Abstract
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.
Highlights
We will scan the phase diagram starting from large, where a PT transition can be fully characterized, to smaller values of, where there appears to be a unique transition with universal exponents, through a Lifshitz point where the intermediate critical phase disappears
The picture that emerges from our results for the commensurate-incommensurate transition of the chiral three-state Potts model is that of a continuous order-disorder transition from the Potts point at = 0 to a Lifshitz point at = 0.169(3), followed by a Pokrovsky-Talapov transition into a critical phase
We have revisited the phase diagram of the chiral three-state Potts model using a tensor network method introduced in the late nineties, the Corner Transfer Matrix Renormalization Group
Summary
Since its introduction by Ostlund [1] and Huse [2] in the context of commensurateincommensurate transitions, the chiral Potts model has been the focus of an uninterrupted activity both in its two-dimensional statistical physics formulation [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27], and in its one-dimensional quantum version [28,29,30,31,32,33,34,35,36,37,38,39,40]. The critical temperature is known exactly from a duality argument, Tc = 3/[2 ln( 3 + 1)], and the correlation length diverges with an exponent ν = 5/6. Away from this point, the chiral perturbation introduced by is relevant, and the transition has to be modified in an essential way. Huse and Fisher suggested in 1982 that the transition could remain a direct commensurate-incommensurate transition up to a Lifschitz point L, but in a new chiral universality class characterized by q ξx → cst > 0, where ξx is the correlation length in the x direction, and q is the incommensurate vector in the high temperature phase [3]. While most numerical results seem to be consistent with a single transition for not too large , some authors have reached a different conclusion [16,20], it has proven exceedingly difficult to determine these critical exponents, either numerically with Monte Carlo [4,11,12], or using finite-size renormalization group [7,10] or finite-size transfer matrix [13,16], and the question remains unsettled as to whether there is a chiral transition, or rather a very narrow floating phase up to the Potts point
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
