1,009 research outputs found
From Bosonic Grand-Canonical Ensembles to Nonlinear Gibbs Measures
In a recent paper, in collaboration with Mathieu Lewin and Phan Th{\`a}nh
Nam, we showed that nonlinear Gibbs measures based on Gross-Pitaevskii like
functionals could be derived from many-body quantum mechanics, in a mean-field
limit. This text summarizes these findings. It focuses on the simplest, but
most physically relevant, case we could treat so far, namely that of the
defocusing cubic NLS functional on a 1D interval. The measure obtained in the
limit, which (almost) lives over H^{1/2} , has been previously shown to be
invariant under the NLS flow by Bourgain.Comment: This is the text of a Laurent Schwartz X-EDP seminar I gave in
November 2014. It summarizes some of the results of arXiv:1410.033
On the stability of 2D dipolar Bose-Einstein condensates
We study the existence of energy minimizers for a Bose-Einstein condensate
with dipole-dipole interactions, tightly confined to a plane. The problem is
critical in that the kinetic energy and the (partially attractive) interaction
energy behave the same under mass-preserving scalings of the wave-function. We
obtain a sharp criterion for the existence of ground states, involving the
optimal constant of a certain generalized Gagliardo-Nirenberg inequality
The Laughlin liquid in an external potential
We study natural perturbations of the Laughlin state arising from the effects
of trapping and disorder. These are N-particle wave functions that have the
form of a product of Laughlin states and analytic functions of the N variables.
We derive an upper bound to the ground state energy in a confining external
potential, matching exactly a recently derived lower bound in the large N
limit. Irrespective of the shape of the confining potential, this sharp upper
bound can be achieved through a modification of the Laughlin function by
suitably arranged quasi-holes.Comment: Typos corrected and one remark added. To be published in Letters in
Mathematical Physic
Higher Dimensional Coulomb Gases and Renormalized Energy Functionals
We consider a classical system of n charged particles in an external
confining potential, in any dimension d larger than 2. The particles interact
via pairwise repulsive Coulomb forces and the coupling parameter scales like
the inverse of n (mean-field scaling). By a suitable splitting of the
Hamiltonian, we extract the next to leading order term in the ground state
energy, beyond the mean-field limit. We show that this next order term, which
characterizes the fluctuations of the system, is governed by a new
"renormalized energy" functional providing a way to compute the total Coulomb
energy of a jellium (i.e. an infinite set of point charges screened by a
uniform neutralizing background), in any dimension. The renormalization that
cuts out the infinite part of the energy is achieved by smearing out the point
charges at a small scale, as in Onsager's lemma. We obtain consequences for the
statistical mechanics of the Coulomb gas: next to leading order asymptotic
expansion of the free energy or partition function, characterizations of the
Gibbs measures, estimates on the local charge fluctuations and factorization
estimates for reduced densities. This extends results of Sandier and Serfaty to
dimension higher than two by an alternative approach.Comment: Structure has slightly changed, details and corrections have been
added to some of the proof
Incompressibility Estimates for the Laughlin Phase
This paper has its motivation in the study of the Fractional Quantum Hall
Effect. We consider 2D quantum particles submitted to a strong perpendicular
magnetic field, reducing admissible wave functions to those of the Lowest
Landau Level. When repulsive interactions are strong enough in this model,
highly correlated states emerge, built on Laughlin's famous wave function. We
investigate a model for the response of such strongly correlated ground states
to variations of an external potential. This leads to a family of variational
problems of a new type. Our main results are rigorous energy estimates
demonstrating a strong rigidity of the response of strongly correlated states
to the external potential. In particular we obtain estimates indicating that
there is a universal bound on the maximum local density of these states in the
limit of large particle number. We refer to these as incompressibility
estimates
On the binding of polarons in a mean-field quantum crystal
We consider a multi-polaron model obtained by coupling the many-body
Schr\"odinger equation for N interacting electrons with the energy functional
of a mean-field crystal with a localized defect, obtaining a highly non linear
many-body problem. The physical picture is that the electrons constitute a
charge defect in an otherwise perfect periodic crystal. A remarkable feature of
such a system is the possibility to form a bound state of electrons via their
interaction with the polarizable background. We prove first that a single
polaron always binds, i.e. the energy functional has a minimizer for N=1. Then
we discuss the case of multi-polarons containing two electrons or more. We show
that their existence is guaranteed when certain quantized binding inequalities
of HVZ type are satisfied.Comment: 28 pages, a mistake in the former version has been correcte
Boundary Behavior of the Ginzburg-Landau Order Parameter in the Surface Superconductivity Regime
We study the 2D Ginzburg-Landau theory for a type-II superconductor in an
applied magnetic field varying between the second and third critical value. In
this regime the order parameter minimizing the GL energy is concentrated along
the boundary of the sample and is well approximated to leading order by a
simplified 1D profile in the direction perpendicular to the boundary. Motivated
by a conjecture of Xing-Bin Pan, we address the question of whether this
approximation can hold uniformly in the boundary region. We prove that this is
indeed the case as a corollary of a refined, second order energy expansion
including contributions due to the curvature of the sample. Local variations of
the GL order parameter are controlled by the second order term of this energy
expansion, which allows us to prove the desired uniformity of the surface
superconductivity layer
Quantum Hall phases and plasma analogy in rotating trapped Bose gases
A bosonic analogue of the fractional quantum Hall eff ect occurs in rapidly
rotating trapped Bose gases: There is a transition from uncorrelated Hartree
states to strongly correlated states such as the Laughlin wave function. This
physics may be described by eff ective Hamiltonians with delta interactions
acting on a bosonic N-body Bargmann space of analytic functions. In a previous
paper [N. Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)] we
studied the case of a quadratic plus quartic trapping potential and derived
conditions on the parameters of the model for its ground state to be
asymptotically strongly correlated. This relied essentially on energy upper
bounds using quantum Hall trial states, incorporating the correlations of the
Bose-Laughlin state in addition to a multiply quantized vortex pinned at the
origin. In this paper we investigate in more details the density of these trial
states, thereby substantiating further the physical picture described in [N.
Rougerie, S. Serfaty, J. Yngvason, Phys. Rev. A 87, 023618 (2013)], improving
our energy estimates and allowing to consider more general trapping potentials.
Our analysis is based on the interpretation of the densities of quantum Hall
trial states as Gibbs measures of classical 2D Coulomb gases (plasma analogy).
New estimates on the mean- field limit of such systems are presented.Comment: Minor modification
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