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Journal of Physics A Mathematical and General

New integrable quantum chains combining different kinds of spins

Hector J. de Vega, Ferenc Woynarovich

Published Sep 7, 1992• 111 citations
Publisher

Abstract

A general procedure to generate new integrable Hamiltonians combining to any kind of spins distributed arbitrarily on the line is given. As a concrete application, anisotropic chains formed by spin-1/2 and spin-1 operators at alternating sites are presented and solved exactly by the Bethe ansatz (BA). The authors compute the ground-state and excitation energies and momentum. The higher-order BA equations are derived. Depending on the choice, these new Hamiltonians exhibit, or not, conformal invariance in their low energy spectrum.

DOI
10.1088/0305-4470/25/17/012

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A general procedure to generate new integrable Hamiltonians combining to any kind of spins distributed arbitrarily on the line is given. As a concrete application, anisotropic chains formed by spin-1/2 and spin-1 operators at alternating sites are presented and solved exactly by the Bethe ansatz (BA). The authors compute the ground-state and excitation energies and momentum. The higher-order BA equations are derived. Depending on the choice, these new Hamiltonians exhibit, or not, conformal invariance in their low energy spectrum.
New exactly integrable quantum spin-1 chains are derived. These new quantum Hamiltonians, like the XXZ chain, have a free parameter (anisotropy) and commute with the z -component of the total magnetization. The eigenvalues and eigenvectors are obtained directly by imposing a generalized coordinate Bethe ansatz. The derived Bethe ansatz equations have an unusual form and the associated R -matrix, has a dependence with the spectral parameter that is not of difference form, like the Hubbard quantum chain.
We have recently constructed a large class of open quantum spin chains which have quantum-algebra symmetry and which are integrable. We show here that these models can be exactly solved using a generalization of the analytical Bethe Ansatz (BA) method. In particular, we determine in this way the spectrum of the transfer matrices of the Uq[su(2)]-invariant spin chains associated with A1(1) and A2(2) in the fundamental representation. The quantum-algebra invariance of these models plays an essential role in obtaining these results. The BA equations for these open chains are “doubled” with respect to the BA equations for the corresponding closed chains.
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.
The spectral properties and phase diagram of the exact integrable spin one quantum chain introduced by Alcaraz and Bariev are presented. The model has a U(1) symmetry and its integrability is associated to an unknown R-matrix whose dependence on the spectral parameters is not of difference form. The associated Bethe ansatz equations, that fix the eigenspectra, are distinct from those associated to other known integrable spin models. The model has a free parameter tpt_ptp​. We show that at the special point tp=1t_p=1tp​=1 the model gets an extra U(1) symmetry and reduces to the deformed SU(3) Perk-Schultz model at a special value of its anisotropy q=exp⁡(i2π/3)q=\exp(i2π/3)q=exp(i2π/3) andin the presence of an external magnetic field. Our analysis is done either by solving the associated Bethe-ansatz equations or by direct diagonalization of thequantum Hamiltonian for small lattice sizes. The phase diagram is calculated by exploring the consequences of conformal invariance on the finite-size corrections of the Hamiltonian eigenspectrum. The model exhibits a critical phase ruled by c=1c=1c=1 conformal field theory separated from a massive phase by first-order phase transitions.
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.