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arXiv:cond-mat/0511272 · Journal of Statistical Mechanics Theory and Experiment

The openXXZ-chain: bosonization, the Bethe ansatz and logarithmic corrections

Jesko Sirker, Michael Bortz

Published Jan 12, 2006• 29 citations
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Abstract

We calculate the bulk and boundary parts of the free energy for an open spin-1/2 XXZ -chain in the critical regime by bosonization. We identify the cut-off independent contributions and determine their amplitudes by comparing with Bethe ansatz calculations at zero temperature T . For the bulk part of the free energy we find agreement with Lukyanov's result (1998 Nucl. Phys. B 522 533). In the boundary part we obtain a cut-off independent term which is linear in T and determines the temperature dependence of the boundary susceptibility in the attractive regime for . We further show that at particular anisotropies where contributions from irrelevant operators with different scaling dimensions cross, logarithmic corrections appear. We give explicit formulae for these terms at those anisotropies where they are most important. We verify our results by comparing with extensive numerical calculations based on a numerical solution of the T = 0 Bethe ansatz equations, the finite temperature Bethe ansatz equations in the quantum transfer matrix formalism, and the density matrix renormalization group applied to transfer matrices.

DOI
10.1088/1742-5468/2006/01/p01007

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We calculate the bulk and boundary parts of the free energy for an open spin-1/2 XXZ -chain in the critical regime by bosonization. We identify the cut-off independent contributions and determine their amplitudes by comparing with Bethe ansatz calculations at zero temperature T . For the bulk part of the free energy we find agreement with Lukyanov's result (1998 Nucl. Phys. B 522 533). In the boundary part we obtain a cut-off independent term which is linear in T and determines the temperature dependence of the boundary susceptibility in the attractive regime for . We further show that at particular anisotropies where contributions from irrelevant operators with different scaling dimensions cross, logarithmic corrections appear. We give explicit formulae for these terms at those anisotropies where they are most important. We verify our results by comparing with extensive numerical calculations based on a numerical solution of the T = 0 Bethe ansatz equations, the finite temperature Bethe ansatz equations in the quantum transfer matrix formalism, and the density matrix renormalization group applied to transfer matrices.
In the first part we calculate the boundary susceptibility χ B in the open XXZ -chain at zero temperature and arbitrary magnetic field h by the Bethe ansatz. We present analytical results for the leading terms when | h | ≪ α, where α is a known scale, and a numerical solution for the entire range of fields. In the second part we calculate susceptibility profiles near the boundary at finite temperature T numerically by using the density-matrix renormalization group for transfer matrices and analytically for T ≪ 1 by field theoretical methods. Finally, we compare χ B at finite temperature with a low-temperature asymptotics which we obtain by combining our Bethe ansatz result with recent predictions from bosonization.
We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.
We consider the open spin- s XXZ quantum spin chain with N sites and general integrable boundary terms for generic values of the bulk anisotropy parameter, and for values of the boundary parameters which satisfy a certain constraint. We derive two sets of Bethe ansatz equations, and find numerical evidence that together they give the complete set of (2 s +1) N eigenvalues of the transfer matrix. For the case of s = 1, we explicitly determine the Hamiltonian, and find an expression for its eigenvalues in terms of Bethe roots.
The density-matrix renormalization-group technique is used to calculate the spin-correlation functions 〈SjxSkx〉〈{S}_{j}^{x}{S}_{k}^{x}〉〈Sjx​Skx​〉 and 〈SjzSkz〉〈{S}_{j}^{z}{S}_{k}^{z}〉〈Sjz​Skz​〉 of the one-dimensional S=12S=\frac{1}{2}S=21​ XXZ\mathrm{XXZ}XXZ model in the gapless regime. The numerical results for open chains of 200 spins are analyzed by comparing them with correlation functions calculated from a low-energy field theory. This gives precise estimates of the amplitudes of the correlation functions in the thermodynamic limit. The exact amplitude recently conjectured by Lukyanov and Zamolodchikov and the logarithmic correction in the Heisenberg model are confirmed numerically.
Exact low-temperature asymptotic behavior of boundary contribution to specific heat and susceptibility in the one-dimensional spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model with exchange anisotropy $1/2<\ensuremath{\Delta}<~1$ is analytically obtained using the Abelian bosonization method. The boundary spin susceptibility is divergent in the low-temperature limit. This singular behavior is caused by the first-order contribution of a bulk leading irrelevant operator to boundary free energy. The result is confirmed by numerical simulations of finite-size systems. The anomalous boundary contributions in the spin isotropic case are universal.