bare temper- th/9709100. [9] D.R.T. Jones, A. Love, and M.A. Moore, J. Phys. C 9, ature g0 at scale of the overall cut-off Λ0 (typically, the 743 (1976). inverse lattice spacing). Due to asymptotic freedom g0 [10] D. Bailin, A. Love, and M.A. Moore, J. Phys.C 10, 1159 goes to zero when taking the continuum limit Λ0 . It is easy to see from (13) that, in this limit, the configu-→∞ (1977). [11] T. Garel and P. Pfeuty, J. Phys.C , 245 (1976). rations contributing to the partition function correspond L9 [12] M. Yosefin and E. Domany, Phys. Rev. B 32, 1778 to SO(N) matrices (up to a normalization): the delta (1985). constraint of (4) is recovered from RG transformations. [13] P. Azaria, B. Delamotte, and Th. Jolicoeur, Phys. Rev. Thus, in the continuum limit, the GLW and NLσ models Lett. 64, 3175 (1990). coincide. The statistical interpretation of this is that the [14] P. Azaria, B. Delamotte, F. Delduc, and Th. Jolicoeur, soft field GLW model appears as the block-spin iterated Nucl. Phys. B408, 485 (1993). NLσ model. [15] B.I. Halperin, T.C. Lubensky, and S.H. Ma, Phys. Rev. These results show that the PC GLW and NLσ mod- Lett. 32, 292 (1974). els belong to the same universality class near two dimen- [16] C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, sions. This is a strong evidence of the validity of the NLσ 1556 (1981). model approach and of the existence of a second order [17] I.D. Lawrie and C. Athorne, J. Phys. A 16, 587 (1983). phase transition near two dimensions. Thus, the criti- [18] J. March-Russel, Phys. Lett. 296B, 364 (1992). cal behaviour of the PC GLW model must change as D [19] S. Hikami, Prog. Theor. Phys. 62, 226 (1979). varies between D =2andD= 4. This, of course, relies [20] T. Dombre and N. Read, Phys. Rev. B 39, 6797 (1989). on the assumptions that our results persist beyond the [21] B.D. Gaulin, Magnetic Systems with Competing Inter- low-temperature expansion and the LPA, and that the actions, (H.T. Diep Editor, World Scientific, 1994).  =4 Dexpansion of the GLW model is meaningful. [22] P. de V. Du Plessis, C.F. Van Doorn, and D.C. Van The change− of critical behaviour could be a general fea- Delden, J. Magn. Mater. 40, 91 (1983). ture of models that are afflicted by the same troubles even [23] H. Kawamura, J. Appl. Phys. 63, 3086 (1988). if their origins – presence of topological excitations, role [24] E. Br´ezin, S. Hikami, and J. Zinn-Justin, Nucl. Phys. of irrelevant operators – certainly depend on the precise B165, 528 (1980). model under study. In any case, analyzing this requires to [25] D.H. Friedan, Ann. Phys. 163, 318 (1985). [26] P.B. Wiegmann, Phys. Lett. , 217 (1984). vary the dimension and to use the next orders of approx- 141B [27] H. Kawamura, cond-mat/9805134. imation in the derivative expansion [36–38].Asomewhat [28] G. Ferretti, Nucl. Phys. B450, 713 (1995). similar study has been performed for superconductors [39] [40] [29] S. Nishigaki, Nucl. Phys. B376, 73 (1996). and for the Kosterlitz-Thouless phase transition .In [30] C. Wetterich, Phys. Lett. 301B, 90 (1993). the context of the PC model, it will be addressed in a [31] U. Ellwanger and L. Vergara, Nucl. Phys. B398,52 future publication. (1993). We thank P.K. Mitter, B. Dou¸cot and G. Zumbach [32] M. Reuter and C. Wetterich, Nucl. Phys. B417, 182 for very useful discussions about the Wilson RG point of (1994). view. We also thank J. Vidal for a careful reading of the [33] G. Zumbach, Nucl. Phys. B413, 771 (1994). manuscript. [34] G. Zumbach, Phys. Lett. 190A, 225 (1994). [35] P.K. Mitter and T.R. Ramadas, Commun. Math. Phys. 122, 575 (1989). [36] T.R. Morris, Phys. Lett. 329B, 241 (1994). [37] R. D. Ball, P.E. Haagensen, J.I. Latorre, and E. Moreno, Phys. Lett. 347B, 80 (1995). [38] J. Comellas, Nucl. Phys. B509, 662 (1998). [1] J. Zinn-Justin, Quantum Field Theory and Critical Phe- [39] B. Bergerhoff, F. Freire, D. Litim, S. Lola, and C. Wet- nomena, (Oxford University Press, New York, 1989). terich, Phys. Rev. B 53, 5734 (1996). [2] K.G. Wilson and J. Kogut, Physics Report C 12,75 [40] M. Grater and C. Wetterich, Phys. Rev. Lett. 75, 378 (1974). (1995). [3] F.J. Wegner and A. Houghton, Phys. Rev. A 8, 401, (1973). [4] J. Polchinski, Nucl. Phys. B231, 269 (1984).

4