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arXiv:cond-mat/9904042

An Exactly Solvable Constrained XXZ Chain

Francisco Castilho Alcaraz, R. Z. Bariev

Published Apr 2, 1999• 19 citations
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Abstract

A new family of exactly solvable models is introduced. These models are generalizations of the XXZ chain where the distance among spins up ($σ^z$-basis) cannot be smaller or equal to t (t=0,1,2,...). The case t=0 recovers the standard XXZ chain. The coordinate Bethe ansatz is applied and the phase diagram is calculated. Exploring the finite-size consequences of conformal invariance the critical exponents are evaluated exactly at the critical regions of the phase diagram

DOI
10.48550/arxiv.cond-mat/9904042

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A new family of exactly solvable models is introduced. These models are generalizations of the XXZ chain where the distance among spins up (σzσ^zσz-basis) cannot be smaller or equal to t (t=0,1,2,...). The case t=0 recovers the standard XXZ chain. The coordinate Bethe ansatz is applied and the phase diagram is calculated. Exploring the finite-size consequences of conformal invariance the critical exponents are evaluated exactly at the critical regions of the phase diagram
New exactly integrable quantum spin-1 chains are derived. These new quantum Hamiltonians, like the XXZ chain, have a free parameter (anisotropy) and commute with the z -component of the total magnetization. The eigenvalues and eigenvectors are obtained directly by imposing a generalized coordinate Bethe ansatz. The derived Bethe ansatz equations have an unusual form and the associated R -matrix, has a dependence with the spectral parameter that is not of difference form, like the Hubbard quantum chain.
The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.
The central charge of the Izergin-Korepin model the corresponding quantum spin chain, and the O (n) model is calculated analytically via the Bethe ansatz. The calculation extends a technique recently developed for the Zamolodchikov-Fateev model. In addition critical exponents and the central charge for these models are obtained from numerical solutions of Bethe ansatz equations for finite systems. As a physical application the authors find the exponents v=12/23 and gamma =53/46 for the Theta -transition of polymers in two dimensions.
We present a new and efficient method for deriving finite-size effects in statistical physics models solvable by Bethe Ansatz. It is based on the study of the functional that maps a function to the sum of its evaluations over the Bethe roots. A simple and powerful constraint is derived when applying this functional to infinitely derivable test functions with compact support, that generalizes then to more general test functions. The method is presented in the context of the simple spin-1/2 XXZ chain for which we derive the finite-size corrections to leading eigenvalues of the Hamiltonian for any configuration of Bethe numbers with real Bethe roots. The expected results for the central charge and conformal dimensions are recovered.