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Physical review. B, Condensed matter

Monte Carlo renormalization group for quantum systems

M. A. Novotny, David Paul Landau

Published Feb 1, 1985• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.31.1449

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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 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.
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