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International Journal of Modern Physics A

SPECTRUM AND COMPLETENESS OF THE INTEGRABLE 3-STATE POTTS MODEL: A FINITE SIZE STUDY

Giuseppe Albertini, Srinandan Dasmahapatra, Barry M. McCoy

Published Apr 1, 1992• 38 citations
Publisher

Abstract

All eigenvalues of the transfer matrix of the integrable 3-state Potts model are computed as polynomials in the spectral variable for chains of length M ≤7. The zeroes of the eigenvalues are known to satisfy a Bethe's Ansatz equation and thus it is of particular interest that we find many solutions whose zeroes do not satisfy the traditional "string hypothesis". We also find many cases where the integers in the logarithmic form of the Bethe equations do not satisfy the monotonicity properties that they are usually assumed to possess. We present a classification of all eigenvalues in terms of sets of roots and show that, for all M, this classification yields a complete set.

DOI
10.1142/s0217751x92003719

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All eigenvalues of the transfer matrix of the integrable 3-state Potts model are computed as polynomials in the spectral variable for chains of length M ≤7. The zeroes of the eigenvalues are known to satisfy a Bethe's Ansatz equation and thus it is of particular interest that we find many solutions whose zeroes do not satisfy the traditional "string hypothesis". We also find many cases where the integers in the logarithmic form of the Bethe equations do not satisfy the monotonicity properties that they are usually assumed to possess. We present a classification of all eigenvalues in terms of sets of roots and show that, for all M, this classification yields a complete set.
In this paper, we consider the parameterization of the spectra of the three-state critical Potts quantum chain with integrable twisted boundary conditions in terms of Bethe ansatz type equations. The Bethe equations are found by investigating the structure of the eigenvalues of the respective twisted transfer matrices, and with the help of certain identities satisfied by the product of transfer matrix operators. We have studied the completeness of the spectrum in terms of the Bethe roots for small lattice sizes and have computed the eigenstate momenta. We found that the spins of the low-lying excitations can have fractional values in accordance with predictions of the underlying conformal field theory. We argue that our framework can be used to build integrable Hamiltonians whose spectra are determined by mixing different toroidal boundary conditions.
In this paper, we consider the parameterization of the spectra of the three-state critical Potts quantum chain with integrable twisted boundary conditions in terms of Bethe ansatz type equations. The Bethe equations are found by investigating the structure of the eigenvalues of the respective twisted transfer matrices, and with the help of certain identities satisfied by the product of transfer matrices operators. We have studied the completeness of the spectrum in terms of the Bethe roots for small lattice sizes and have computed the eigenstate momenta. We found that the spins of the low-lying excitations can have fractional values in accordance with predictions of the underlying conformal field theory. We argue that our framework can be used to build integrable Hamiltonians whose spectra are determined by mixing different toroidal boundary conditions.
The spectra of recently constructed auxiliary matrices for the six-vertex model respectively the spin s = 1/2 Heisenberg chain at roots of unity are investigated. Two conjectures are formulated both of which are proved for N = 3 and are verified numerically for several examples with N > 3. The first conjecture identifies an Abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable. The zeros of these polynomials are shown to fall into two sets. One consists of the solutions to the Bethe ansatz equations which determine the eigenvalues of the six-vertex transfer matrix. The other set of zeros contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry present at roots of 1. The second conjecture then states a polynomial identity which relates the complete string centres to the Bethe roots allowing one to determine the dimension of the degenerate eigenspaces. Its proof for N = 3 involves the derivation of a new functional equation for the auxiliary matrices and the six-vertex transfer matrix. Moreover, it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the Drinfeld polynomial. The latter is used to classify the finite-dimensional irreducible representations of the loop algebra . This suggests that the constructed auxiliary matrices not only enable one to solve the six-vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry.
The transfer matrix of the XXZ open spin-½ chain with general integrable boundary conditions and generic anisotropy parameter ( q is not a root of unity and | q | = 1) is diagonalized using the representation theory of the q -Onsager algebra. Similarly to the Ising and superintegrable chiral Potts models, the complete spectrum is expressed in terms of the roots of a characteristic polynomial of degree d = 2 N . The complete family of eigenstates are derived in terms of rational functions defined on a discrete support which satisfy a system of coupled recurrence relations. In the special case of linear relations between left and right boundary parameters for which Bethe-type solutions are known to exist, our analysis provides an alternative derivation of the results of Nepomechie et al and Cao et al . In the latter case the complete family of eigenvalues and eigenstates splits into two sets, each associated with a characteristic polynomial of degree d < 2 N . Numerical checks performed for small values of N support the analysis.
The most general cyclic representations of the quantum integrable τ 2 -model are analyzed. The complete characterization of the τ 2 -spectrum (eigenvalues and eigenstates) is achieved in the framework of Sklyanin's separation of variables (SOV) method by extending and adapting previous results of one of the authors: (i) the determination of the τ 2 -spectrum is reduced to the classification of the solutions of a given functional equation in a class of polynomials; (ii) the determination of the τ 2 -eigenstates is reduced to the classification of the solutions of an associated Baxter equation. These last solutions are proved to be polynomials for a quite general class of τ 2 -self-adjoint representations and the completeness of the associated Bethe ansatz type equations is derived. Finally, the following results are derived for the inhomogeneous chiral Potts model: (i) simplicity of the spectrum, for general representations; (ii) complete characterization of the chiral Potts spectrum (eigenvalues and eigenstates) and completeness of Bethe ansatz type equations, for the self-adjoint representations of the τ 2 -model on the chiral Potts algebraic curves.