A quantum simulation approach for a three-dimensional Ising spin model—Comparison to mean field theory

Zhaosen Liu,Orion Ciftja

doi:10.1063/1.4983212

Abstract

We use a recently developed quantum simulation approach to study the properties of a three-dimensional Ising model consisting of S = 1/2 quantum spins localized at the sites of a simple cubic lattice. We assume nearest-neighbor interaction between spins with an exchange interaction that can be either ferromagnetic or antiferromagnetic. It is found that the computational method quickly converges towards the expected equilibrium spin configurations. The resulting spontaneous magnetization curves corresponding to the two types of magnetic interactions under consideration were found to be almost identical to the ones obtained via quantum mean field theory at all temperatures. The derived total energies, total free energies, magnetic entropies and specific heats per mole of spins show no sizeable differences from known theoretical values. Furthermore, the results of the simulations for two different 3D Ising systems containing 4×4×4 and 20×20×20 spins localized at the sites of a simple cubic lattice were found to be almost identical to each other. This finding suggests that the self-consistent algorithm approach of the current simulation method allows one to obtain the physical bulk properties of a large magnetic system by relying on simulations of a much smaller spin system sample. Therefore, the method presently considered appears to be not only very accurate as gauged by comparison to mean field theory, but also able to greatly increase the speed of simulations.

Highlights

The three dimensional (3D) Ising spin model has been of great interest since the time of Ernst Ising[1] and has been the subject of many studies that continue to these days.[2,3,4,5,6,7] In this model, spins are treated as unit vectors assuming only discrete values of +1 and 1
It is obvious that many classical methods including the classical Monte Carlo (MC) method do not have much flexibility when it comes to the description of complicated magnetic systems such as those consisting of different types of spins in a crystal unit cell.[17]
We used the self-consistent algorithm (SCA) approach to carry out detailed simulations for FM and AFM 3D Ising spin systems consisting of a given number of S = 1/2 quantum spins localized at sites of simple cubic lattices

Summary

INTRODUCTION

We successfully applied the SCA approach to study magnetic nanosystems consisting of rare-earth elements[19,20] and 3d transition metals.[21,22,23,24] In particular, the simulated magnetic structure of a DyNi2B2C nanoball using the SCA approach agrees well with the one observed in the bulk sample For such a case, we found that the magnetic moments on an ab-plane inside the core align in a ferromagnetic (FM) way along the [110] direction, whereas the two adjacent ab layers order in an antiferromagnetic (AFM) manner below the transition temperature. Availability of analytical results is quite helpful on gauging the accuracy of any given computational method

MEAN FIELD THEORY FOR A 3D ISING MODEL

COMPUTATIONAL SPEED AND EFFICIENCY OF THE SCA APPROACH

CONCLUSIONS AND DISCUSSIONS