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arXiv:1809.05371 · Physical Review A

Non-Gibbs states on a Bose-Hubbard lattice

Alexander Yu. Cherny, Thomas Engl, Sergej Flach A

Published Feb 1, 2019• 16 citations
arXivHTMLPDF

Abstract

We study the equilibrium properties of the repulsive quantum Bose-Hubbard model at high temperatures in arbitrary dimensions, with and without disorder. In its microcanonical setting the model conserves energy and particle number. The microcanonical dynamics is characterized by a pair of two densities: energy density $\ensuremath{\varepsilon}$ and particle number density $n$. The macrocanonical Gibbs distribution also depends on two parameters: the inverse non-negative temperature $\ensuremath{\beta}$ and the chemical potential $\ensuremath{\mu}$. We prove the existence of non-Gibbs states, that is, pairs $(\ensuremath{\varepsilon},n)$ which cannot be mapped onto $(\ensuremath{\beta},\ensuremath{\mu})$. The separation line in the density control parameter space between Gibbs and non-Gibbs states $\ensuremath{\varepsilon}\ensuremath{\sim}{n}^{2}$ corresponds to infinite temperature $\ensuremath{\beta}=0$. The non-Gibbs phase cannot be cured into a Gibbs one within the standard Gibbs formalism using negative temperatures.

DOI
10.1103/physreva.99.023603

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We study the equilibrium properties of the repulsive quantum Bose-Hubbard model at high temperatures in arbitrary dimensions, with and without disorder. In its microcanonical setting the model conserves energy and particle number. The microcanonical dynamics is characterized by a pair of two densities: energy density \ensuremathε\ensuremath{\varepsilon}\ensuremathε and particle number density nnn. The macrocanonical Gibbs distribution also depends on two parameters: the inverse non-negative temperature \ensuremathβ\ensuremath{\beta}\ensuremathβ and the chemical potential \ensuremathμ\ensuremath{\mu}\ensuremathμ. We prove the existence of non-Gibbs states, that is, pairs (\ensuremathε,n)(\ensuremath{\varepsilon},n)(\ensuremathε,n) which cannot be mapped onto (\ensuremathβ,\ensuremathμ)(\ensuremath{\beta},\ensuremath{\mu})(\ensuremathβ,\ensuremathμ). The separation line in the density control parameter space between Gibbs and non-Gibbs states \ensuremathε\ensuremath∼n2\ensuremath{\varepsilon}\ensuremath{\sim}{n}^{2}\ensuremathε\ensuremath∼n2 corresponds to infinite temperature \ensuremathβ=0\ensuremath{\beta}=0\ensuremathβ=0. The non-Gibbs phase cannot be cured into a Gibbs one within the standard Gibbs formalism using negative temperatures.
We study mathematically the equilibrium properties of the Bose-Hubbard Hamiltonian in the limit of a vanishing hopping amplitude. This system conserves the energy and the number of particles. We establish the equivalence between the microcanonical and the grand-canonical ensembles for all allowed values of the density of particles ρ and density of energy ε. Moreover, given ρ, we show that the system undergoes a transition as ε increases, from a usual positive temperature state to the infinite temperature state where a macroscopic excess of energy condensates on a single site. Analogous results have been obtained by S. Chatterjee (2017) for a closely related model. We introduce here a different method to tackle this problem, hoping that it reflects more directly the basic understanding stemming from statistical mechanics. We discuss also how, and in which sense, the condensation of energy leads to a glassy dynamics.
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.
A formal thermodynamic mapping is established between the attractive Fermi–Hubbard model and the repulsive Bose–Hubbard model at finite temperature and at imaginary chemical potential μ=iθ. By utilizing a large N-expansion, it is shown that the partition functions of the two models are related by a plain shift θ→θ+π. This condition maps the BCS–BEC crossover of attractive fermions to a Bose–Fermi crossover (fermion-like occupation) of repulsive bosons. A central feature of this correspondence is the thermal kernel g(βE,ϕ) (with β the inverse absolute temperature, E the energy scale, and ϕ the phase angle), whose analytic continuation gB(βE,ϕ)=gF(βE,ϕ+π) governs the bosonic (B) and fermionic (F) sectors. Interestingly, the particular angles ϕ=2π/3 and 4π/3 for fermions correspond to ϕ=π/3 and 5π/3 for bosons, marking the boundaries of an universal thermal window. It is further argued that the present mechanism shows how an emergent, fermionization-like phenomenon can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. It is emphasized that this does not imply a transmutation of quantum statistics at the operator level, but rather a thermodynamic exclusion-like behavior driven by the imaginary chemical potential, unlike the Tonks–Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Effectively, the phase ϕ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behavior without hard-core constraints or infinite repulsion through purely thermodynamic mechanisms. The gap equation and number equation for the bosonic model are derived, highlighting the role of the imaginary chemical potential as a statistical regulator. The results obtained here provide a unified framework for understanding crossovers in interacting lattice systems.
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.
Preparing Gibbs states on a quantum computer offers a way to computationally probe how microscopic interactions give rise to macroscopic phenomena at finite temperature. However, an uncomfortable tension has set in: our strongest general guarantees for Gibbs state preparation hold in regimes where the resulting states can also be simulated classically. We propose investigating physical models with competing energy scales. The Fermi--Hubbard model is a natural target: the competition between electron hopping, on-site repulsion and chemical potential encapsulates many mysteries of condensed matter physics, including Mott insulators, striping and superconductivity. Our main result identifies a simple guiding principle: the temperature scale for efficient preparation is set by the couplings between sites, while the on-site energy scale is unrestricted. For the Fermi--Hubbard model on an nnn-site nearest-neighbor lattice of fixed dimension, we construct a quantum Gibbs sampler that mixes in O(log⁡(n/ε))\mathcal{O}(\log(n/ε))O(log(n/ε)) time whenever the temperature is at least a constant factor larger than the hopping strength, but is allowed to be independent of the repulsion and chemical potential. We obtain analogous guarantees for local qubit systems with bounded intersite interactions and arbitrary external fields. As a consequence, our algorithms can efficiently prepare Fermi--Hubbard Gibbs states that are provably outside the convex hull of fermionic Gaussian states and qubit Gibbs states for locally interacting Hamiltonians outside the set of product states.
We find a mapping between the attractive Fermi-Hubbard model and the repulsive Bose-Hubbard model at finite temperature and at imaginary chemical potential μ=iθμ=iθμ=iθ. We show, by using a large NNN-expansion, that the partition functions of the two models are related by a simple shift θ→θ+πθ\to θ+ πθ→θ+π. This condition maps the BCS--BEC crossover of attractive fermions to a Bose--Fermi crossover (fermion-like occupation) of repulsive bosons. Central feature of this correspondence plays the thermal kernel g(βE,ϕ),g(βE,ϕ),g(βE,ϕ), whose analytic continuation gB(βE,ϕ)=gF(βE,ϕ+π)g_B(βE,ϕ) = g_F(βE,ϕ+π)gB​(βE,ϕ)=gF​(βE,ϕ+π) governs the bosonic and fermionic sectors. Interestingly, we are able to find that the special angles ϕ=2π/3,4π/3ϕ= 2π/3,4π/3ϕ=2π/3,4π/3 for fermions correspond to ϕ=π/3,5π/3ϕ= π/3,5π/3ϕ=π/3,5π/3 for bosons, marking the boundaries of a universal thermal window. We further argue that the present mechanism shows that fermionization can occur at finite interaction strength through a thermodynamic effect induced by the imaginary chemical potential. This suggests that it is a new way of fermionization (not a change in statistics but a fermion-like behaviour) unlike the Tonks--Girardeau limit, where fermionization arises from an infinite repulsive interaction and anyonic or Floquet-engineered systems where transmutation emerges from modified statistics or dynamics. Essentially, the phase ϕϕϕ is a statistical parameter; by twisting the thermal phase, it generates fermion-like behaviour without hard-core constraints or infinite repulsion but only by using thermodynamics. We derive the gap equation and number equation for the bosonic model, highlighting the role of the imaginary chemical potential as a statistical regulator. Our results provide a unified framework for understanding crossovers in interacting lattice systems.