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International Journal of Modern Physics B

FUNCTIONAL RELATIONS FOR TRANSFER MATRICES OF THE CHIRAL POTTS MODEL

R. J. Baxter, Vladimir V. Bazhanov, Jacques H. H. Perk

Published Apr 1, 1990• 155 citations
Publisher

Abstract

It has recently been shown that the solvable N-state chiral Potts model is related to a vertex model with N-state spins on vertical edges, two-state spins on horizontal edges. Here we generalize this to a “j-state by N-state” model and establish three sets of functional relations between the various transfer matrices. The significance of the “super-integrable” case of the chiral Potts model is discussed, and results reported for its finite-size corrections at criticality.

DOI
10.1142/s0217979290000395

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FUSED POTTS MODELS

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Interfacial tension of the chiral Potts model

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We obtain the interfacial tension of the general solvable N-state chiral Potts model. It has exponent mu =(N+2)/(2N) in both the horizontal and vertical directions, in agreement with the scaling relation 2 mu =2- alpha .

Journal of Physics A Mathematical and General•Mar 21, 1994•23 citations
It has recently been shown that the solvable N-state chiral Potts model is related to a vertex model with N-state spins on vertical edges, two-state spins on horizontal edges. Here we generalize this to a “j-state by N-state” model and establish three sets of functional relations between the various transfer matrices. The significance of the “super-integrable” case of the chiral Potts model is discussed, and results reported for its finite-size corrections at criticality.
By the Baxter's Q72Q_{72}Q72​-operator method, we demonstrate the equivalent theory between the generalized τ(2)τ^{(2)}τ(2)-model (other than two special cases with a pseudovacuum state) and the NNN-state chiral Potts model with two alternating vertical rapidities, where the degenerate models are included. As a consequence, the theory of the XXZ chain model associated to cyclic representations (with the parameter ςςς) of Uq(sl2)U_{\sf q}(sl_2)Uq​(sl2​) with qN=1{\sf q}^N=1qN=1 for odd NNN is identified with either (for ςN=1ς^N=1ςN=1) the chiral Potts model with two superintegrable vertical rapidities, or (for ςN≠1ς^N \neq 1ςN=1) the degenerate model for the selfdual solution of the star-triangle relation. In all these identifications, the transfer matrices T,T^T, \hat{T}T,T^ of the chiral Potts model (including the degenerate ones) serve as the QR,QLQ_R, Q_LQR​,QL​-operators of the corresponding τ(2)τ^{(2)}τ(2)-model, so that the functional relations hold as in the solvable NNN-state chiral Potts model.
Generalizing the mapping between the Potts model with nearest neighbor interaction and the six-vertex model, we build a family of “fused Potts models” related to the spin k/2 U q su (2) invariant vertex model and quantum spin chain. These Potts models still have variables taking values 1, …,Q[Formula: see text] but they have a set of complicated multispin interactions. The general technique to compute these interactions, the resulting lattice geometry, symmetries, and the detailed examples of k=2, 3 are given. For Q>4, spontaneous magnetizations are computed on the integrable first order phase transition line, generalizing Baxter’s results for k=1. For Q≤4, we discuss the full phase diagram of the spin 1 (k=2) anisotropic and U q su (2) invariant quantum spin chain [it reduces in the limit Q=4 (q=1) to the much studied phase diagram of the isotropic spin 1 quantum spin chain], Several critical lines and massless phases are exhibited. The appropriate generalization of the valence bond state method of Affleck et al. is worked out.