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arXiv:2405.04055 · Documenta Mathematica

Expansion of the Many-body quantum Gibbs state of the Bose–Hubbard model on a finite graph

Zied Ammari, Shahnaz Farhat, Sören Petrat

Published Mar 20, 2025• 0 citations
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Abstract

We consider the many-body quantum Gibbs state for the Bose–Hubbard model on a finite graph at positive temperature. We scale the interaction with the inverse temperature, corresponding to a mean-field limit where the temperature is of the order of the average particle number. For this model it is known that the many-body Gibbs state converges, as temperature goes to infinity, to the Gibbs measure of a discrete nonlinear Schrödinger equation, i.e., a Gibbs measure defined in terms of a one-body theory. In this article we extend these results by proving an expansion to any order of the many-body Gibbs state with inverse temperature as a small parameter. The coefficients in the expansion can be calculated as vacuum expectation values using a recursive formula, and we compute the first two coefficients explicitly.

DOI
10.4171/dm/1001

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We consider the many-body quantum Gibbs state for the Bose–Hubbard model on a finite graph at positive temperature. We scale the interaction with the inverse temperature, corresponding to a mean-field limit where the temperature is of the order of the average particle number. For this model it is known that the many-body Gibbs state converges, as temperature goes to infinity, to the Gibbs measure of a discrete nonlinear Schrödinger equation, i.e., a Gibbs measure defined in terms of a one-body theory. In this article we extend these results by proving an expansion to any order of the many-body Gibbs state with inverse temperature as a small parameter. The coefficients in the expansion can be calculated as vacuum expectation values using a recursive formula, and we compute the first two coefficients explicitly.
While recent advances have established efficient quantum algorithms for preparing Gibbs states of finite-dimensional systems, comparable complexity results for bosonic and other infinite-dimensional models remain unexplored. We introduce the first general rigorous Gibbs sampling framework for bosonic many-body systems, showing that physically relevant bosonic models admit gapped dissipative generators, enabling efficient preparation of thermal states, provided that the spectral gap scales favourably with the number of modes. Although our results hold for broad classes of models, we illustrate them using Bose-Hubbard Hamiltonians, both within and beyond the mean-field regime. In both cases, we show that the associated dissipative generators maintain a positive spectral gap, thereby implying exponential convergence to the thermal state. For the Gibbs sampler corresponding to the full Bose-Hubbard model, we obtain a spectral-gap lower bound Ce−cnCe^{-cn}Ce−cn, where nnn is the number of modes. We apply our results to provide a Gibbs-state preparation algorithm on qubit hardware, with runtime polynomial in the number of modes and the inverse spectral gap, and thereby obtain a quantum algorithm to compute thermal properties of the model. This provides the first mathematically controlled route to Gibbs sampling in infinite-dimensional systems, with implications for quantum simulation, thermalization, and many-body complexity, where quantum advantages may arise.
We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d ⩽ 3 d \leqslant 3 . For d > 1 d > 1 the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the L r L^r -norm with optimal exponent r r . Moreover, we prove the convergence in the L ∞ L^\infty -norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.
We consider the homogeneous mean-field Bose gas at temperatures proportional to the critical temperature of its Bose-Einstein condensation phase transition. We prove a trace norm approximation for the grand canonical Gibbs state in terms of a reference state, which is given by a convex combination of products of coherent states and Gibbs states associated with certain temperature-dependent Bogoliubov Hamiltonians. The convex combination is expressed as an integral over a Gibbs distribution of a one-mode Φ4Φ^4Φ4-theory describing the condensate. This result justifies an analogue of Lee and Yang's extension of Bogoliubov theory to positive temperatures, and it allows us to derive various limiting distributions for the number of particles in the condensate, as well as precise formulas for the one- and two-particle density matrices of the Gibbs state. Key ingredients of our proof, which are of independent interest, include two novel abstract correlation inequalities. The proof of one of them is based on an application of an infinite-dimensional version of Stahl's theorem.
Abstract We prove that the flow of the discrete nonlinear Schrödinger equation (DNLS) is the mean field limit of the quantum dynamics of the Bose–Hubbard model for N interacting particles. In particular, we show that the Wick symbol of the annihilation operators evolved in the Heisenberg picture converges, as N becomes large, to the solution of the DNLS. A quantitative LpL^pLp L p -estimate, for any p≥1p \ge 1p≥1 p ≥ 1 , is obtained with a linear dependence on time due to a Gaussian measure on initial data coherent states.
We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures. Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by high-precision Monte-Carlo simulations of the Bose-Hubbard model. In addition, our Green’s function approach allows for calculating the excitation spectrum both for zero and finite temperature and for determining the effective masses of particles and holes.
We use a diagrammatic hopping expansion to calculate finite-temperature Green functions of the Bose-Hubbard model which describes bosons in an optical lattice. This technique allows for a summation of subsets of diagrams, so the divergence of the Green function leads to non-perturbative results for the boundary between the superfluid and the Mott phase for finite temperatures. Whereas the first-order calculation reproduces the seminal mean-field result, the second order goes beyond and shifts the phase boundary in the immediate vicinity of the critical parameters determined by the latest high-precision Monte-Carlo simulations of the Bose-Hubbard model. In addition, our Green's function approach allows for calculating the excitation spectrum at finite temperature and for determining the effective masses of particles and holes.