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Journal of Physics A Mathematical and General

Finite-size corrections for ground states of the XXZ Heisenberg chain

Chris J. Hamer

Published Nov 11, 1986• 97 citations
Publisher

Abstract

The method of de Vega and Woynarovitch (1985) is used to calculate finite-size corrections to the ground-state energy in different sectors for the XXZ Heisenberg chain. Finite-size scaling amplitudes and correction-to-scaling exponents in the critical region are derived. Using conformal invariance, a scaling dimension x=( pi - gamma )/2 pi is extracted corresponding to the electric field operator in the 8-vertex model: this confirms a conjecture of Baxter and Kelland (1974). Finite-size scaling properties near the Kosterlitz-Thouless critical point Delta =-1 are discussed.

DOI
10.1088/0305-4470/19/16/030

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The method of de Vega and Woynarovitch (1985) is used to calculate finite-size corrections to the ground-state energy in different sectors for the XXZ Heisenberg chain. Finite-size scaling amplitudes and correction-to-scaling exponents in the critical region are derived. Using conformal invariance, a scaling dimension x=( pi - gamma )/2 pi is extracted corresponding to the electric field operator in the 8-vertex model: this confirms a conjecture of Baxter and Kelland (1974). Finite-size scaling properties near the Kosterlitz-Thouless critical point Delta =-1 are discussed.
The methods of de Vega and Woynarovitch (1985) are used to calculate finite-size corrections to the ground state energy in different sectors for the XXZ Heisenberg chain, in the critical region -1< Delta <1. The finite-size scaling amplitude for the mass gap between the lowest lying sectors is derived. Using conformal invariance, the scaling dimension is extracted for an associated operator, corresponding to the electric field operator in the 8-vertex model. The conjecture of Baxter and Kelland (1974) for the electric field exponent is confirmed.
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.