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Journal of Physics A Mathematical and General

Finite-size effects in the integrable XXZ Heisenberg model with arbitrary spin

Holger Frahm, Naichang Yu

Published Jun 7, 1990• 33 citations
Publisher

Abstract

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 .

DOI
10.1088/0305-4470/23/11/032

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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 .
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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.
We compute the finite size spectrum for the spin 1/2 XXZ chain with twisted boundary conditions, for anisotropy in the regime 0 < γ < π/2, and arbitrary twist θ. The string hypothesis is employed for treating complex excitations. The Bethe Ansatz equtions are solved within a coupled non-linear integral equation approach, with one equation for each type of string. The root-of-unity quantum group invariant periodic chain reduces to the XXZ1/2 chain with a set of twist boundary conditions (π/γ ∈ ZZ, θ an integer multiple of γ). For this model, the restricted Hilbert space corresponds to an unitary conformal field theory, and we recover all primary states in the Kač table in terms of states with specific twist and strings.
We compute the finite-size spectrum for the spin- chain with twisted boundary conditions, for anisotropy in the regime , and arbitrary twist . The string hypothesis is employed for treating complex excitations. The Bethe ansatz equtions are solved within a coupled nonlinear integral equation approach, with one equation for each type of string. The root-of-unity quantum group invariant periodic chain reduces to the chain with a set of twist boundary conditions ( , an integer multiple of ). For this model, the restricted Hilbert space corresponds to a unitary conformal field theory, and we recover all primary states in the Kac table in terms of states with specific twist and strings.
The Bethe-ansatz equations for the integrable spin-s isotropic Heisenberg antiferromagnet are solved numerically at a finite number of sites N. The vacuum and the lowest singlet and triplet solutions are presented for s=1/2, . . ., 2 up to sN=240. Through extrapolation of the finite-size data, the central charge and anomalous dimensions of the scaling operators for the underlying conformal field theory are calculated and found to agree with the expected theoretical values. The status of the Bethe string hypothesis about the structure of the solutions is discussed based on the obtained computer data up to s=9/2.
In an earlier work [1] J. Suzuki proposed a set of nonlinear integral equations (NLIE) to describe the excited state spectrum of the integrable spin-1 XXZ chain in its repulsive regime. In this paper we extend his equations for the attractive regime of the model, and calculate analytically the conformal spectrum of the spin chain. We also discuss the typical root configurations of the thermodynamic limit as well as the 2-string deviations of certain excited states of the model. Special objects appearing in the NLIE are also treated with special care.