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A large deviation principle in many-body quantum dynamics

Kirkpatrick Kay, Rademacher Simone, Schlein Benjamin
26 Oct 2020 math-ph math.MP arxiv.org/abs/2010.13754

We consider the many-body quantum evolution of a factorized initial data, in the mean-field regime. We show that fluctuations around the limiting Hartree dynamics satisfy large deviation estimates, that are consistent with central limit theorems that have been established in the last years.

Correlation Energy of a Weakly Interacting Fermi Gas

Benedikter Niels, , Porta Marcello, Schlein Benjamin, Seiringer Robert
18 May 2020 math-ph cond-mat.quant-gas cond-mat.str-el math.MP arxiv.org/abs/2005.08933

We derive rigorously the leading order of the correlation energy of a Fermi gas in a scaling regime of high density and weak interaction. The result verifies the prediction of the random-phase approximation. Our proof refines the method of collective bosonization in three dimensions. We approximately diagonalize an effective Hamiltonian describing approximately bosonic collective excitations around the Hartree-Fock state, while showing that gapless and non-collective excitations have only a negligible effect on the ground state energy.

Landau-Pekar equations and quantum fluctuations for the dynamics of a strongly coupled polaron

Leopold Nikolai, Mitrouskas David, Rademacher Simone, Schlein Benjamin, Seiringer Robert
05 May 2020 math-ph math.MP arxiv.org/abs/2005.02098

We consider the Fr\"ohlich Hamiltonian with large coupling constant $\alpha$. For initial data of Pekar product form with coherent phonon field and with the electron minimizing the corresponding energy, we provide a norm approximation of the evolution, valid up to times of order $\alpha^2$. The approximation is given in terms of a Pekar product state, evolved through the Landau-Pekar equations, corrected by a Bogoliubov dynamics taking quantum fluctuations into account.

Bose-Einstein Condensation Beyond the Gross-Pitaevskii Regime

Adhikari Arka, Brennecke Christian, Schlein Benjamin
09 Feb 2020 math-ph math.MP arxiv.org/abs/2002.03406

We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order $N^{-1+\kappa}$, for $\kappa>0$. Assuming that $\kappa\in (0;1/43)$, we show that low-energy states of the system exhibit complete Bose-Einstein condensation by providing explicit bounds on the expectation and on higher moments of the number of excitations.

A path-integral analysis of interacting Bose gases and loop gases

, Knowles Antti, Schlein Benjamin, Sohinger Vedran
31 Jan 2020 math-ph cond-mat.quant-gas math.MP arxiv.org/abs/2001.11714

We review some recent results on interacting Bose gases in thermal equilibrium. In particular, we study the convergence of the grand-canonical equilibrium states of such gases to their mean-field limits, which are given by the Gibbs measures of classical field theories with quartic Hartree-type self-interaction, and to the Gibbs states of classical gases of point particles. We discuss various open problems and conjectures concerning, e.g., Bose-Einstein condensation, polymers and $\vert \boldsymbol{\phi} \vert^{4}$-theory.

The mean-field limit of quantum Bose gases at positive temperature

, Knowles Antti, Schlein Benjamin, Sohinger Vedran
06 Jan 2020 math-ph math.AP math.MP math.PR arxiv.org/abs/2001.01546

We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schr\"odinger 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 $d \leq 3$. For $d > 1$ 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 $L^r$-norm with optimal exponent $r$. Moreover, we prove the convergence in the $L^\infty$-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.

The Landau-Pekar equations: Adiabatic theorem and accuracy

Leopold Nikolai, Rademacher Simone, Schlein Benjamin, Seiringer Robert
29 Apr 2019 math-ph math.MP arxiv.org/abs/1904.12532

We prove an adiabatic theorem for the Landau-Pekar equations. This allows us to derive new results on the accuracy of their use as effective equations for the time evolution generated by the Fr\"ohlich Hamiltonian with large coupling constant $\alpha$. In particular, we show that the time evolution of Pekar product states with coherent phonon field and the electron being trapped by the phonons is well approximated by the Landau-Pekar equations until times short compared to $\alpha^2$.

Central Limit Theorem for Bose-Einstein Condensates

Rademacher Simone, Schlein Benjamin
01 Mar 2019 math-ph math.MP arxiv.org/abs/1903.00365

We consider a Bose gas trapped in the unit torus in the Gross-Pitaevskii regime. In the ground state, we prove that fluctuations of bounded one-particle observables satisfy a central limit theorem.

Optimal Rate for Bose-Einstein Condensation in the Gross-Pitaevskii Regime

Boccato Chiara, Brennecke Christian, Cenatiempo Serena, Schlein Benjamin
07 Dec 2018 math-ph math.MP arxiv.org/abs/1812.03086

We consider systems of bosons trapped in a box, in the Gross-Pitaevskii regime. We show that low-energy states exhibit complete Bose-Einstein condensation with an optimal bound on the number of orthogonal excitations. This extends recent results obtained in \cite{BBCS1}, removing the assumption of small interaction potential.

Optimal Upper Bound for the Correlation Energy of a Fermi Gas in the Mean-Field Regime

Benedikter Niels, , Porta Marcello, Schlein Benjamin, Seiringer Robert
06 Sep 2018 math-ph cond-mat.quant-gas cond-mat.str-el math.MP arxiv.org/abs/1809.01902

While Hartree-Fock theory is well established as a fundamental approximation for interacting fermions, it has been unclear how to describe corrections to it due to many-body correlations. In this paper we start from the Hartree-Fock state given by plane waves and introduce collective particle-hole pair excitations. These pairs can be approximately described by a bosonic quadratic Hamiltonian. We use Bogoliubov theory to construct a trial state yielding a rigorous Gell-Mann-Brueckner-type upper bound to the ground state energy. Our result justifies the random phase approximation in the mean-field scaling regime, for repulsive, regular interaction potentials.