Comparison of Calculations for the Hubbard model obtained with Quantum-Monte-Carlo, exact and stochastic Diagonalization

TL;DR

This paper compares Quantum-Monte-Carlo, exact, and stochastic diagonalization methods for the Hubbard model, finding good agreement in ground state energies and correlations, while analyzing convergence issues and the impact of the minus sign problem.

Contribution

It provides a systematic comparison of numerical methods for the Hubbard model, highlighting their agreement and limitations, especially regarding convergence and the minus sign problem.

Findings

Quantum-Monte-Carlo agrees with diagonalization for small to moderate interactions.

Convergence issues arise in Quantum-Monte-Carlo for large interactions due to the minus sign problem.

Superconducting correlations are consistent across methods, despite fluctuations in Monte-Carlo results.

Abstract

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.