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Journal of Physics Condensed Matter

The longitudinal susceptibility of an XY spin-chain

Ibha Chatterjee

Published Aug 16, 2000• 3 citations
Publisher

Abstract

A previously unsolved problem has been solved numerically using transfer matrix renormalization group analysis, based on a density matrix renormalization group idea. This calculation leads to a high-precision prediction of the value of susceptibility in the x - or y - directions of an XY spin-chain, because in the calculation the transfer matrix idea is incorporated and a thermodynamic limit is ensured. From this calculation the susceptibility at zero temperature is predicted for a spin-½ antiferromagnetic XY model in one dimension.

DOI
10.1088/0953-8984/12/35/102

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A previously unsolved problem has been solved numerically using transfer matrix renormalization group analysis, based on a density matrix renormalization group idea. This calculation leads to a high-precision prediction of the value of susceptibility in the x - or y - directions of an XY spin-chain, because in the calculation the transfer matrix idea is incorporated and a thermodynamic limit is ensured. From this calculation the susceptibility at zero temperature is predicted for a spin-½ antiferromagnetic XY model in one dimension.
By extending White's density-matrix renormalization group (DMRG) technique to finite temperatures, we calculate the impurity susceptibility for the following systems: (a) a spin-1/2 impurity coupled to a single site of an antiferromagnetic Heisenberg spin-1/2 chain and (b) an impurity coupled symmetrically to the two spins at the open ends of the chain. We also perform a Monte Carlo simulation and show that the DMRG adaptation gives reliable results. The calculated impurity susceptibility shows Pauli-like behavior at low temperatures in the former case while a ln T divergence is found in the latter case, in agreement with predictions based on conformal field theory.
The density matrix renormalization group (DMRG) method is applied to the anisotropic Heisenberg chain at finite temperatures. The free energy of the system is obtained using the quantum transfer matrix which is iteratively enlarged in the imaginary time direction. The magnetic susceptibility and the specific heat are calculated down to T=0.01J and compared with the Bethe ansatz results. The agreement including the logarithmic correction in the magnetic susceptibility at the isotropic point is fairly good.
The density matrix renormalization group (DMRG) is a numerical method for studying low dimensional strongly correlated models. The study of these models is one of the most active areas of condensed matter physics today and has been spurred not only by the discovery of high temperature superconductivity but also by experiments on new compounds. In this thesis we study fundamentals of the density matrix renormalization group method as well as applications of the method to Heisenberg spin chains. The introductory text is meant as a brief overview of the DMRG as well as Heisenberg spin systems and provides an introduction to the papers. In papers I and II we investigate the density matrix renormalization group with the aim to better understand the underlying structure of the method. We find that if the renormalization converges to a fixed point in the thermodynamic limit, the resulting basis states can be written as a matrix product form. We also show an explicit relation of the DMRG to variational calculations using the fact that the matrix product states can be rederived through a variational ansatz without any reference to the DMRG calculation. It is then shown how the matrix product states can be used to define a set of Bloch waves for describing elementary excitations of the Heisenberg spin-1 chain. The low lying energy spectrum is calculated. In papers III and IV we investigate the thermodynamics of impurities in a spin-1/2 chain. For this purpose we have shown that a recently developed transfer matrix DMRG method can be used to calculate impurity properties. Both impurity corrections to thermodynamic properties and local properties at finite temperature can be obtained. The method gives quicker and more accurate results than quantum Monte Carlo. The numerical results show good agreement with field theory calculations for a number of impurity configurations and the cross-over function for the magnetic susceptibility from high- to low-temperature behavior shows the expected data collapse. Our results also give quantitative predictions for experiments on doped spin-1/2 chains.
A special limit of an antiferromagnetic XYZ\mathrm{XYZ}XYZ chain was recently shown to exhibit interesting bulk as well as surface spin-flop transitions at T=0.T=0.T=0. Here we provide a complete calculation of the thermodynamics of the bulk transition using a transfer-matrix renormalization-group method that addresses directly the thermodynamic limit of quantum spin chains. We also shed some light on certain spin-wave anomalies at low temperature predicted earlier by Johnson and Bonner.
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.