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Christian Hainzl

Publications and source records attributed to Christian Hainzl.

At least 37 records · Page 2Linked to original sources
arXiv2020

The dilute Fermi gas via Bogoliubov theory

We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej. In this paper we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density.

math-ph↗
arXiv2019

On the correlation energy of interacting fermionic systems in the mean-field regime

We consider a system of $N\gg 1$ interacting fermionic particles in three dimensions, confined in a periodic box of volume $1$, in the mean-field scaling. We assume that the interaction potential is bounded and small enough. We prove upper and lower bounds for the correlation energy, which are optimal in their $N$-dependence. Moreover, we compute the correlation energy at leading order in the interaction potential, recovering the prediction of second order perturbation theory. The proof is based on the combination of methods recently introduced for the study of fermionic many-body quantum dynamics together with a rigorous version of second-order perturbation theory, developed in the context of non-relativistic QED.

math-ph↗
arXiv2019

Lower Bound on the Hartree-Fock Energy of the Electron Gas

The Hartree-Fock ground state of the Homogeneous Electron Gas is never translation invariant, even at high densities. As proved by Overhauser, the (paramagnetic) free Fermi Gas is always unstable under the formation of spin or charge density waves. We give here the first explicit bound on the energy gain due to the breaking of translational symmetry. Our bound is exponentially small at high density, which justifies posteriori the use of the non-interacting Fermi Gas as a reference state in the large-density expansion of the correlation energy of the Homogeneous Electron Gas. We are also able to discuss the positive temperature phase diagram and prove that the Overhauser instability only occurs at temperatures which are exponentially small at high density. Our work sheds a new light on the Hartree-Fock phase diagram of the Homogeneous Electron Gas.

cond-mat.str-el↗
arXiv2017

The BCS critical temperature in a weak homogeneous magnetic field

We show that, within a linear approximation of BCS theory, a weak homogeneous magnetic field lowers the critical temperature by an explicit constant times the field strength, up to higher order terms. This provides a rigorous derivation and generalization of results obtained in the physics literature from WHH theory of the upper critical magnetic field. A new ingredient in our proof is a rigorous phase approximation to control the effects of the magnetic field.

math-ph↗
arXiv2016

Note on a Family of Monotone Quantum Relative Entropies

Given a convex function $φ$ and two hermitian matrices $A$ and $B$, Lewin and Sabin study in [M. Lewin, J. Sabin, {\it A Family of Monotone Quantum Relative Entropies}, Lett. Math. Phys. \textbf{104} (2014), 691-705.] the relative entropy defined by $\mathcal{H}(A,B)=\text{Tr} [ φ(A) - φ(B) - φ'(B)(A-B) ]$. Amongst other things, they prove that the so-defined quantity is monotone if and only if $φ'$ is operator monotone. The monotonicity is then used to properly define $\mathcal{H}(A,B)$ for self-adjoint bounded operators acting on an infinite-dimensional Hilbert space by a limiting procedure. More precisely, for an increasing sequence of finite-dimensional projections $\lbrace P_n \rbrace_{n=1}^{\infty}$ with $P_n \to 1$ strongly, the limit $\lim_{n \to \infty} \mathcal{H}(P_n A P_n, P_n B P_n)$ is shown to exist and to be independent of the sequence of projections $\lbrace P_n \rbrace_{n=1}^{\infty}$. The question whether this sequence converges to its "obvious" limit, namely $\text{Tr} [ φ(A)- φ(B) - φ'(B)(A-B) ]$, has been left open. We answer this question in principle affirmatively and show that $\lim_{n \to \infty} \mathcal{H}(P_n A P_n, P_n B P_n) = \text{Tr}[ φ(A) - φ(B) - \frac{\text{d}}{\text{d} α} φ( αA + (1-α)B )|_{α= 0} ]$. If the operators $A$ and $B$ are regular enough, that is $(A-B)$, $φ(A)-φ(B)$ and $φ'(B)(A-B)$ are trace-class, the identity $\text{Tr}[ φ(A) - φ(B) - \frac{\text{d}}{\text{d} α} φ( αA + (1-α)B )|_{α= 0} ] = \text{Tr} [ φ(A)- φ(B) - φ'(B)(A-B) ]$ holds.

math-ph↗
arXiv2016

Comparing the full time-dependent Bogoliubov--de-Gennes equations to their linear approximation: A numerical investigation

In this paper we report on the results of a numerical study of the nonlinear time-dependent Bardeen--Cooper--Schrieffer (BCS) equations, often also denoted as Bogoliubov--de--Gennes (BdG) equations, for a one-dimensional system of fermions with contact interaction. We show that, even above the critical temperature, the full equations and their linear approximation give rise to completely different evolutions. In contrast to its linearization, the full nonlinear equation does not show any diffusive behavior in the order parameter. This means that the order parameter does not follow a Ginzburg--Landau-type of equation, in accordance with a recent theoretical results. We include a full description on the numerical implementation of the partial differential BCS\ BdG equations.

math-ph↗
arXiv2016

Incompatibility of time-dependent Bogoliubov--de-Gennes and Ginzburg--Landau equations

We study the time-dependent Bogoliubov--de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg--Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time. The full non-linear structure of the equations is necessary to understand this behavior.

math-ph↗
arXiv2015

Bogolubov-Hartree-Fock theory for strongly interacting fermions in the low density limit

We consider the Bogolubov-Hartree-Fock functional for a fermionic many-body system with two-body interactions. For suitable interaction potentials that have a strong enough attractive tail in order to allow for two-body bound states, but are otherwise sufficiently repulsive to guarantee stability of the system, we show that in the low-density limit the ground state of this model consists of a Bose-Einstein condensate of fermion pairs. The latter can be described by means of the Gross-Pitaevskii energy functional.

math-ph↗
arXiv2015

The BCS functional of superconductivity and its mathematical properties

We review recent results concerning the mathematical properties of the Bardeen-Cooper-Schrieffer (BCS) functional of superconductivity, which were obtained in a series of papers partly in collaboration with R. Frank, E. Hamza, S. Naboko, and J.P. Solovej. Our discussion includes, in particular, an investigation of the critical temperature for a general class of interaction potentials, as well as a study of its dependence on external fields. We shall explain how the Ginzburg-Landau model can be derived from the BCS theory in a suitable parameter regime.

math-ph↗
arXiv2015

Microscopic Derivation of the Ginzburg-Landau Model

We present a summary of our recent rigorous derivation of the celebrated Ginzburg-Landau (GL) theory, starting from the microscopic Bardeen-Cooper-Schrieffer (BCS) model. Close to the critical temperature, GL arises as an effective theory on the macroscopic scale. The relevant scaling limit is semiclassical in nature, and semiclassical analysis, with minimal regularity assumptions, plays an important part in our proof.

math-ph↗
arXiv2014

The external field dependence of the BCS critical temperature

We consider the Bardeen-Cooper-Schrieffer free energy functional for particles interacting via a two-body potential on a microscopic scale and in the presence of weak external fields varying on a macroscopic scale. We study the influence of the external fields on the critical temperature. We show that in the limit where the ratio between the microscopic and macroscopic scale tends to zero, the next to leading order of the critical temperature is determined by the lowest eigenvalue of the linearization of the Ginzburg-Landau equation.

math-ph↗
arXiv2014

A Spatially Homogeneous and Isotropic Einstein-Dirac Cosmology

We consider a spatially homogeneous and isotropic cosmological model where Dirac spinors are coupled to classical gravity. For the Dirac spinors we choose a Hartree-Fock ansatz where all one-particle wave functions are coherent and have the same momentum. If the scale function is large, the universe behaves like the classical Friedmann dust solution. If however the scale function is small, quantum effects lead to oscillations of the energy-momentum tensor. It is shown numerically and proven analytically that these quantum oscillations can prevent the formation of a big bang or big crunch singularity. The energy conditions are analyzed. We prove the existence of time-periodic solutions which go through an infinite number of expansion and contraction cycles.

math-ph↗
arXiv2014

On the well-posedness and scattering for the Gross-Pitaevskii hierarchy via quantum de Finetti

We prove the existence of scattering states for the defocusing cubic Gross-Pitaevskii (GP) hierarchy in ${\mathbb R}^3$. Moreover, we show that an energy growth condition commonly used in the well-posedness theory of the GP hierarchy is, in a specific sense, necessary. In fact, we prove that without the latter, there exist initial data for the focusing cubic GP hierarchy for which instantaneous blowup occurs.

math-ph↗
arXiv2014

On the BCS gap equation for superfluid fermionic gases

We present a rigorous derivation of the BCS gap equation for superfluid fermionic gases with point interactions. Our starting point is the BCS energy functional, whose minimizer we investigate in the limit when the range of the interaction potential goes to zero.

math-ph↗
arXiv2013

Translation-invariant quasi-free states for fermionic systems and the BCS approximation

We study translation-invariant quasi-free states for a system of fermions with two-particle interactions. The associated energy functional is similar to the BCS functional but includes also direct and exchange energies. We show that for suitable short-range interactions, these latter terms only lead to a renormalization of the chemical potential, with the usual properties of the BCS functional left unchanged. Our analysis thus represents a rigorous justification of part of the BCS approximation. We give bounds on the critical temperature below which the system displays superfluidity.

math-ph↗