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 − <i;j> α=1 <i;j> 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.