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Journal of Physics A Mathematical and General

Central charge for the integrable higher-spin XXZ model

Henrik Johannesson

Published Jun 7, 1988• 13 citations
Publisher

Abstract

The central charge for the integrable higher-spin XXZ model is derived, exploiting the exact Bethe ansatz solution by Babujian and Tsvelick (1986). For integer values of v, where gamma = pi /v is the anisotropy parameter (0< gamma < pi /2S), we find c=3S/(S+1), with S the magnitude of the spin. Hence, contrary to expectation, the U(1)-invariant critical point of the theory does not renormalise onto a free massless scalar field when S>1/2.

DOI
10.1088/0305-4470/21/11/008

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The central charge for the integrable higher-spin XXZ model is derived, exploiting the exact Bethe ansatz solution by Babujian and Tsvelick (1986). For integer values of v, where gamma = pi /v is the anisotropy parameter (0< gamma < pi /2S), we find c=3S/(S+1), with S the magnitude of the spin. Hence, contrary to expectation, the U(1)-invariant critical point of the theory does not renormalise onto a free massless scalar field when S>1/2.
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.
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It is known that the standard method for calculating finite-size corrections in Bethe ansatz solvable systems is not applicable to the Takhtajan-Babujian model and its anisotropic XXZ generalisation. The authors develop a new analytic method explicitly avoiding root densities and associated problems. Nonlinear integral equations are derived whose solutions yield the correct central charges c=1 and c=3/2 for the spin-1/2 and spin-1 XXZ chains, respectively. In the spin-1 case the authors obtain as a by-product the finite-size deviation of the Bethe ansatz roots from the 2-string formation.
The Bethe-Ansatz approach is used to solve a nineteen-vertex model equivalent to a spin model with O(n) symmetry on the square lattice. The bulk free energy is determined exactly on four branches of critical points. On two of these branches, the central charge is derived from finite-size corrections. On the two other branches, the nature of the Bethe Ansatz solutions prevents us from computing the central charge analytically. In this case, our approach still allows the exact numerical determination of the free energy on strips of hundreds of sites wide.
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 .
We present a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisemberg chain we derive, for arbitrary values of the anysotropy, a {\bf single} non--linear integral equation from which the free energy can be exactly calculated. The high--temperature expansion follows in a sistematic and relatively simple way. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around T=0T=0T=0. Furthermore, we derive a single non-linear integral equation describing the finite--size ground--state energy of the Sine--Gordon quantum field theory. PACS: 05.30, 03.70. 75.10.5