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International Journal of Modern Physics B

MULTILEVEL BLOCKING MONTE CARLO SIMULATIONS FOR QUANTUM DOTS

Reinhold Egger, Chi H. Mak

Published May 10, 2001• 3 citations
Publisher

Abstract

This article provides an introduction to the ideas behind the multilevel blocking (MLB) approach to the fermion sign problem in path-integral Monte Carlo simulations, and also gives a detailed discussion of MLB results for quantum dots. MLB can turn the exponential severity of the sign problem into an algebraic one, thereby enabling numerically exact studies of otherwise inaccessible systems. Low-temperature simulation results for up to eight strongly correlated electrons in a parabolic 20 quantum dot are presented.

DOI
10.1142/s021797920100591x

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This article provides an introduction to the ideas behind the multilevel blocking (MLB) approach to the fermion sign problem in path-integral Monte Carlo simulations, and also gives a detailed discussion of MLB results for quantum dots. MLB can turn the exponential severity of the sign problem into an algebraic one, thereby enabling numerically exact studies of otherwise inaccessible systems. Low-temperature simulation results for up to eight strongly correlated electrons in a parabolic 20 quantum dot are presented.
This article provides an introduction to the ideas behind the multilevel blocking (MLB) approach to the fermion sign problem in path-integral Monte Carlo simulations, and also gives a detailed discussion of MLB results for quantum dots. MLB can turn the exponential severity of the sign problem into an algebraic one, thereby enabling numerically exact studies of otherwise inaccessible systems. Low-temperature simulation results for up to eight strongly correlated electrons in a parabolic 2D quantum dot are presented.
This article gives an introduction to the multilevel blocking (MLB) approach to both the fermion and the dynamical sign problem in path-integral Monte Carlo simulations. MLB is able to substantially relieve the sign problem in many situations. Besides an exposition of the method, its accuracy and several potential pitfalls are discussed, providing guidelines for the proper choice of certain MLB parameters. Simulation results are shown for strongly interacting electrons in a 2D parabolic quantum dot, the real-time dynamics of several simple model systems, and the dissipative two-state dynamics (spin-boson problem).
This paper provides an analysis of the recently proposed multilevel blocking (MLB) algorithm for the treatment of the fermion sign problem in path-integral Monte Carlo simulations. The basic concepts of MLB and its practical implementation, as well as its accuracy and error estimates, are discussed in detail. MLB calculations on a simple exactly solvable model are used to illustrate the accuracy of the method under different circumstances. This paper is aimed at providing practical guidelines for the proper choice of certain MLB parameters.
A general algorithm toward the solution of the fermion sign problem in finite-temperature quantum Monte Carlo simulations has been formulated for discretized fermion path integrals with nearest-neighbor interactions in the Trotter direction. This multilevel approach systematically implements a simple blocking strategy in a recursive manner to synthesize the sign cancellations among different fermionic paths throughout the whole configuration space. The practical usefulness of the method is demonstrated for interacting electrons in a quantum dot.
The crossover from weak to strong correlations in parabolic quantum dots at zero magnetic field is studied by numerically exact path-integral Monte Carlo simulations for up to eight electrons. By the use of a multilevel blocking algorithm, the simulations are carried out free of the fermion sign problem. We obtain a universal crossover governed only by the density parameter rs{r}_{s}rs​. For rs>rc{r}_{s}>{r}_{c}rs​>rc​, the data are consistent with a Wigner molecule description, while, for rs<rc{r}_{s}<{r}_{c}rs​<rc​, Fermi liquid behavior is recovered. The crossover value ${r}_{c}\ensuremath{\approx}4$ is surprisingly small.