Bethe Ansatz for XXX chain with negative spin (original) (raw)

XXX Spin Chain: from Bethe Solution to Open Problems

2007

We present some open problems in the field of exactly solvable models. Two of the problems are related to the correlation functions of the XXX spin chain and the XXZ spin chain, one to the entropy of subsystems and one to the six vertex model with domain wall boundary conditions.

Origin of the singular Bethe ansatz solutions for the Heisenberg XXZ spin chain

Physica A: Statistical Mechanics and its Applications, 2000

We investigate symmetry properties of the Bethe ansatz wave functions for the Heisenberg XXZ spin chain. The XXZ Hamiltonian commutes simultaneously with the shift operator T and the lattice inversion operator V in the space of Ω = ±1 with Ω the eigenvalue of T . We show that the Bethe ansatz solutions with normalizable wave functions cannot be the eigenstates of T and V with quantum number (Ω, Υ) = (±1, ∓1) where Υ is the eigenvalue of V . Therefore the Bethe ansatz wave functions should be singular for nondegenerate eigenstates of the Hamiltonian with quantum number (Ω, Υ) = (±1, ∓1). It is also shown that such states exist in any nontrivial down-spin number sector and that the number of them diverges exponentially with the chain length.

Bethe ansatz solution for quantum spin-1 chains with boundary terms

Nuclear Physics B, 2002

The procedure for obtaining integrable open spin chain Hamiltonians via reflection matrices is explicitly carried out for some three-state vertex models. We have considered the 19-vertex models of Zamolodchikov-Fateev and Izergin-Korepin, and the Z2-graded 19-vertex models with sl(2|1) and osp(1|2) invariances. In each case the eigenspectrum is determined by application of the coordinate Bethe Ansatz.

Bethe Ansatz Solutions for Temperley–Lieb Quantum Spin Chains

International Journal of Modern Physics A, 2000

We solve the spectrum of quantum spin chains based on representations of the Temperley–Lieb algebra associated with the quantum groups [Formula: see text] for Xn=A1, Bn, Cn and Dn. The tool is a modified version of the coordinate Bethe ansatz through a suitable choice of the Bethe states which give to all models the same status relative to their diagonalization. All these models have equivalent spectra up to degeneracies and the spectra of the lower-dimensional representations are contained in the higher-dimensional ones. Periodic boundary conditions, free boundary conditions and closed nonlocal boundary conditions are considered. Periodic boundary conditions, unlike free boundary conditions, break quantum group invariance. For closed nonlocal cases the models are quantum group invariant as well as periodic in a certain sense.

Thermodynamic Bethe ansatz for the spin-1/2 staggered XXZ-model

Nuclear Physics B, 2003

We develop the technique of Thermodynamic Bethe Ansatz to investigate the ground state and the spectrum in the thermodynamic limit of the staggered XXZ models proposed recently as an example of integrable ladder model. This model appeared due to staggered inhomogeneity of the anisotropy parameter ∆ and the staggered shift of the spectral parameter. We give the structure of ground states and lowest lying excitations in two different phases which occur at zero temperature.

Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model

2014

The implementation of the algebraic Bethe ansatz for the XXZ Heisenberg spin chain in the case, when both reflection matrices have the upper-triangular form is analyzed. The general form of the Bethe vectors is studied. In the particular form, Bethe vectors admit the recurrent procedure, with an appropriate modification, used previously in the case of the XXX Heisenberg chain. As expected, these Bethe vectors yield the strikingly simple expression for the off-shell action of the transfer matrix of the chain as well as the spectrum of the transfer matrix and the corresponding Bethe equations. As in the XXX case, the so-called quasi-classical limit gives the off-shell action of the generating function of the corresponding trigonometric Gaudin Hamiltonians with boundary terms.

Spectrum and transition rates of the XX chain analyzed via Bethe ansatz

Physical Review B, 2004

As part of a study that investigates the dynamics of the s = 1 2 XXZ model in the planar regime |∆| < 1, we discuss the singular nature of the Bethe ansatz equations for the case ∆ = 0 (XX model). We identify the general structure of the Bethe ansatz solutions for the entire XX spectrum, which include states with real and complex magnon momenta. We discuss the relation between the spinon or magnon quasiparticles (Bethe ansatz) and the lattice fermions (Jordan-Wigner representation). We present determinantal expressions for transition rates of spin fluctuation operators between Bethe wave functions and reduce them to product expressions. We apply the new formulas to two-spinon transition rates for chains with up to N = 4096 sites.