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International Journal of Theoretical Physics

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

Detlef Lehmann

Published Mar 11, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1007/s10773-026-06253-0

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

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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.
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.
Abstract As a typical quantum many body problem, we consider the time evolution of density matrix elements in the Bose-Hubbard model. For an arbitrary initial state, these quantities can be obtained from an SDE or stochastic differential equation system. To this SDE system, a Girsanov transformation can be applied. This has the effect that all the information from the initial state moves into the drift part, into the mean field part, of the transformed system. In the large N limit with g=UNg=UNg=UN g = U N fixed, the diffusive part of the transformed system vanishes and as a result, the exact quantum dynamics is given by an ODE system which turns out to be the time dependent discrete Gross Pitaevskii equation. For the two site Bose-Hubbard model, the GP equation reduces to the mathematical pendulum and the difference of expected number of particles at the two lattice sites is equal to the velocity of that pendulum which is either oscillatory or it can have rollovers which then corresponds to the self trapping or insulating phase. As a by-product, we also find an equivalence of the mathematical pendulum with a quartic double well potential. Collapse and revivals are a more subtle phenomenom, in order to see these the diffusive part of the SDE system or quantum corrections have to be taken into account. This can be done with an approximation and collapse and revivals can be reproduced, numerically and also through an analytic calculation. Since expectation values of Fresnel or Wiener diffusion processes, we write the density matrix elements exactly in this way, can be obtained from parabolic second order PDEs, we also obtain various exact PDE representations. The paper has been written with the goal to come up with an efficient calculation scheme for quantum many body systems and as such the formalism is generic and applies to arbitrary dimension, arbitrary hopping matrices and, with suitable adjustments, to fermionic models.
Abstract The study of ground-state properties of the Fermi-Hubbard model is a long-lasting task in the research of strongly correlated systems, and is crucial for the understanding of notable quantum phenomena including superconductivity and magnetism. Owing to the exponentially growing complexity of the system, a quantitative analysis usually demands high computational cost and is restricted to small samples, especially in two or higher dimensions. Here, we introduce a variational method in the frame of fermionic Gaussian states, and obtain the ground states of one- and two-dimensional attractive Hubbard models via imaginary-time evolution. We calculate the total energy and benchmark the results in a wide range of interaction strength and filling factor with those obtained via exact two-body results, the density matrix renormalization group based on matrix product states (MPS), and projector Quantum Monte Carlo (QMC) method. For both 1D and 2D cases, the Gaussian variational method presents accurate results for total energy with a maximum systematic error ∼ 4 % in the intermediate interaction region. The accuracy of these results has negligible dependence on the system size. We further calculate the double occupancy and find excellent agreement with MPS and QMC, as well as the experimental results of cold quantum gases in optical lattices. The results suggest that the Gaussian pairing state is a good approximation to the ground states of attractive Hubbard model, in particular in the strong and weak coupling limits. Moreover, we generalize the method to the attractive Hubbard model with a finite spin-polarization, which can be mapped to the repulsive interaction case via particle-hole transformation, and obtain accurate results of ground state energy and double occupancy. Our work demonstrates the ability of the Gaussian variational method to extract ground state properties of strongly correlated many-body systems with negligible computational cost, especially of large size and in higher dimensions.
By precisely writing down the matrix element of the local Boltzmann operator (e(−τhe^(-{\tau}he(−τh), where h is the paired operator formed by two interacting terms conjugate to each other), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. Our formula presents remarkable outcomes such that: 1) the Hubbard-Stratonovich transformation is not necessary, and the auxiliary field is no longer needed, which can improve the computational efficiency. 2) The new formula improves the accuracy of the Suzuki-Trotter decomposition. Our method also provides a potential theoretical framework for analyzing and solving the fermion sign problem. As an example, we studied the one-dimensional half-filled Hubbard model at a finite temperature. The obtained results were in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be used in various studies.
Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/tU/tU/t from 2 to 8, for a range of densities including half-filling and n=0.3,0.5,0.6,0.75n=0.3,0.5,0.6,\phantom{\rule{0.16em}{0ex}}0.75n=0.3,0.5,0.6,0.75, and 0.8750.8750.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations.Using the two-dimensional half-filled Hubbard model at weak coupling as a testing ground, we perform a comparative study of a comprehensive set of state-of-the-art quantum many-body methods.Upon cooling into its insulating antiferromagnetic ground state, the model hosts a rich sequence of distinct physical regimes with crossovers between a high-temperature incoherent regime, an intermediate-temperature metallic regime, and a low-temperature insulating regime with a pseudogap created by antiferromagnetic fluctuations.We assess the ability of each method to properly address these physical regimes and crossovers through the computation of several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantum Monte Carlo methods) serving as a benchmark.By combining computational results and analytical insights, we elucidate the nature and role of spin fluctuations in each of these regimes.Based on this analysis, we explain how quasiparticles can coexist with increasingly longrange antiferromagnetic correlations and why dynamical mean-field theory is found to provide a remarkably accurate approximation of local quantities in the metallic regime.We also critically