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Communications in Theoretical Physics

The Numerical Study of the S = 1/2 XXZ Chain

Fei Ye, Guo-Hui Ding, Xu Bo-wei

Published Oct 15, 2001• 0 citations
Publisher

Abstract

We adopt the Lanczos method combined with the quantum conformal field theory to investigate the S = 1/2 XXZ chain in detail. The bulk-limit ground state energy, the anomalous scaling dimension of the spin operators, the Fermi velocity and the zero-temperature susceptibility are numerically calculated. The results agree to the exact solution well.

DOI
10.1088/0253-6102/36/4/483

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We adopt the Lanczos method combined with the quantum conformal field theory to investigate the S = 1/2 XXZ chain in detail. The bulk-limit ground state energy, the anomalous scaling dimension of the spin operators, the Fermi velocity and the zero-temperature susceptibility are numerically calculated. The results agree to the exact solution well.
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.
Highly precise numerical calculations using a variation of the Lanczos method devised by Paige were performed for s=1/2 antiferromagnetic Heisenberg finite chains up to sizes N=28. Several interesting physical quantities, including the ground-state energy, the mass gap, and the spin-wave velocity, were computed and fitted for logarithmic finite-size corrections, as suggested by conformal invariance and Bethe ansatz calculations. A crossover size for a maximum of the spin-wave velocity is predicted for N\ensuremath{\sim}50, well above the current available sizes in a typical Lanczos simulation. A list of our numerical results is given along with extrapolated values using standard algorithms.
We discuss the ground state of the Heisenberg spin-1/2 chain in one dimension, considering nearest neighbor interactions. In the thermodynamic limit the energy is calculated analytically in several ways. We start by applying the Bethe ansatz to calculate the exact ground state energy of the XXX Heisenberg model. Next we study small perturbations of the spin anisotropy, the parameter for the interaction in the z-direction for the XXZ model, by applying time-independent perturbation theory. In addition we expand an exact integral expression of the energy of the XXZ model around the spin anisotropy in the thermodynamic limit. The results are compared to numerical solutions, which are obtained by means of exact diagonalization up to twelve lattice sites. We find that for ten or more lattice sites the numerics agree well with the expectation from the analytical calculations. Finally the concept of integrability is discussed in the context of the Heisenberg model.