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arXiv:0803.2618 · Journal of Physics A Mathematical and Theoretical

Critical points in coupled Potts models and critical phases in coupled loop models

Paul Fendley, Jesper Lykke Jacobsen

Published May 6, 2008• 25 citations
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Abstract

We show how to couple two critical Q-state Potts models to yield a new self-dual critical point. We also present strong evidence of a dense critical phase near this critical point when the Potts models are defined in their completely packed loop representations. In the continuum limit, the new critical point is described by an SU(2) coset conformal field theory, while in this limit of the the critical phase, the two loop models decouple. Using a combination of exact results and numerics, we also obtain the phase diagram in the presence of vacancies. We generalize these results to coupling two Potts models at different Q.

DOI
10.1088/1751-8113/41/21/215001

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Making complex CFTs real: The two-dimensional Potts model for Q>4Q>4Q>4 and complex QQQ
The two-dimensional QQQ-state Potts model with real couplings has a first-order transition for Q>4Q>4Q>4. Starting from a triangular-lattice Potts model with two- and three-spin interactions, we study an equivalent loop model in which QQQ is a continuous parameter. By a combination of analytical and numerical arguments, we show that this loop model allows for the collision of a critical and a tricritical fixed point at Q=4Q=4Q=4. These then emerge as a pair of complex conformally invariant theories at Q>4Q>4Q>4, or even complex QQQ, for suitable complex coupling constants. We conjecture that all conformal data (such as the central charge, critical exponents, and three-point structure constants) can be obtained by analytic continuation of known exact results for the loop model with Q≤4Q \le 4Q≤4. This conjecture is checked, both for real Q>4Q>4Q>4 and for Q∈CQ \in \mathbb{C}Q∈C, by extensive transfer-matrix computations and comparison to previous studies for Q=5Q=5Q=5.
We establish explicit duality transformations for systems of M q-state Potts models coupled through their local energy density, generalizing known results for M=1,2,3. The M-dimensional space of coupling constants contains a self-dual submanifold of dimension D(M)=[M/2]. For the case M=4, the variation of the effective central charge along the self-dual surface is investigated by numerical transfer matrix techniques. Evidence is given for the existence of a family of critical points, corresponding to conformal field theories with an extended S(M) symmetry algebra.
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Boundary critical phenomena are studied in the three-state Potts model in two dimensions using conformal field theory, duality and renormalization group methods. A presumably complete set of boundary conditions is obtained using both fusion and orbifold methods. Besides the previously known free, fixed and mixed boundary conditions a new one is obtained. This illustrates the necessity of considering fusion with operators that do not occur in the bulk spectrum, to obtain all boundary conditions. It is shown that this new boundary condition is dual to the mixed ones. The phase diagram for the quantum chain version of the Potts model is analysed using duality and renormalization group arguments.
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Boundary critical phenomena are studied in the three-state Potts model in two dimensions using conformal field theory, duality and renormalization group methods. A presumably complete set of boundary conditions is obtained using both fusion and orbifold methods. Besides the previously known free, fixed and mixed boundary conditions a new one is obtained. This illustrates the necessity of considering fusion with operators that do not occur in the bulk spectrum, to obtain all boundary conditions. It is shown that this new boundary condition is dual to the mixed ones. The phase diagram for the quantum chain version of the Potts model is analysed using duality and renormalization group arguments.