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arXiv:hep-th/0004099 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Properties of the massive Thirring model from the XYZ spin chain

Marko Kolanović, S. Pallua, Predrag Dominis Prester

Published Jun 27, 2000• 2 citations
arXivHTMLPDF

Abstract

We consider here the massive Thirring model regularized with the $\mathrm{XYZ}$ spin chain. We numerically calculate the mass ratios of particles which lie in the discrete part of the spectrum and obtain results in accordance with the DHN formula and in disagreement with recent calculations in the literature based on the numerical Bethe ansatz and infinite momentum frame methods. We also analyze the short distance behavior of these states and evaluate the conformal dimensions. This paper, taken together with the previous one for the sine-Gordon model, confirms the duality relation between two models formulated by Klassen and Melzer [Int. J. Mod. Phys. A 8, 4131 (1993)].

DOI
10.1103/physrevd.62.025021

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We consider here the massive Thirring model regularized with the XYZ\mathrm{XYZ}XYZ spin chain. We numerically calculate the mass ratios of particles which lie in the discrete part of the spectrum and obtain results in accordance with the DHN formula and in disagreement with recent calculations in the literature based on the numerical Bethe ansatz and infinite momentum frame methods. We also analyze the short distance behavior of these states and evaluate the conformal dimensions. This paper, taken together with the previous one for the sine-Gordon model, confirms the duality relation between two models formulated by Klassen and Melzer [Int. J. Mod. Phys. A 8, 4131 (1993)].
We consider here the XXZ spin chain perturbed by the operator ${\ensuremath{\sigma}}^{x}$ (``in a transverse field'') which is a lattice regularization of the sine-Gordon model. This can be shown using conformal perturbation theory. We calculate the mass ratios of particles which lie in a discrete part of the spectrum and obtain results in accord with the DHN formula and in disagreement with recent calculations in the literature based on the numerical Bethe ansatz and infinite momentum frame methods. We also analyze the short distance behavior of these states (UV or conformal limit). Our result for conformal dimension of the second breather state is different from that conjectured by Klassen and Melzer [Int. J. Mod. Phys. A 8, 4131 (1993)] and is consistent with this paper for other states.
The thermodynamics of the massive Thirring model is discussed with the help of the Bethe Ansatz approach in a way different from Fowler and Zotos (1982). In particular a finite Debye cut-off is introduced to investigate effects of the lattice discreteness. The specific heat seems to have a structure as a sum of the Debye-type specific heat and a 'soliton' peak, when the authors take into account the relation of the massive Thirring model to the quantum sine-Gordon model. The finite temperature excitations are calculated. The excitation energy proposed by Fowler and Zotos is shown to have significance which is insensitive to the way of formulation.
The excitation spectrum of massive fermions with self-interaction moving in one space dimension is calculated, using exact solutions for the spin-1/2 chain problem. This spectrum is shown to be the same as in the massive Thirring model, sine-Gordon equation, and Tomonaga-Luttinger with gap, with suitable relations between coupling constants. Solitons and bound solitons occur, corresponding to magnons and bound magnons in the spin chain. The soliton mass is calculated as a function of lattice spacing, and the ratio of binding energy to soliton mass is shown to be given correctly by the WKB approximation.
We calculate for the first time the finite size corrections in the massive Thirring model. This is done by numerically solving the equations of periodic boundary conditions of the Bethe ansatz solution. It is found that the corresponding central charge extracted from the 1/L1/L1/L term is around 0.4 for the coupling constant of ${g}_{0}=\ensuremath{-}\ensuremath{\pi}/4$ and decreases down to zero when ${g}_{0}=\ensuremath{-}\ensuremath{\pi}/3.$ This is quite different from the predicted central charge of the sine-Gordon model as well as the light cone six vertex model.