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Journal of Physics A Mathematical and General

Finite-size corrections for spin-S Heisenberg chains and conformal properties

B -D Dorfel

Published Jul 7, 1989• 8 citations
Publisher

Abstract

The author compares recent results on finite-size corrections for the isotropic (XXX) Heisenberg chain (and its integrable generalisations) with former numerical and analytic calculations. For the ground state and its lowest excitations, leading and non-leading terms are considered. The analysis confirms the predictions of conformal invariance and sheds some new light on the fine structure of finite-size corrections.

DOI
10.1088/0305-4470/22/13/022

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The author compares recent results on finite-size corrections for the isotropic (XXX) Heisenberg chain (and its integrable generalisations) with former numerical and analytic calculations. For the ground state and its lowest excitations, leading and non-leading terms are considered. The analysis confirms the predictions of conformal invariance and sheds some new light on the fine structure of finite-size corrections.
Leading and next-to-leading-order finite-size corrections to the ground and first excited states are calculated for the spin-1/2 anisotropic Heisenberg model in the critical region. The analytic results are compared to numerical data obtained for chains up to a length of N=1024. It is found that, near the isotropic point, the asymptotic region where the results obtained for N to infinity are applicable sets in at very large N values, and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms, even at large (N>100) chain lengths.
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 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.