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Nuclear Physics B

A comparison of Langevin and microcanonical simulations for fermions

Hong-Qiang Ding, Olivier Martin

Published Jan 1, 1987• 4 citations
Publisher

Abstract

Langevin and microcanonical methods are studied on a quantum mechanical system which exhibits symmetry breaking. We introduce a dynamical step-size adjustment to improve the numerical accuracy of the simulations. Also, we present a heat bath method on the pseudofields which decreases the autocorrelation time and enforces the ergodicity of the microcanonical simulations.

DOI
10.1016/0550-3213(87)90158-1

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Langevin and microcanonical methods are studied on a quantum mechanical system which exhibits symmetry breaking. We introduce a dynamical step-size adjustment to improve the numerical accuracy of the simulations. Also, we present a heat bath method on the pseudofields which decreases the autocorrelation time and enforces the ergodicity of the microcanonical simulations.
In the first part of this thesis, a stochastic adaptation of the microcanonical simulation method is applied to the numerical simulation of the Su-Schrieffer-Heeger Hamiltonian for polyacetylene, a one-dimensional polymer where fermion-boson interactions play a dominant role in the dynamics of the system. The pure microcanonical simulation method fails in the marginally ergodic case and a stochastic adaptation, the hybrid microcanonical method, is employed to resolve problems with ergodicity. The hybrid method is shown to be an efficient method for higher dimensional fermionic quantum systems. In the second part of this thesis, a numerical simulation of the evolution of a network of global cosmic strings in an expanding Robertson-Walker universe is carried out. The system is quenched through an order-disorder phase transition and the nature of the string distribution is examined. While the string distribution observed at the phase transition is in good agreement with earlier estimates, the simulation reveals that the dynamics of the strings are suppressed by interactions with the Goldstone field. The network decays by topological annihilation and no spatial correlations are observed at any point in the simulation.
A comparative study of the efficiency of different algorithms (Metropolis, Langevin, and Hybrid) for the simulation of a one-dimensional spinless fermion model with a half-filled band is presented. The Metropolis updating of the fields is performed with an algorithm introduced by Blankenbecler, Scalapino, and Sugar. The density-density correlation function, the order parameter, and the autocorrelation are calculated. A study of the dependence of the systematic errors on the temperature and the step size for the Langevin and hybrid algorithms is presented. The relation between the accuracy and the speed of the algorithms is studied.
Based on the microcanonical ensemble theory in statistical mechanics, we devise a method that can be used to enhance the capability of numerical calculation on statistical models with large lattice sizes. In our method, we take the expectation value of the energy, as defined in quantum mechanics, instead of the eigenvalue as the energy of a physical system. We show mathematically that the relevant physical quantities obtained in this way are unchanged in the thermodynamical limit and we apply this method to numerical calculations. In this paper, we present our numerical results with the one-dimensional spinless fermionic model as a first test of our method. The numerical calculations are done to a 4096 lattice size using a computer with a speed of about 40 mflops. Our numerical data agree quite well with the exact values. Also, the fluctuation of data is small, in contrast to that obtained using the quantum Monte Carlo method. \textcopyright{} 1996 The American Physical Society.
We generalize the microcanonical algorithm developed by Creutz et al. (1986), and make a detailed comparison with the exact solution in the case of a two-dimensional Ising model at finite volume. We present a new numerical method to compute the temperature in the microcanonical ensemble. This allows us to define a `thermalization` criterion to estimate the point where the differences between canonical and microcanonical results are the smallest. This criterion is shown to work well in the case of the two-dimensional Ising system.