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Progress of Theoretical Physics

Monte Carlo Simulation of Quantum Spin Systems. I

Masuo Suzuki, Seiji Miyashita, A. Kuroda

Published Nov 1, 1977• 271 citations
Publisher

Abstract

A general explicit formulation of Monte Carlo simulation for quantum systems is given in this paper on the basis of the previous fundamental proposal by Suzuki. This paper also demonstrates explicitly the possibility of it and gives new interesting physical results on the two-dimensional XY-model. That is, the present preliminary simulation seems to indicate a phase transition with a divergent susceptibility, and a very weak singularity of specific heat if it exists, and without long-range order.

DOI
10.1143/ptp.58.1377

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A general explicit formulation of Monte Carlo simulation for quantum systems is given in this paper on the basis of the previous fundamental proposal by Suzuki. This paper also demonstrates explicitly the possibility of it and gives new interesting physical results on the two-dimensional XY-model. That is, the present preliminary simulation seems to indicate a phase transition with a divergent susceptibility, and a very weak singularity of specific heat if it exists, and without long-range order.
The generalized Trotter formula is used to map the two-dimensional spin-l/2 XY model onto a three-dimensional lsing model with complicated many-spin interactions. This relationship sused to construct an efficient Monte Carlo algorithm. Simulation data for the specific heat and vortex correlation functions give strong evidence for the existence of a phase transition. Although it is well known that there cannot exist long-range order in the two-dimensional (2D) spin-1/2 XY model at non-zero temperature [1], high-temper-ature series expansions [2] suggest he existence of critical behavior. Little is known about the quantum model mainly because most techniques which have been succesfully applied to the classical model (also referred to as planar rotator model) are very difficult to extend to the quantum case. In particular, quantum renormalization group analYsis of the 2D spin-1/2 XY model have given either contradictive or unreliable re-suits [3-8]. An alternative method to calculate the thermody-namic properties of this modelis to exploit the formal analogy between d-dimensional quantum spin systems and (d + 1)-dimensional Ising-spin models [9]. Using ~ ' This paper was first submitted to Physical Review Letters
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.
Monte Carlo simulations are performed for the spin-\textonehalf{} XY\mathrm{XY}XY model in two dimensions for large (up to 24\ifmmode\times\else\texttimes\fi{}24 sites) lattices. Results are obtained over a wide temperature range which includes the critical temperature Tc{T}_{c}Tc​, estimated to be 0.4-0.5. The energy, specific heat, vortex density, and derivative of the helicity modulus are given as functions of temperature. As the lattice size is increased, the specific heat per spin approaches a finite value and does not diverge.
A new efficient scheme for the finite-size-scaling study of ground-state phase transitions in quantum spin systems is proposed on the basis of the quantum Monte Carlo method. That is, Trotter numbers and simulated temperatures are scaled in accordance with system sizes, and only one simulation is required for each system size. The transition point is estimated independently of the critical exponents using the non-equilibrium relaxation of the order parameter in a very large system. This scheme is confirmed by the estimation of the critical phenomenon of the one-dimensional S =1/2 asymmetric XY model.
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.