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Journal of Applied Physics

Computer simulation of a 1d quantum ground state

M. A. Novotny, David Paul Landau

Published Mar 15, 1984• 3 citations
Publisher

Abstract

The development of a Monte Carlo renormalization group (MCRG) method to study quantum systems is reported. This method can be used either to study the ground state of the system or to study the system at a finite temperature. In this paper we illustrate the MCRG method by applying it to the ground state of the one-dimensional Ising model in a transverse magnetic field. Very rapid convergence to the known critical exponents is found even for small lattice sizes.

DOI
10.1063/1.333690

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The development of a Monte Carlo renormalization group (MCRG) method to study quantum systems is reported. This method can be used either to study the ground state of the system or to study the system at a finite temperature. In this paper we illustrate the MCRG method by applying it to the ground state of the one-dimensional Ising model in a transverse magnetic field. Very rapid convergence to the known critical exponents is found even for small lattice sizes.
A Monte Carlo renormalization-group (MCRG) method developed to study quantum systems is presented. This MCRG method is the synthesis of the Monte Carlo method for quantum systems using the Trotter transformation with the MCRG developed for classical systems. It is applicable to the study of either the ground-state or the finite-temperature behavior of a quantum system. The convergence properties of this MCRG are illustrated by applying it to the ground state of the d=1 Ising model in a transverse magnetic field. Very rapid convergence to the known critical exponents is exhibited even for small lattice sizes.
We report current progress on the synthesis of methods to alleviate two major difficulties in implementing a Monte Carlo Renormalization Group (MCRG) for quantum systems. In particular, we have utilized the loop-algorithm to reduce critical slowing down, and we have implemented an MCRG method in which the symmetries of the classical equivalent model need not be fully understood, since the Renormalization Group is given by the Monte Carlo simulation. We report preliminary results obtained when the resulting MCRG method is applied to the d=1 XXZ model. Our results are encouraging. However, since this model has a Kosterlitz-Thouless transition, it does not yet provide a full test of our MCRG method.
We have undertaken a large-scale computer simulation of the 3d Ising model using the Monte Carlo Renormalization Group (MCRG) method. Our calculation improves on past calculations in three ways. Firstly, larger lattices are used -- we have run on 128{sup 3} as well as 64{sup 3}. Secondly, many more spin operators -- 53 even and 46 odd -- are measured. Lastly, we have incorporated more efficient ways of spin update -- Wolff's single-cluster variant of the Swendsen and Wang algorithm. In addition to calculating the critical temperature the thermal and magnetic exponents, and improving the estimate of the correction-to-scaling exponent, we are aiming to verify the presence of redundant operators. We do not yet have sufficient statistics to give final results so here we describe our calculation and present preliminary results which demonstrate that we have control over both finite size effects and systematic errors arising from truncation. 16 refs., 7 figs.
We implement a momentum-space renormalization-group procedure into the Monte Carlo renormalization-group (MCRG) method. Usually the symmetries of the system must be known and incorporated into the renormalization group, but here we let the renormalization group be given by the Monte Carlo simulation itself. This allows the MCRG method to be applicable to systems where the underlying symmetries may not be known, such as in Monte Carlo studies of quantum systems. As a test of this method, we apply it to the study of the two-dimensional square-lattice q-state Potts model, both for integer values of q and for noninteger values of q near q=4.