Ordered phase and scaling in Zn models and the three-state antiferromagnetic Potts model in three dimensions Masaki Oshikawa Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan (July 20, 1999) the spontaneously broken Zn symmetry and the high- Based on a Renormalization-Group picture of Zn symmet- temperature disordered phase. The intermediate phase is ric models in three dimensions, we derive a scaling law for the O(2) symmetric and corresponds to the low-temperature Zn order parameter in the ordered phase. An existing Monte phase of the XY model. Carlo calculation on the three-state antiferromagnetic Potts On three dimensional (3D) case, Blankschtein et al.3 model, which has the effective Z6 symmetry, is shown to be in 1984 proposed an RG picture of the Z models, to consistent with the proposed scaling law. It strongly supports 6 the Renormalization-Group picture that there is a single mas- discuss the STI model. They suggested that the transi- sive ordered phase, although an apparently rotationally sym- tion between the ordered and disordered phases belongs metric region in the intermediate temperature was observed to the (3D) XY universality class, and that the ordered numerically. phase reflects the symmetry breaking to Z6 in a large enough system. It means that there is no finite region of 75.10.Hk, 05.50+q rotationally symmetric phase which is similar to the or- dered phase of the XY model. Unfortunately, their paper is apparently not widely known in the related fields. It might be partly because their discussion was very brief I. INTRODUCTION and not quite clear. In fact, there has been a long-standing controversy on The symmetry and the dimensionality are important the the three-state antiferromagnetic Potts (AFP) model factors to determine the universality class of critical phe- on a simple cubic lattice, defined by the Hamiltonian nomena. The O(2) symmetry is the simplest among the continuous symmetry, and statistical models with the H =+ δσj σk , (2) O(2) symmetry has been studied intensively. A natu- hXj,ki ral question then would be the effect of the symmetry breaking from the continuous O(2) to the discrete Z . A where σj = 0, 1, 2 and j, k runs over nearest neighbor n h i simple spin model with Zn symmetry is the n-state clock pairs on a simple cubic lattice. The order parameter of model with a Hamiltonian this model is not evident. However, previous studies re- vealed that the low-temperature ordered phase, which 4 H = cos(θj θk), (1) is called as Broken Sublattice Symmetry (BSS) phase , − − hXj,ki corresponds to a spontaneous breaking of the Z6 sym- metry. Thus the effective symmetry of this model may where j, k runs over nearest neighbors, and θj takes be regarded5 as Z , although it is not apparent in the h i 6 integral multiples of 2π/n. The standard XY model with model. It is now widely accepted that there is a phase O(2) symmetry is defined by the Hamiltonian of the same transition with critical exponents characterized by the form; the only difference is that θ takes continuous values. 3D XY universality class8–12, at temperature T 1.23 c ∼ The Zn symmetry is fundamentally different from O(2) (we set the Boltzmann constant kB = 1.) On the other because of its discrete nature. On the other hand, for hand, according to numerical calculations, there appears large n, it is natural to expect the Zn symmetry to have to be an intermediate phase below Tc and above the similar effects to that of the O(2) symmetry. Under- low-temperature phase. While there have been various standing these two apparently contradictory aspects is proposals5–7 for the intermediate region, most reliable an interesting problem. Besides the theoretical motiva- numerical results at present indicates that the interme- tion, there are some possible experimental realizations diate region appears to be rotationally symmetric phase arXiv:cond-mat/9907388v1 [cond-mat.stat-mech] 26 Jul 1999 of the effective Zn symmetry. For example, the stacked which is similar to the ordered phase of the 3D XY triangular antiferromagnetic Ising (STI) model with ef- model10–12. However, the “transition” between the in- fective Z6 symmetry may correspond to materials such termediate region and the low-temperature phase is not 1 as CsMnI3. well understood. According to the suggestion in Ref. 3, In two dimensions, the phase diagram of the Zn model the intermediate “phase” would be rather a crossover to is well understood2 in the framework of the renormal- the low-temperature massive phase. ization group (RG). For n 5, there is an intermediate On the other hand, there has been a claim of an inter- ≥ phase between the low-temperature ordered phase with mediate phase13 also in the 6-state clock (6CL) model, 1 which has the manifest Z6 symmetry. In a recent de- to be 4 in O(ǫ) In fact, n = 2 and n = 3 corresponds tailed numerical study, Miyashita14 found that the inter- to the 3D Ising and 3-state (ferromagnetic) Potts model, mediate region appears to have a rotationally symmetric which do not belong to XY universality class. Thus nc character, as found in the AFP model. However, through is expected to be at least 4. This is consistent with the a careful examination of the system size dependence, he above result from O(ǫ). However, extrapolating the low- concluded that it is just a crossover to the massive low- est order result in ǫ to 3D (ǫ = 1) is not quite reliable; temperature phase, and that the rotationally symmetric the true value of nc might be larger than 4. On the other XY phase does not exist in the thermodynamic limit. His hand, we can make following observation. For n 6, λn conclusion is consistent with the suggestion in Ref. 3. is marginal or irrelevant at the 3D Gaussian fixed≥ point In this article, based on the RG picture, we derive a (g = 0). Thus it is natural to expect them to be irrelevant scaling law of an order parameter which measures the ef- at the more stable 3D XY fixed point, namely n 6. In c ≤ fect of symmetry breaking from O(2) to Zn. We demon- fact, the numerical observation of the 3D XY universality strate that the Monte Carlo results on the AFP model in class in 6CL and AFP model strongly suggests that λ6 Ref. 11 is consistent with the scaling law, supporting the is irrelevant at the XY fixed point and hence nc 6. In RG picture with a single phase transition. the following, we restrict the discussion to the irrelevant≤ case n n . ≥ c For the O(2) symmetric case λn = 0, low-temperature II. RENORMALIZATION-GROUP PICTURE phase u< 0 is renormalized to the low-temperature fixed point. It describes the massless Nambu-Goldstone (NG) Since the discussion of the RG picture in Ref. 3 was modes on the groundstate with the spontaneously broken rather brief, it would be worthwhile to present the RG O(2) symmetry. Let us call the low-temperature fixed picture here, with some clarifications and more details. point as NG fixed point. In terms of the field theory, it is We also make a straightforward extension to general in- described by the O(2) sigma model (free massless boson teger n from the n = 6 case. field) A generic Zn symmetric model may be mapped, in the 4 3 K 2 long-distance limit, to the following Φ -type field theory S = d x (∂µφ) (5) with the Euclidean action Z 2 where φ is the angular variable Φ Φ eiφ. Namely, only 3 2 2 4 n n S = d x ∂µΦ + u Φ + g Φ λn(Φ + Φ¯ ) (3) ∼ | | Z | | | | | | − the angular mode φ remains gapless as a NG boson. In three dimensions, the coupling constant K renormalizes with the complex field Φ and its conjugate Φ.¯ The λn- proportional to the scale l, and goes to infinity in the low- term is the lowest order term in Φ which breaks the energy limit. The coupling constant may be absorbed by symmetry from O(2) to Zn. The phase transition cor- using the rescaled field θ = √K(φ φ0) so that the action 3 2 − responds to the vanishing of (the renormalized value of) is always written as d x (∂µθ) /2. the parameter u. The temperature T in the Zn statistical Now let us considerR effects of the symmetry break- system roughly corresponds to u as u T Tc where Tc ing λn. The symmetry breaking term can be written is the critical temperature. ∼ − as λ (Φn + Φ¯ n) = λ Φ n cos nφ. Using the rescaled − n − n| | In the absence of the symmetry breaking λn, the tran- field θ, the total effective action at scale l becomes sition belongs to the so-called 3D XY universality class. 3 1 2 3 3 θ Its stability under the symmetry breaking to Zn is de- S = d x (∂µθ) λnK d x cos [n(φ0 + )], Z 2 − Z √ termined by the scaling dimension of λn at the 3D XY K fixed point. It may be estimated with the standard ǫ- (6) expansion method. The lowest order result in ǫ can be easily obtained from where the factor K3 l3 comes from the scale trans- ∼ the Operator Product Expansion (OPE) coefficients15. formation of the integration measure. In the thermody- As a result, we obtain the scaling dimension y of λ in namic limit, we should take K limit.