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Journal of the Physical Society of Japan

Exact Diagonalization Study of the Spin 1/2XXZModel on the 4×4 Square Lattice

Macoto Kikuchi, Yutaka Okabe

Published Feb 15, 1989• 14 citations
Publisher

Abstract

We study the thermodynamic properties of the spin 1/2 quantum XXZ model with the XY -like anisotropy on the 4×4 square lattice by diagonalizing the Hamiltonian numerically. The temperature dependence of the energy, the specific heat and the uniform susceptibility are reported. The staggered and the in-plane magnetizations at the ground state are also calculated.

DOI
10.1143/jpsj.58.679

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We study the thermodynamic properties of the spin 1/2 quantum XXZ model with the XY -like anisotropy on the 4×4 square lattice by diagonalizing the Hamiltonian numerically. The temperature dependence of the energy, the specific heat and the uniform susceptibility are reported. The staggered and the in-plane magnetizations at the ground state are also calculated.
The spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model on an infinite square lattice at zero temperature has been studied and is presented here. Our methods using finite square lattices have been the exact diagonalization and quantum Monte Carlo methods. The physical properties estimated per lattice vertex include ground-state and first-excited-state energy, staggered magnetization, and susceptibility in the parallel and perpendicular directions, spin stiffness, and spin-wave velocity. In the XXZ\mathrm{XXZ}XXZ model range from the Ising model to the Heisenberg model our estimates of physical properties compare very well with estimates by several other methods published in several other papers. In the XXZ\mathrm{XXZ}XXZ range between the Heisenberg and XY\mathrm{XY}XY models there is very little in the literature to compare with our finite lattice estimates of physical properties.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.
Ground-state and thermodynamic properties of the spin-1/2 two-dimensional easy-plane XXZ model are investigated by both a Green's-function approach and by Lanczos diagonalization on lattices with up to 36 sites. We calculate the spatial and temperature dependences of various spin correlation functions, as well as the wave-vector dependence of the spin susceptibility for all anisotropy parameters delta. In the easy plane ferromagnetic region (-1 < delta < 0), the longitudinal correlators of spins at distance r change sign at a nite temperature T0 (delta, r). This transition, observed in the 2D case for the first time, can be interpreted as a quantum to classical crossover.
This paper describes an improvement in the method of exact diagonalization of Hamiltonians of quantum spin models on finite square lattices and the statistical analysis of the data so obtained to estimate the physical properties of the models on the infinite square lattices at zero temperature. The geometry and topology of finite square lattices are described. The models studied are the spin one-half XY and Heisenberg antiferromagnets using 28 finite square lattices with up to 32 vertices. Our estimates of the energy and magnetization on each model on the infinite lattice at zero temperature compare very well with recent estimates using quantum Monte Carlo, series expansion, and spin wave estimates. Estimates of spin wave velocity and transverse susceptibilities are more scattered.PACS No.: 75.10J