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Physical Review Letters

Conformal invariance and the Heisenberg model with arbitrary spin

Francisco Castilho Alcaraz, Marcio Jose Martins

Published Aug 14, 1989• 23 citations
Publisher

Abstract

The eigenspectrum of the critical anisotropic Heisenberg model (or XXZ model), with arbitrary spin S in its disordered-ferroelectric regime (0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}\ensuremath{\pi}/2S) is solved by the Bethe-Ansatz method. The amplitudes of the leading finite-size corrections are calculated analytically and numerically. Using conformal invariance we give exact results for the conformal anomaly and scaling dimensions. Our results indicate that, for all spin S, the critical behavior is governed by a conformal field theory with central charge c=1.

DOI
10.1103/physrevlett.63.708

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The eigenspectrum of the critical anisotropic Heisenberg model (or XXZ model), with arbitrary spin S in its disordered-ferroelectric regime (0\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}\ensuremath{\pi}/2S) is solved by the Bethe-Ansatz method. The amplitudes of the leading finite-size corrections are calculated analytically and numerically. Using conformal invariance we give exact results for the conformal anomaly and scaling dimensions. Our results indicate that, for all spin S, the critical behavior is governed by a conformal field theory with central charge c=1.
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 eigenenergies for the spin SSS generalization of the critical XXZ\mathrm{XXZ}XXZ model are calculated by solving numerically, for finite chains, their associated Bethe-Ansatz equations. The conformal anomaly and scaling dimensions are obtained from the conformal invariance of the infinite chain. Our results reveal that these models are governed by a conformal field theory with central charge c=3S(S+1)c=\frac{3S}{(S+1)}c=(S+1)3S​ and having continuously varying critical exponents, like the eight-vertex model. We also show that they are related to a large class of unitary conformal field theories.
The finite-size effects in the spectrum of the integrable XXZ Heisenberg chain with arbitrary spin-S are studied analytically on the basis of the string hypothesis for bound states and numerically by solving the associated Bethe ansatz equations. The finite-size corrections to the spectrum and to the low-temperature behaviour of the free energy are found to be similar to the universal predictions for conformally invariant theories, although the model is, in general, not Lorentz invariant, since it can have an arbitrary number of branches for low-energy excitations with different Fermi velocities depending on the value of S and the anisotropy parameter gamma .
The critical O(n)\mathrm{O}(n)O(n) model on a finite honeycomb lattice is solved by the Bethe-Ansatz method. Amplitudes of the dominant finite-size corrections to part of the eigenspectrum are obtained analytically. A comparison with Cardy's predictions from the theory of conformal invariance leads to the exact results for the conformal anomaly and scaling dimensions.
The effect of anisotropic interactions on the eigenvalue spectrum of the row-to-row transfer matrix of critical lattice spin models is investigated. It is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta =(a y /a x ) sin theta where a y and a x are lattice spacings and theta is an angle describing the anisotropy. For exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices. This is done for the eight-vertex model, hard hexagons and interacting hard squares and it is found that theta = pi u/ lambda , 10u/3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter. For each of these models, the amplitude of the finite-size corrections to the free energy at criticality is found to be of the form pi zeta c/6N 2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c=1, 4/5 and 7/10 respectively. This is an analytic result for the eight-vertex model. For the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to N=48. Finally, the authors argue that the anisotropy angle for magnetic hard squares and the q-state Potts models is also given by theta = pi u/ lambda , so this result is quite general.