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Le Journal de Physique Colloques

MONTE CARLO STUDY OF LOW-TEMPERATURE PROPERTIES OF QUANTUM FERROMAGNETIC XY MODEL ON THE TRIANGULAR LATTICE

Masako Takasu, Seiji Miyashita, Masuo Suzuki, Yasumasa Kanada

Published Dec 1, 1988• 0 citations
Publisher

Abstract

The S = 1/2 ferromagnetic XY model on the triangular lattice is investigated by quantum Monte Carlo methods based on the Suzuki-Trotter formula. The squared magnetization (M2x) / N2 is calculated for finite temperatures and extrapolated to T = 0. The existence of the long-range order is discussed. Kubo's susceptibility χxx is also calculated and compared with the squared magnetization.

DOI
10.1051/jphyscol:19888637

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Le Journal de Physique Colloques•Dec 1, 1988•1 citation
The S = 1/2 ferromagnetic XY model on the triangular lattice is investigated by quantum Monte Carlo methods based on the Suzuki-Trotter formula. The squared magnetization (M2x) / N2 is calculated for finite temperatures and extrapolated to T = 0. The existence of the long-range order is discussed. Kubo's susceptibility χxx is also calculated and compared with the squared magnetization.
The Suzuki-Trotter transformation has been used to transform NNN-spin S=12S=\frac{1}{2}S=21​ Heisenberg and XY\mathrm{XY}XY chains into two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) classical Ising models with complicated interactions. A Monte Carlo method using multispin flipping techniques is then used to study these models for $\frac{\mathrm{kT}}{|J|}\ensuremath{\ge}0.0125$ and $m\ensuremath{\le}10$. Finite-size scaling is used to analyze the low-temperature susceptibility of the ferromagnetic Heisenberg chain, and we find a power-law divergence $\ensuremath{\chi}={T}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}=1.32\ifmmode\pm\else\textpm\fi{}0.13$. Difficulties in the method, particularly due to singular bonding, are discussed.
Spin-1/2 nearest neighbor Heisenberg antiferromagnet and XY model on a square lattice are studied via large scale quantum Monte Carlo simulations using a fast and efficient multispin coding algorithm on the Caltech/JPL MarkIIIfp parallel supercomputer, based on the Suzuki-Trotter transformation. We performed simulations with very good statistics on lattices as large as 128x128 spins, in the temperature range from 0.1 to 2.5 in units of the effective exchange coupling J. We calculated energy, specific heat, magnetic susceptibilities and also spin correlation functions from which we deduce the correlation lengths. For the Heisenberg model, at temperatures higher than J the results are in excellent agreement with high-temperature series expansion. At low temperatures the long wavelength behavior is essentially classical. Our data show that the correlation length and staggered susceptibility are quantitatively well described by the renormalized classical picture at the 2-loop level of approximation. From the divergence of the correlation length, we deduce the value of the quantum renormalized spin stiffness, [rho][subscript s]/J = 0.199(2). We give evidence that the correlation function is of the Ornstein-Zernicke type. By comparing the largest measured correlation lengths with neutron scattering experiments on La2CuO4, we deduce the value of effective exchange coupling J = 1450±30 K. By measuring the imaginary time-dependent correlation functions, we show that the dynamics of the model can be well understood within a Bose liquid-type picture. The spin waves are rather sharp throughout most of the Brillouin zone and the damping is weakly dependent on the wave vector. In the case of the XY model, convincing numerical evidence is obtained on square lattices as large as 96x96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless (KT) phase transition at kTc/J = 0.350(4). The correlation length and in-plane susceptibility diverge at Tc precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly on heating near Tc and exhibits a peak around kT/J = 0.45. We also measure the spin stiffness and the correlation function exponent below the transition temperature. Within the statistical accuracy of the measurements, the results are well described by the square root singularity (with a nonuniversal amplitude) below Tc, and they have the universal values in agreement with KT theory at Tc.
We present high-precision quantum Monte Carlo results for the S=12S=\frac{1}{2}S=21​ XY\mathrm{XY}XY model on a two-dimensional square lattice, in the ground state as well as at finite temperature. The energy, the spin stiffness, the magnetization, and the susceptibility are calculated and extrapolated to the thermodynamic limit. For the ground state, we test a variety of finite-size scaling predictions of effective Lagrangian theory and find good agreement and consistency between the finite-size corrections for different quantities. The low-temperature behavior of the susceptibility and the internal energy is also in good agreement with theoretical predictions.
We analyse the low-temperature behaviour of the classical isotropic ferromagnetic Heisenberg model on a two-dimensional square lattice of finite size. Presence of a residual magnetization in a finite-size system enables us to use a low-temperature approximation, which is however more restricting than the usual spin-wave approximation known to give reliable results for the XY model at low temperatures T . For the system considered, we find that the spin–spin correlation function decays as 1/ r η( T ) for large separations r bringing about the presence of a quasi-long-range ordering. We give analytic estimates for the exponent η( T ) in different regimes and support our findings by Monte Carlo simulations of the model on lattices of different sizes at different temperatures.
We consider the zero-temperature properties of four different spin 1/2 models on two-dimensional lattices: the XY ferromagnet, the XY antiferromagnet, the Heisenberg antiferromagnet, and the Dzyaloshinsky–Moriya models. Most of this article is a review of previously published work, but a few previously unpublished results are included. The relation between three of the models on bipartite lattices is described. The properties of the XY ferromagnet in two dimensions, especially those derived from extrapolation of finite lattice results, are reviewed. A numerical factor by which spin-wave and finite-lattice estimates of the long-range order parameter differ is discussed. For frustrated models on the triangular lattice the possibility of a chirality phase transition instead of, or in addition to, a magnetic phase transition is considered.