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arXiv:1002.4857 · Journal of Physics A Mathematical and Theoretical

Phase diagram and spectral properties of a new exactly integrable spin-1 quantum chain

Francisco Castilho Alcaraz, Gilberto M Nakamura

Published Mar 25, 2010• 8 citations

Abstract

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 $t_p$. We show that at the special point $t_p=1$ 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 $q=\exp(i2π/3)$ 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 $c=1$ conformal field theory separated from a massive phase by first-order phase transitions.

DOI: 10.1088/1751-8113/43/15/155002

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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

t_p

. We show that at the special point

t_p=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)

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=1

conformal field theory separated from a massive phase by first-order phase transitions.

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 study an integrable spin chain with an alternating array of spins S=1/2, 1 in external magnetic fields using the Bethe ansatz exact solution. The calculated magnetization possesses a cusp structure at a critical magnetic field H=H_{C}, at which the specific heat shows a divergence property. We also calculate finite-size corrections to the energy spectrum, and obtain the critical exponents of correlation functions with the use of conformal field theory (CFT). Low-energy properties of the model are described by two c=1 U(1) CFTs in HH_{C}.

The one-dimensional S = 1 quantum spin chain with a linear and bilinear nearest-neighbour interaction of equal strength is integrable by the nested Bethe ansatz. In this paper the model is studied in the presence of an external magnetic field and an internal crystal field. By solving the Bethe ansatz equations for chains up to length N = 360 we construct the complete phase diagram of the system. We discuss how the magnetization curves depend on the internal field. In the SU(3) phase, where all three densities of atoms with -component -1, 0, +1 are non-vanishing, lines of constant density can be approximately parametrized by modified hypocycloides.

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 authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.

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 .