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arXiv:cond-mat/9803210 · Journal of Physics A Mathematical and General

Dispersion relations and speeds of sound in special sectors for the integrable chain with alternating spins

B-D Dörfel, St Meißner

Published Jan 1, 1999• 1 citation
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Abstract

We investigate the anisotropic integrable chain consisting of spins and s = 1. Depending on the anisotropy and the ratio of the coupling constants it has different (antiferromagnetic) ground states, manifesting themselves in a different string content in the Bethe ansatz framework. In this paper we continue the study of those regions with different string content, below called sectors. First we compare the dispersion relations for the sectors with infinite Fermi zones. Further we calculate the speeds of sound for regions close to sector borders, where the Fermi radii either vanish or diverge, and compare the results.

DOI
10.1088/0305-4470/32/1/006

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We investigate the anisotropic integrable chain consisting of spins and s = 1. Depending on the anisotropy and the ratio of the coupling constants it has different (antiferromagnetic) ground states, manifesting themselves in a different string content in the Bethe ansatz framework. In this paper we continue the study of those regions with different string content, below called sectors. First we compare the dispersion relations for the sectors with infinite Fermi zones. Further we calculate the speeds of sound for regions close to sector borders, where the Fermi radii either vanish or diverge, and compare the results.
We investigate the anisotropic integrable spin chain consisting of spins and s = 1 by means of thermodynamic Bethe ansatz for the anisotropy , where the analysis of the Takahashi conditions leads to a more complicated string picture. We give the phase diagram with respect to the two real coupling constants and , which contain a new region where the ground state is formed by strings with infinite Fermi zones. In this region the velocities of sound for the two physical excitations have been calculated from the dressed energies. This leads to an additional line of conformal invariance not known before.
An anisotropic integrable spin chain, consisting of spins s = 1 and , is investigated [1]. It is characterized by two real parameters and , the coupling constants of the spin interactions. For the case and the ground-state configuration is obtained by means of thermodynamic Bethe ansatz. Furthermore, the low excitations are calculated. It turns out that apart from free magnon states being the holes in the ground-state rapidity distribution, there exist bound states given by special string solutions of Bethe ansatz equations (BAE) in analogy to [13]. The dispersion law of these excitations is calculated numerically.
An anisotropic integrable spin chain, consisting of spins s=1s=1s=1 and s=12s=\frac{1}{2}s=21​, is investigated \cite{devega}. It is characterized by two real parameters cˉ\bar{c}cˉ and c~\tilde{c}c~, the coupling constants of the spin interactions. For the case cˉ<0\bar{c}<0cˉ<0 and c~<0\tilde{c}<0c~<0 the ground state configuration is obtained by means of thermodynamic Bethe ansatz. Furthermore the low excitations are calculated. It turns out, that apart from free magnon states being the holes in the ground state rapidity distribution, there exist bound states given by special string solutions of Bethe ansatz equations (BAE) in analogy to \cite{babelon}. The dispersion law of these excitations is calculated numerically.
In this paper we continue the investigation of an anisotropic integrable spin chain, consisting of spins s = 1 and , started in our paper (Meiß ner St and Dörfel B-D 1996 J. Phys. A: Math. Gen. 29 1949) . The thermodynamic Bethe ansatz is analysed especially for the case, when the signs of the two couplings and differ. For the conformally invariant model we have calculated heat capacity and magnetic susceptibility at low temperature. In the isotropic limit our analysis is carried out further and susceptibilities are calculated near phase transition lines (at T = 0).
Abstract Calculations of low lying excited states in even length L → ∞ chains begun in 2019 J. Phys. Commun. 3 025007 in the total spin S = 0 sector are here extended to S > 0. Bethe string configurations that were observed to transition from their definition at small L to different ‘apparent’ configurations at large L are now understood to be a manifestation of the approximate chiral symmetry that develops with increasing chain length. It is conjectured that when chiral symmetry is built into the configuration definitions a complete explicit labelling of all low lying excited states by strings is obtained. The structure is of the form of a product of Bethe strings defining left and right chiral states of spin sL and sR and a central complex ‘string’ necessary to bind <?CDATA s⃗L{\vec{{\rm{s}}}}_{L}sL​?> s ⃗ L and <?CDATA s⃗R{\vec{{\rm{s}}}}_{R}sR​?> s ⃗ R into total <?CDATA S⃗.\vec{S}.S.?> S ⃗ . Rules based on the string content of a state yield its momentum and asymptotic energy, including the amplitude of the leading ln(L) correction.
The exact solution of an integrable anisotropic Heisenberg spin chain with nearest-neighbor, next-nearest-neighbor and scalar chirality couplings is studied, where the boundary condition is the antiperiodic one. The detailed construction of Hamiltonian and the proof of integrability are given. The antiperiodic boundary condition breaks the U(1)-symmetry of the system and we use the off-diagonal Bethe Ansatz to solve it. The energy spectrum is characterized by the inhomogeneous T−Q relations and the contribution of the inhomogeneous term is studied. The ground state energy and the twisted boundary energy in different regions are obtained. We also find that the Bethe roots at the ground state form the string structure if the coupling constant J=−1 although the Bethe Ansatz equations are the inhomogeneous ones.