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physica status solidi (b)

Quantum Monte Carlo: What can random numbers tell us about correlated quantum systems?

S. Rombouts, K. Heyde

Published Apr 24, 2003• 5 citations
Publisher

Abstract

Abstract Over the past decades, quantum Monte Carlo methods have proven to be very valuable for the study of quantum‐mechanical many‐body systems. The basic ingredient of theses methods is a decomposition of the imaginary‐time evolution operator into path integrals. These paths can then be sampled using Monte Carlo techniques. Though very successful for bosons, these methods often have troubles in dealing with fermions: the fermionic antisymmetry leads to a ‘sign problem’. One approach, the auxiliary‐field method, has yielded slightly better results in dealing with the sign problem. Some recent developments on this method are highlighted. As a case study, it is applied to a pairing model for Fe nuclei, where indications are found for a phase transition related to nucleon–nucleon pairing.

DOI
10.1002/pssb.200301787

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Abstract Over the past decades, quantum Monte Carlo methods have proven to be very valuable for the study of quantum‐mechanical many‐body systems. The basic ingredient of theses methods is a decomposition of the imaginary‐time evolution operator into path integrals. These paths can then be sampled using Monte Carlo techniques. Though very successful for bosons, these methods often have troubles in dealing with fermions: the fermionic antisymmetry leads to a ‘sign problem’. One approach, the auxiliary‐field method, has yielded slightly better results in dealing with the sign problem. Some recent developments on this method are highlighted. As a case study, it is applied to a pairing model for Fe nuclei, where indications are found for a phase transition related to nucleon–nucleon pairing.
We discuss finite temperature quantum Monte Carlo methods in the framework of the interacting nuclear shell model. The methods are based on a representation of the imaginary-time many-body propagator as a superposition of one-body propagators describing non-interacting fermions moving in fluctuating auxiliary fields. Fermionic Monte Carlo calculations have been limited by a ``sign'' problem. A practical solution in the nuclear case enables realistic calculations in much larger configuration spaces than can be solved by conventional methods. Good-sign interactions can be constructed for realistic estimates of certain nuclear properties. We present various applications of the methods for calculating collective properties and level densities.
The auxiliary-field quantum Monte Carlo method is reviewed. The Hubbard-Stratonovich transformation converts an interacting Hamiltonian into a non-interacting Hamiltonian in a time-dependent stochastic field, allowing calculation of the resulting functional integral by Monte Carlo methods. The method is presented in a sufficiently general form to be applicable to any Hamiltonian with one- and two-body terms, with special reference to the Heisenberg model and one- and many-band Hubbard models. Many physical correlation functions can be related to correlation functions of the auxiliary field; general results are given here. Issues relating to the choice of auxiliary fields are addressed; operator product identities change the relative dimensionalities of the attractive and repulsive parts of the interaction. Frequently the integrand is not positive-definite, rendering numerical evaluation unstable. If the auxiliary field violates time-reversal invariance, the integrand is complex and this sign problem becomes a phase problem. The origin of this sign or phase is examined from a number of geometrical and other viewpoints and illustrated by simple examples: the phase problem by the spin (1/2) Heisenberg model, and the sign problem by the attractive SU(N) Hubbard model on a triangular molecule with negative hopping integrals. In the latter case, widely studied in the Jahn-Teller literature, the sign is due neither to fermions nor spin, but to frustration. This system is used to illustrate a number of suggested interpretations of the sign problem.
Abstract This paper describes a different Q11C method for solving the Schrödinger equation, one unlike variational, diffusion, and Green’s function methods. It can be applied to many-body systems to produce solutions for ground states. For systems of fermions, antisymmetrization can be imposed to overcome the problem of node location, but the instability of the method is a serious shortcoming, so that the method as described is limited to treating relatively simple systems. Nevertheless, this auxiliary field quantum 1,fonte Carlo (AFMC) method was demonstrated to be successful for test cases reported in this paper, and in time has been improved to allow treatment of realistic systemsa. The method makes use of the imaginary-time propagation of an initial wavefunction toward the ground-state function. It requires use of a trial wavefunction which is the product of single-particle orbitals for bosons or an antisymmetrized product for fermions. A Hubbard-Stratonovich transformation of the propagator changes the many-body problem to a problem of a collection of single particles interacting with a fluctuating (auxiliary) field which replaces the many-body interactions. The authors developed an integral expression for the energy which, after discretization, could be evaluated by a Metropolis Monte Carlo integration forward in time. The algorithm was tested for one-dimensional systems of particles: 6 to 20 bosons with varied interactions, 2 to 6 neutron-proton pairs, others with repulsive potentials. Importance sampling was found to be critical and computational requirements to be large. For spin systems on a lattice the AFMC method was considered feasible, but for general multidimensional systems the method appeared to be limited by its computation requirements.
We describe an algorithm for the numerical simulation of quantum spin and boson systems. The method is based on the Trotter decomposition in imaginary time and a decoupling by auxiliary Ising spins. It can be applied, in principle, to arbitrary (random) spin systems, however, in general it suffers from the ``minus-sign problem.'' This problem is absent in the case of the Ising model in a transverse field in arbitrary dimensions and geometries. We show test results for the spin-1/2 XY\mathrm{XY}XY model, the one-dimensional transverse Ising model with disorder, and the phase transition induced by a transverse field in the two-dimensional Ising model.
We discuss finite temperature quantum Monte Carlo methods in the framework of the interacting nuclear shell model. The methods are based on a representation of the imaginary-time many-body propagator as a superposition of one-body propagators describing non-interacting fermions moving in fluctuating auxiliary fields. Fermionic Monte Carlo calculations have been limited by a "sign" problem. A practical solution in the nuclear case enables realistic calculations in much larger configuration spaces than can be solved by conventional methods. Good-sign interactions can be constructed for realistic estimates of certain nuclear properties. We present various applications of the methods for calculating collective properties and level densities.