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Journal of Physics A Mathematical and General

Fractional strings hypothesis and non-simple laced integrable models

Marcio Jose Martins

Published Feb 7, 1991• 12 citations
Publisher

Abstract

The author discusses the ground state and some elementary excitations of the B p non-simple laced integrable models. The zeros of the associated Bethe ansatz equations are characterized by a special distribution in the complex plane, which is different from the usual string hypothesis. The introduction of this new solution makes it possible to calculate exactly various properties in the thermodynamic limit. The finite-size corrections to the eigenspectrum strongly suggest that the B p Wess-Zumino-Witten-Novikov models are the underlying conformal field theory for these non-simple laced models.

DOI
10.1088/0305-4470/24/3/011

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The author discusses the ground state and some elementary excitations of the B p non-simple laced integrable models. The zeros of the associated Bethe ansatz equations are characterized by a special distribution in the complex plane, which is different from the usual string hypothesis. The introduction of this new solution makes it possible to calculate exactly various properties in the thermodynamic limit. The finite-size corrections to the eigenspectrum strongly suggest that the B p Wess-Zumino-Witten-Novikov models are the underlying conformal field theory for these non-simple laced models.
The Bethe ansatz equations of the fundamental Sp (2 N ) integrable model are solved by a peculiar configuration of roots leading us to determine the nature of the excitations. They consist of N elementary generalized spinons and N -1 composite excitations made by special convolutions between the spinons. This fact is essential to determining the low-energy behaviour which is argued to be described in terms of 2 N Majorana fermions. Our results have practical applications to spin-orbital systems and also shed new light on the connection between integrable models and Wess-Zumino-Witten field theories.
We present a computational framework not based on the string hypothesis for the thermodynamics of statistical and quantum field theory models solvable by the Bethe ansatz. In the cases of XXZ Heisenberg chain and the sine-Gordon quantum field theory we derive a single nonlinear integral equation which determines the free energy (or ground-state scaling function). Our approach is very effective at high temperature, and correctly reproduces the low-temperature central charge and the analytic structure of the corrections.
This paper is concerned with the study of properties of the exact solution of the fundamental integrable G2 vertex model. The model R-matrix and respective spin chain are presented in terms of the basis generators of the G2 Lie algebra. This formulation permits us to relate the number of the Bethe roots of the respective Bethe equations with the eigenvalues of the U(1) conserved charges from the Cartan subalgebra of G2. The Bethe equations are solved by a peculiar string structure which combines complex three-strings with real roots allowing us to determine the bulk properties in the thermodynamic limit. We argue that G2 spin chain is gapless but the low-lying excitations have two different speeds of sound and the underlying continuum limit is therefore not strictly Lorentz invariant. We have investigated the finite-size corrections to the ground state energy and proposed that the critical properties of the system should be governed by the product of two c=1 conformal field theories. By combining numerical and analytical methods we have computed the bulk free-energy of the G2 vertex model. We found that there are three regimes in the spectral parameter in which the free-energy is limited and continuous. There exists however at least two sharp corner points in which the bulk-free energy is not differentiable.
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.
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
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 .