Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Series in contemporary applied mathematics

Numerical Methods for Quantum Monte Carlo Simulations of the Hubbard Model

Zhaojun Bai, Wenbin Chen, Richard T. Scalettar, Ichitaro Yamazaki

Published Jun 1, 2009• 21 citations
Publisher

Abstract

One of the core problems in materials science is how the interactions between electrons in a solid give rise to properties like
DOI
10.1142/9789814273268_0001

Related papers

Quantum Monte Carlo simulation of the two-dimensional Hubbard modell

Joong-Bae Kim

•Jan 1, 2001•0 citations

Comparison of Calculations for the Hubbard Model Obtained with Quantum-Monte-Carlo, Exact, and Stochastic Diagonalization

Thomas Husslein, Werner Fettes, I. Morgenstern

In this paper we compare numerical results for the ground state of the Hubbard model obtained by Quantum-Monte-Carlo simulations with results from exact and stochastic diagonalizations. We find good agreement for the ground state energy and superconducting correlations for both, the repulsive and attractive Hubbard model. Special emphasis lies on the superconducting correlations in the repulsive Hubbard model, where the small magnitude of the values obtained by Monte-Carlo simulations gives rise to the question, whether these results might be caused by fluctuations or systematic errors of the method. Although we notice that the Quantum-Monte-Carlo method has convergence problems for large interactions, coinciding with a minus sign problem, we confirm the results of the diagonalization techniques for small and moderate interaction strengths. Additionally we investigate the numerical stability and the convergence of the Quantum-Monte-Carlo method in the attractive case, to study the influence of the minus sign problem on convergence. Also here in the absence of a minus sign problem we encounter convergence problems for strong interactions.

arXiv:cond-mat/9705026•International Journal of Modern Physics C•Apr 1, 1997•31 citations

The extended Hubbard model: Theory, simulations, and applications

Rudolf Torsten Clay

The Hubbard model is one of the most widely studied many-body models for interacting electrons in solids. Here we present a study of the Extended Hubbard model, which provides a more realistic description of real materials by including some longer-range Coulomb interactions. We discuss the derivation of the model, numerical methods for many-body systems, the ground state of the one-dimensional model, and an application of the model to organic charge-transfer solids. Our discussion of numerical methods includes details of using the Constrained Path Monte Carlo procedure to treat extended interactions and possible extension of the method for imaginary times. The chapter on the one-dimensional model focuses on the region V>U, where previous work had suggested the possibility of superconductivity. Our results show that instead phase separation is prevalent in that parameter regime. The second application is to model the organic charge-transfer salts, which are intermediate between one and two dimensions. Here we find a coexistence of charge and spin ordering that helps to explain the results of recent experiments on these materials.

•Jan 1, 1999•0 citations

Quantum Monte Carlo simulations of Hubbard type models

Fakher F. Assaad, R. Preuss, A. Muramatsu +1 more

Journal of Low Temperature Physics•Apr 1, 1994•4 citations

Computational Studies Of Strongly Correlated Quantum Matter

Hao Shi

The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.

•Jan 1, 2017•0 citations

QUANTUM MONTE CARLO FOR REALISTIC AND MODEL SOLIDS

Paola Gori‐Giorgi, ALESSANDRO FEDERICO, Giovanni B. Bachelet

Recent advances in computational•Feb 1, 2002•0 citations
In this paper we compare numerical results for the ground state of the Hubbard model obtained by Quantum-Monte-Carlo simulations with results from exact and stochastic diagonalizations. We find good agreement for the ground state energy and superconducting correlations for both, the repulsive and attractive Hubbard model. Special emphasis lies on the superconducting correlations in the repulsive Hubbard model, where the small magnitude of the values obtained by Monte-Carlo simulations gives rise to the question, whether these results might be caused by fluctuations or systematic errors of the method. Although we notice that the Quantum-Monte-Carlo method has convergence problems for large interactions, coinciding with a minus sign problem, we confirm the results of the diagonalization techniques for small and moderate interaction strengths. Additionally we investigate the numerical stability and the convergence of the Quantum-Monte-Carlo method in the attractive case, to study the influence of the minus sign problem on convergence. Also here in the absence of a minus sign problem we encounter convergence problems for strong interactions.
The Hubbard model is one of the most widely studied many-body models for interacting electrons in solids. Here we present a study of the Extended Hubbard model, which provides a more realistic description of real materials by including some longer-range Coulomb interactions. We discuss the derivation of the model, numerical methods for many-body systems, the ground state of the one-dimensional model, and an application of the model to organic charge-transfer solids. Our discussion of numerical methods includes details of using the Constrained Path Monte Carlo procedure to treat extended interactions and possible extension of the method for imaginary times. The chapter on the one-dimensional model focuses on the region V>U, where previous work had suggested the possibility of superconductivity. Our results show that instead phase separation is prevalent in that parameter regime. The second application is to model the organic charge-transfer salts, which are intermediate between one and two dimensions. Here we find a coexistence of charge and spin ordering that helps to explain the results of recent experiments on these materials.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.