Thermodynamics of the Fermi-Hubbard Model through Stochastic Calculus and Girsanov Transformation

TL;DR

This paper introduces a Girsanov transformation-based approach to analyze thermodynamic correlation functions in the Fermi-Hubbard model, simplifying dependence on initial factorization choices and enabling new analytical and numerical insights.

Contribution

The authors develop a Girsanov transformation method that reduces dependence on initial factorization in the Fermi-Hubbard model, facilitating analysis of correlations and energies.

Findings

Analytical proof of antiferromagnetic sign of spin-spin correlations at half-filling.

Approximate ground state energies obtained from an ODE system match benchmark data.

Method is generic and applicable to various quantum many-body models.

Abstract

We apply the methodology of our recent paper 'The Dynamics of the Hubbard Model through Stochastic Calculus and Girsanov Transformation' [1] to thermodynamic correlation functions in the Fermi-Hubbard model. They can be obtained from a stochastic differential equation (SDE) system. To this SDE system, a Girsanov transformation can be applied. This has the effect that the usual determinant or pfaffian which shows up in a pfaffian quantum Monte Carlo (PfQMC) representation [2] basically gets absorbed into the new integration variables and information from that pfaffian moves into the drift part of the transformed SDE system. While the PfQMC representation depends heavily on the choice of how the quartic interaction has been factorized into quadratic quantities in the beginning, the Girsanov transformed formula has the very remarkable property that it is nearly independent of that choice, the drift part of the transformed SDE system as well as a remaining exponential which has the obvious meaning of energy are always the same and do not depend on the Hubbard-Stratonovich details. The resulting formula may serve as a starting point for further theoretical or numerical investigations. Here we consider the spin-spin correlation at half-filling on a bipartite lattice and obtain an analytical proof that the signs of these correlations have to be of antiferromagnetic type, at arbitrary temperatures. Also, by checking against available benchmark data [3], we find that approximate ground state energies can be obtained from an ODE system. This may even hold for exact ground state energies, but future work would be required to prove or disprove this. As in [1], the methodology is generic and can be applied to arbitrary quantum many body or quantum field theoretical models.