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Progress of Theoretical Physics

Theoretical Basis of the Decoupled Cell Monte Carlo Simulation for Quantum System

H. Matsuda, Kazuyuki ISHII, Shigeo Homma, N. Ogita

Published Oct 1, 1988• 1 citation
Publisher

Abstract

As a theoretical basis of the decoupled cell Monte Carlo simulation (DCM) for quantum systems, it is proved that the error caused in the transition probabilities of the simulation by decoupling the cell with radius ν is at most of the order of βν+1 in the limit of β=1/kT →0. Relative merits of DCM and the Trotter method are discussed.

DOI
10.1143/ptp.80.583

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As a theoretical basis of the decoupled cell Monte Carlo simulation (DCM) for quantum systems, it is proved that the error caused in the transition probabilities of the simulation by decoupling the cell with radius ν is at most of the order of βν+1 in the limit of β=1/kT →0. Relative merits of DCM and the Trotter method are discussed.
The properties of the spin-1/2 Heisenberg model on 1, 2, and 3-dimensional lattices are calculated using the Decoupled Cell Method of Homma et al., and these results are compared with high temperature and spin-wave expansions, and with other numerical approaches. The DCM has advantages over other Monte Carlo methods currently in wide use in that the transition probability is positive definite, there is no need to introduce an additional imaginary time, or Trotter, dimension, and the acceptance rate for transitions is comparable to that of classical lattice models. We find very good agreement between the DCM and the high temperature expansion in the temperature region where the high temperature expansion is valid, and reasonably good agreement at low temperatures with spin wave theory. The DCM fails for temperatures T < Tc which decreases with the size of the cell.
The Quantum Transfer Matrix and Its Application to Quantum Spin Chains (K Kubo) Transfer Matrices in Quantum Many-Body Systems (T Koma) Monte Carlo Calculations of Elementary Excitation (M Takahashi) The Decoupled Cell Method of Quantum Monte Carlo Calculation (S Homma) Decoupled Cell Monte Carlo Study of the Critical Properties of the Spin-1/2 Ferromagnetic Heisenberg Model in Three Dimensions (R J Creswick & C J Sisson) Variational Monte Carlo Studies of Correlated Electrons (H Shiba) Quantum Monte Carlo Simulation of Multiband Fermion Systems and Its Application to Superconductivity (K Kuroki & H Aoki) Quantum Monte Carlo in the Infinite Dimensional Limit (M Jarrell et al) Aspects of the Sign Problem (J H Samson) Ground-State Projection Using Auxiliary Fields (S Fahy) Fermion Simulations of Correlated Systems (M Imada) Dirty Bosons in 2D: Phases and Phase Transitions (N Trivedi & M Makivi'c). (Part Contents).
Decoupled cell method is introduced as a method of Monte Carlo simulation for quantum spin systems. Based on this method we calculated the thermodynamic quantities such as an internal energy, in-plane and out-of-plane susceptibilities and spin-spin correlation functions in one dimensional XY model. The agreement with the exact results is excellent above a certain demarcation temperature. This method is also applied to the antiferromagnetic Heisenberg model on the triangular lattice. The results indicate the existence of a sublattice structure composed of three sublattices, although its spatial and temporal behaviours are short ranged.