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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Duality relations for M coupled Potts models

Jesper Lykke Jacobsen

Published Jul 1, 2000• 33 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.62.r1

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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.
We establish explicit duality transformations for systems of M q-state Potts models coupled through their local energy density, generalising known results for M=1,2,3. The M-dimensional space of coupling constants contains a selfdual sub-manifold of dimension D_M = [M/2]. For the case M=4, the variation of the effective central charge along the selfdual 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.
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.
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.
This paper is concerned with the critical properties of a family of self-dual two-dimensional Z(N) models whose bulk free energy is exactly known at the self-dual point. The author's analysis is performed by studying the finite-size behaviour of the corresponding one-dimensional quantum Hamiltonians which also possess an exact solution at their self-dual point. By exploring finite-size scaling ideas and the conformal invariance of the critical infinite system the author calculates, for N up to 8, the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra. The results strongly suggest that the Z(N) quantum field theory of Zamolodchikov and Fateev is the underlying field theory associated with these statistical mechanical systems. The author also tests for the Z(5) case, the conjecture that these models correspond to the bifurcation points in the phase diagram of the General Z(N) spin model, where a massless phase originates.
Introduces a family of D-dimensional q-state quantum Potts models with multi-spin interactions, which are self-dual for any D. These models are introduced using their (d=D+1)-dimensional classical formulations and the transfer matrix technique. The self-duality is proved on the quantum Hamiltonian. In the limit q to infinity the ground state energy is obtained exactly and the phase transition is first-order. A 1/q expansion allows one to get an approximate expression for the line in the (q,D)-plane separating regions where the transition changes from first- to second-order.
The author discusses a new class of toroidal boundary conditions for one-dimensional quantum Hamiltonians with S n symmetry which are related to two-dimensional n-state Potts models in the extreme anisotropic Hamiltonian limit. At their self-dual point (a point where a second-order phase transition occurs for n=2, 3, 4) the duality transformation is shown to be an additional symmetry giving rise to a new class of 'duality twisted' toroidal boundary conditions. The corresponding Hamiltonians are given in terms of generators of the periodic Temperley-Lieb algebra with an odd number of generators. The author discusses as an example the critical Ising model. Here the complete spectrum for the new boundary conditions can be obtained from a projection mechanism in the spin-1/2 XXZ Heisenberg chain.