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arXiv:2205.02010 · International Journal of Theoretical Physics

The Dynamics of the Hubbard Model Through Stochastic Calculus and Girsanov Transformation

Detlef Lehmann

Published May 28, 2024• 1 citation
arXivHTMLPDF

Abstract

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 $$g=UN$$ 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.

DOI
10.1007/s10773-024-05649-0

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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 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.
The Monte Carlo trajectory sampling of stochastic differential equations based on the quasiprobability distribution functions, such as the Glauber–Sudarshan P, Wigner, and Husimi Q functions, enables us to investigate bosonic open quantum many-body dynamics described by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation. In this method, the Monte Carlo samplings for the initial distribution and stochastic noises incorporate quantum fluctuations, and thus, we can go beyond the mean-field approximation. However, description using stochastic differential equations is possible only when the corresponding Fokker–Planck equation has a positive-semidefinite diffusion matrix. In this work, we analytically derive the stochastic differential equations for arbitrary Hamiltonian and jump operators based on the path-integral formula, independently of the derivation of the Fokker–Planck equation. In the course of the derivation, we formulate the path-integral representation of the GKSL equation by using the -ordered quasiprobability distribution function, which systematically describes the aforementioned quasiprobability distribution functions by changing the real parameter . The essential point of this derivation is that we employ the Hubbard–Stratonovich transformation in the path integral, and its application is not always feasible. We find that the feasible condition of the Hubbard–Stratonovich transformation is identical to the positive-semidefiniteness condition of the diffusion matrix in the Fokker–Planck equation. In the benchmark calculations, we confirm that the Monte Carlo simulations of the obtained stochastic differential equations well reproduce the exact dynamics of physical quantities and non-equal time correlation functions of numerically solvable models, including the Bose–Hubbard model. This work clarifies the applicability of the approximation and gives systematic and simplified procedures to obtain the stochastic differential equations to be numerically solved.
In the context of out of equilibrium dynamics of isolated interacting quantum many-body systems, we study the mean-field discretized and nonlinear Gross-Pitaevskii equation (GPE) of motion for a fully connected (or all-to-all coupled) Bose-Hubbard model describing a large population of bosons on a lattice of generic dimension V, with potential experimental applications ranging from ultra-cold atomic gases in optical traps to systems of Josephson junctions. Considering a quench on the system Hamiltonian, we describe the time evolution of a system initialized to the Mott insulating phase and taken in the strong superfluid regime, finding a sharp change in the dynamical behaviour when varying the quench intensity across a critical value, that is a dynamical phase transition. Our results generalize the study presented in some previous papers focusing on the particular cases of V = 2, V = 3 and 1D lattice, and place particular emphasis on the relaxation of some dynamical order parameters in the large V limit. On the side of the dynamical phase transition corresponding to the strong superfluid regime, we find such relaxation to consist of a π-synchronization of the mean field bosonic phases, that we find to compete against the eventual site-dependent disorder of the Hamiltonian in what, in analogy with the notorious Kuramoto model for nonlinear coupled oscillators, can be referred to as a synchronization transition. Additionally, in support of the validity of the mean field approximation far from the transition, we report a study beyond mean field based on the Bogolioubov-de Gennes method.
We critically explore the applicability of a recently proposed framework to sample the quantum dynamics of a many-body quantum system interacting with light by stochastic trajectories, applying it to the closed and open Tavis-Cummings model (TCM). The stochastic differential equations (SDEs) sample the positive P phase-space representation by analog complex-valued dynamical variables that are linked to the quantum operators. Statistical average over the stochastic trajectories yields the evolution of the quantum mechanical expectation values. However, numerical implementation of these SDEs for the TCM indicates divergent solutions, also known from other phase-space methods. This limits the applicability of the framework to finite propagation times, that are strongly dependent on the physical parameters and initial conditions of the system. We outline the underlying mathematical reason for these divergences and show that their contribution to the averages are, however, essential. To attempt to regularize the divergences, we transform the SDEs to an equivalent set of SDEs with different noise realisations, thereby pushing the valid time boundary. Quantum collapse and revival of the TCM, however, cannot be recovered by the stochastic trajectory approach, pointing to the general difficulty of the applicability of stochastic phase-space sampling methods to systems with strong quantum features.
Abstract This paper is a first step toward rigorously establishing the validity of mean-field approximations for the dynamics of quantum systems in a large dimensional limit. Our result is a derivation of the large d mean-field limit of the Bose–Hubbard model on a square lattice. The Bose–Hubbard model effectively describes bosons on a lattice with on-site interactions and nearest-neighbour hopping, serving as a foundational framework for understanding strong particle interactions and the superfluid to Mott insulator phase transition. We prove a trace norm estimate between the one-lattice-site reduced density of the Schrödinger dynamics and the mean-field dynamics in the limit of large dimension. Here, the mean-field approximation is in the hopping amplitude and not in the interaction, leading to a very rich and non-trivial mean-field equation. This mean-field equation does not only describe the condensate, as is the case when the mean-field description comes from a large particle number limit averaging out the interaction, but it allows for a phase transition since it contains the full non-trivial interaction. Our work is a rigorous justification of a simple case of the highly successful dynamical mean-field theory (DMFT) for bosons, which somewhat surprisingly yields many qualitatively correct results in three dimensions.