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Physical review. B, Condensed matter

Vectorized coding for Monte Carlo simulation of the one-dimensional quantum spin system

Yutaka Okabe, Macoto Kikuchi

Published Dec 1, 1986• 19 citations
Publisher

Abstract

We implement a high-speed algorithm for Monte Carlo simulation of the quantum spin system using the Suzuki-Trotter formula. The calculation is fully vectorized by decomposing into appropriate sublattices. We introduce a ``plaquette'' spin representation using logical commands. In order to check the efficiency of our algorithm, we study the one-dimensional XY model for the Trotter number m\ensuremath{\le}32. Excellent agreement with the exact solution is obtained.

DOI
10.1103/physrevb.34.7896

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We implement a high-speed algorithm for Monte Carlo simulation of the quantum spin system using the Suzuki-Trotter formula. The calculation is fully vectorized by decomposing into appropriate sublattices. We introduce a ``plaquette'' spin representation using logical commands. In order to check the efficiency of our algorithm, we study the one-dimensional XY model for the Trotter number m\ensuremath{\le}32. Excellent agreement with the exact solution is obtained.
An improved transfer matrix method is proposed for calculating the partition function of quantum spin systems, which is expressed by the M-th approximant of the Suzuki-Trotter transformation with Trotter number M. The point is to incorporate the symmetry property of the Trotter subsystem in calculation. The size of our transfer matrix is reduced very much. Thermodynamics is described by the maximum eigenvalue of the transfer matrix. Our method is applied to the XY model with spin 1/2. We consider the cases of pair and tri spin decompositions with Trotter number up to 4 (M≤4). We can do our calculation analytically in those cases. In the pair spin decomposition we find results which have been obtained by the Monte Carlo simulation and by a transfer matrix method by computer. We find nice results in the tri spin decomposition. Our result in the case M=4 is as good as the result in M=8 by the previous Monte Carlo simulation on comparing them with the exact result of Katsura.
A method for simulating quantum spin systems with nearest neighbour interaction is reviewed. In the framework of a path-integral approach based on the Trotter formula, the quantum spin system is mapped into the limit of a sequence of classical spin systems with one extra dimension. Numerical questions connected to this limit are discussed. The role of particle and winding numbers is clarified. An overview of the problems investigated by our group is given. In particular, two new results are presented. A simulation of the one-dimensional spin S isotropic Heisenberg antiferromagnet, S=1/2, 1, 3/2, 2, supports Haldane's conjecture that the mass gap is zero for half­ integer S and nonzero for integer S. For the two-dimensional spin 1/2 xy model we discuss evidence for a Kosterlitz-Thouless transition. In particular, we show that the helicity modulus can be computed directly from our simulaton.
I present a new and efficient disconnected cluster decomposition to perform Monte Carlo Simulations based on the Trotter-Suzuki method for 1-d quantum systems. As opposed to the normal checkerboard world line algorithm this mapping allows one to simulate models with interactions up to next nearest neighbours or even one-dimensional double layers. I have used this cluster decomposition to simulate the one-dimensional t-J-J' model at α=J'/J=0.5 and have investigated the sign problem in this model. At half band filling the system is dimerised into singlet spin pairs when α=0.5 due to frustration. I believe that the spin gap is no longer present around quarter band filling
The authors develop a quantum Monte Carlo procedure for the xxz spin chain. The quantum model in one dimension is mapped by a path integral method based on the Trotter formula into the infinite anisotropy limit of a classical two-dimensional spin system. The expectation values of the energy density, susceptibility and static s z -s z structure function are computed in the classical model for finite but increasing anisotropy. The computation method is a Monte Carlo simulation of the classical system. The results are systematically extrapolated to the infinite anisotropy limit.
The resolution of dynamics in out of equilibrium quantum spin systems lies at the heart of fundamental questions among Quantum Information Processing, Statistical Mechanics and Nano-Technologies. Efficient computational simulations of interacting many-spin systems are extremely valuable tools for tackling such questions. Here, we use the Trotter-Suzuki (TS) algorithm, a well-known strategy that provides the evolution of quantum systems, to address the spin dynamics. We present a GPU implementation of a particular TS version, which has been previously implemented on single cores in CPUs. We develop a massive parallel version of this algorithm and compare the efficiency between CPU and GPU implementations. This method reduces the execution time considerably and is capable of dealing with systems of up to 27 spins (only limited by GPU memory).