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Physical review. B, Condensed matter

Comparative study of the discrete and the continuous Hubbard-Stratonovich transformation for a one-dimensional spinless fermion model

G. M. Buendía

Published Mar 1, 1986• 26 citations
Publisher

Abstract

A comparative study of the discrete and the continuous Hubbard-Stratonovich transformation is presented. With the use of the Metropolis algorithm, the Monte Carlo time correlation functions for a one-dimensional spinless fermion model are calculated for both methods. The results show the discrete method to be more efficient.
DOI
10.1103/physrevb.33.3519

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We discuss a new functional-integral formulation of interacting fermion systems on a lattice such as the Anderson or Hubbard models: the fermion-fermion interaction is eliminated by introducing auxiliary Ising variables. The resulting model for a ddd-dimensional quantum system is a (d+1d+1d+1)-dimensional Ising model with complicated interactions. The new transformation is particularly useful for performing numerical studies of these models using Monte Carlo techniques. We study its convergence properties and compare it with the usual Gaussian formulation for the case of the Hubbard model.
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.
Efficient continuous-time quantum Monte Carlo (CT-QMC) algorithms that do not suffer from time discretization errors have become the state of the art for most discrete quantum models. They have not been widely used yet for fermionic quantum lattice models, such as the Hubbard model, nor other fermionic lattice systems due to a suboptimal scaling of $O({\ensuremath{\beta}}^{3})$ with inverse temperature $\ensuremath{\beta}$, compared to the linear scaling of discrete-time algorithms. Here we present a CT-QMC algorithm for fermionic lattice systems that matches the scaling of discrete-time methods but is more efficient and free of time discretization errors. This provides an efficient simulation scheme that is free from the systematic errors opening an avenue to more precise studies of large systems at low and zero temperature.
Continuous-time determinantal algorithm is proposed for the quantum Monte Carlo simulation of the interacting fermions. The scheme does not invoke Hubbard-Stratonovich transformation. The fermionic action is divided into two parts. One of them contains the interaction and certain additional terms; another one is purely Gaussian. The first part is considered as a perturbation. Terms of the series expansion for the partition function are generated in a random walk process. The sign problem and the complexity of the algorithm are analyzed. We argue that the scheme should be useful particularly for the systems with non-local interaction.
A numerically exact continuous-time quantum Monte Carlo algorithm for finite fermionic systems with nonlocal interactions is proposed. The scheme is particularly applicable for general multiband time-dependent correlations, since it does not invoke Hubbard-Stratonovich transformation. The present determinantal grand-canonical method is based on a stochastic series expansion for the partition function in the interaction representation. The results for the Green function and for the time-dependent susceptibility of multiorbital supersymmetric impurity model with a spin-flip interaction are presented.
We present a numerically exact continuous-time quantum Monte Carlo algorithm for fermions with a general interaction nonlocal in space-time. The new determinantal grand-canonical scheme is based on a stochastic series expansion for the partition function in the interaction representation. The method is particularly applicable for multiband, time-dependent correlations since it does not invoke the Hubbard-Stratonovich transformation. The test calculations for exactly solvable models, as well results for the Green function and for the time-dependent susceptibility of the multiband supersymmetric model with a spin-flip interaction are discussed.