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Physical review. B, Condensed matter

Phase boundaries of the isotropic helical Potts model on a square lattice

Mehran Kardar A

Published Sep 1, 1982• 14 citations
Publisher

Abstract

The helical Potts model, a generalization of the $q$-state Potts model relevant to the commensurate-incommensurate transition, is studied on the square lattice. Close to zero temperature this model can be mapped onto the six-vertex Baxter model in direct fields. This mapping indicates the presence of an incommensurate phase, characterized by algebraic decay of correlations with an exponent $\ensuremath{\eta}\ensuremath{\rightarrow}\frac{3}{{q}^{2}}$ as $T\ensuremath{\rightarrow}0$. A Lifshitz-like point occurs when the ferromagnetic, modulated, and disordered phases meet.

DOI
10.1103/physrevb.26.2693

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The helical Potts model, a generalization of the qqq-state Potts model relevant to the commensurate-incommensurate transition, is studied on the square lattice. Close to zero temperature this model can be mapped onto the six-vertex Baxter model in direct fields. This mapping indicates the presence of an incommensurate phase, characterized by algebraic decay of correlations with an exponent \ensuremathη\ensuremath→3q2\ensuremath{\eta}\ensuremath{\rightarrow}\frac{3}{{q}^{2}}\ensuremathη\ensuremath→q23​ as T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0. A Lifshitz-like point occurs when the ferromagnetic, modulated, and disordered phases meet.
The anisotropic triangular nearest-neighbor Ising model, with antiferromagnetic interactions, is studied. The phase diagram, as a function of temperature and field, has a complex structure. Two commensurate ordered phases are found. The transition to one is Ising-type. The second is reached from the disordered phase in one of two ways: either via an intermediate uniaxial incommensurate phase---which is reached at high temperatures by a Kosterlitz-Thouless transition and at low temperatures undergoes an incommensurate-commensurate transition---or by a single continuous transition. The latter transition is in the same universality class as the chiral three-state Potts model, with thermal exponent within 4% of that of the three-state Potts model. These two transition regimes are found to be separated by a Lifshitz point. The phase diagram and critical behavior were determined by analytical (symmetry analysis and free-fermion approximation) as well as numerical (phenomenological renormalization-group and Monte Carlo) methods.
Using the corner-transfer matrix renormalization group approach, we revisit the three-state chiral Potts model on the square lattice, a model proposed in the eighties to describe commensurate-incommensurate transitions at surfaces, and with direct relevance to recent experiments on chains of Rydberg atoms. This model was suggested by Huse and Fisher to have a chiral transition in the vicinity of the Potts point, a possibility that turned out to be very difficult to definitely establish or refute numerically. Our results confirm that the transition changes character at a Lifshitz point that separates a line of Pokrosky-Talapov transitions far enough from the Potts point from a line of direct continuous order-disorder transition close to it. Thanks to the accuracy of the numerical results, we have been able to base the analysis entirely on effective exponents to deal with the crossovers that have hampered previous numerical investigations. The emerging picture is that of a new universality class with exponents that do not change between the Potts point and the Lifshitz point, and that appear to be consistent with those of a self-dual version of the model, namely correlation lengths exponents νx=2/3 in the direction of the asymmetry and νy=1 perpendicular to it, an incommensurability exponent β¯=2/3, a specific heat exponent that keeps the value α=1/3 of the three-state Potts model, and a dynamical exponent z=3/2. These results are in excellent agreement with experimental results obtained on reconstructed surfaces in the nineties, and shed light on recent Kibble-Zurek experiments on the period-3 phase of chains of Rydberg atoms.
Motivated by experiments on nonmagnetic triangular-lattice Mott insulators, we study one candidate paramagnetic phase, namely the columnar dimer (or valence-bond) phase. We apply variants of the bond-operator theory to a dimerized and spatially anisotropic spin-1/2 Heisenberg model and determine its zero-temperature phase diagram and the spectrum of elementary triplet excitations (triplons). Depending on model parameters, we find that the minimum of the triplon energy is located at either a commensurate or an incommensurate wave vector. Condensation of triplons at this commensurate-incommensurate transition defines a quantum Lifshitz point, with effective dimensional reduction that possibly leads to nontrivial paramagnetic (e.g., spin-liquid) states near the closing of the triplet gap. We also discuss the two-particle decay of high-energy triplons, and we comment on the relevance of our results for the organic Mott insulator EtMe3{}_{3}3​P[Pd(dmit)2{}_{2}2​]2{}_{2}2​.
We investigate the classical chiral Ashkin-Teller model on a square lattice with the corner transfer matrix renormalization group algorithm. We show that the melting of the period-4 phase in the presence of a chiral perturbation takes different forms depending on the coefficient of the four-spin term in the Ashkin-Teller model. Close to the clock limit of two decoupled Ising models, the system undergoes a two-step commensurate-incommensurate transition as soon as the chirality is introduced, with an intermediate critical floating phase bounded by a Kosterlitz-Thouless transition at high temperature and a Pokrovsky-Talapov transition at low temperature. By contrast, close to the four-states Potts model, we argue for the existence of a unique commensurate-incommensurate transition that belongs to the chiral universality class, and for the presence of a Lifshitz point where the ordered, disordered, and floating phases meet. Finally, we map the whole phase diagram, which turns out to be in qualitative agreement with the 40-year-old prediction by Huse and Fisher.
Abstract We compute the exact partition function for the 3-state Potts model on square lattices of several sizes larger than previously accessible. Making comparison with the exactly solved Ising model we show that, for aspects of the analytic structure close to the ferromagnetic transition point, these lattices are large enough to approach the thermodynamic limit. Subject to certain assumptions this allows for computation of estimates for the specific heat critical exponent. We thus obtain an estimate for this exponent. The estimate is consistent with the known result, thus demonstrating the potential use of this method for other models. We also discuss the antiferromagnetic transition.