arXiv subjects

Nicolas Rougerie

Publications and source records attributed to Nicolas Rougerie.

At least 19 recordsLinked to original sources

arXiv2026

Free-energy variations for determinantal 2D plasmas with holes

We study the Gibbs equilibrium of a classical 2D Coulomb gas in the determinantal case $β$ = 2. The external potential is the sum of a quadratic term and the potential generated by individual charges pinned in several extended groups. This leads to an equilibrium measure (droplet) with flat density and macroscopic holes. We consider ''correlation energy'' (free energy minus its mean-field approximation) expansions, for large particle number N. Under the assumptions that the holes are sufficiently small, separated, and far from the droplet's outer boundary, we prove that (i) the correlation energy up to order 1 is independent of the holes' locations and orientations, and (ii) the difference between the correlation energies of systems differing by their number of holes essentially consists of ``topological'' O(log N) and O (1) terms.

math-ph↗

arXiv2026

$Φ^4_2$ theory limit of a many-body bosonic free energy

We consider the quantum Gibbs state of an interacting Bose gas on the 2D torus. We set temperature, chemical potential and coupling constant in a regime where classical field theory gives leading order asymptotics. In the same limit, the repulsive interaction potential is set to be short-range: it converges to a Dirac delta function with a rate depending polynomially on the other scaling parameters. We prove that the free-energy of the interacting Bose gas (counted relatively to the non-interacting one) converges to the free energy of the $Φ^4_2$ non-linear Schrödinger-Gibbs measure, thereby revisiting recent results and streamlining proofs thereof. We combine the variational method of Lewin-Nam-Rougerie to connect, with controled error, the quantum free energy to a classical Hartree-Gibbs one with smeared non-linearity. The convergence of the latter to the $Φ^4_2$ free energy then follows from arguments of Fröhlich-Knowles-Schlein-Sohinger. This derivation parallels recent results of Nam-Zhu-Zhu.

math.AP↗

arXiv2026

From bosonic canonical ensembles to non-linear Gibbs measures

We study the mean-field limit of the 1D bosonic canonical ensemble in a superharmonic trap. This is the regime with temperature proportional to particle number, both diverging to infinity, and correspondingly scaled interactions. We prove that the limit model is a classical field theory based on a non-linear Schr{ö}dinger-Gibbs measure conditioned on the L2 mass, thereby obtaining a canonical analogue of previous results for the grand-canonical ensemble. We take advantage of this set-up with fixed mass to include focusing/attractive interactions/non-linearities in our study.

math.AP↗

arXiv2026

Statistical mechanics of the radial focusing nonlinear Schrödinger equation in general traps

In this paper, we investigate the Gibbs measures associated with the focusing nonlinear Schrödinger equation with an anharmonic potential. We establish a dichotomy for normalizability and non-normalizability of the Gibbs measures in one dimension and higher dimensions with radial data. This extends a recent result of the third and fourth authors with Robert and Seong (2022), where the focusing Gibbs measures with a harmonic potential were addressed. Notably, in the case of a subharmonic potential, we identify a novel critical nonlinearity (below the usual mass-critical exponent) for which the Gibbs measures exhibit a phase transition. The primary challenge emerges from the limited understanding of eigenvalues and eigenfunctions of the Schrödinger operator with an anharmonic potential. We overcome the difficulty by employing techniques related to a recent work of the first two authors (2022).

math.AP↗

arXiv2026

The 2D Chern-Simons-Schr{ö}dinger system reduced to 1D

We study a mean-field model for a system of 2D abelian anyons, given by the dynamics of a Schr{ö}dinger matter field coupled to a Chern-Simons gauge field. We derive an effective 1D equation by adding a strongly anisotropic trapping potential (wave-guide) acting on the Schr{ö}dinger field, and tracing out the tight confinement direction. The effective dynamics in the loose direction of the wave-guide turns out to be governed by the classical 1D quintic NLS equation.

math.AP↗

arXiv2026

Magnetic Thomas-Fermi theory for 2D abelian anyons

Two-dimensional abelian anyons are, in the magnetic gauge picture, represented as fermions coupled to magnetic flux tubes. For the ground state of such a system in a trapping potential, we theoretically and numerically investigate a Hartree approximate model, obtained by restricting trial states to Slater determinants and introducing a self-consistent magnetic field, locally proportional to matter density. This leads to a fermionic variant of the Chern-Simons-Schr{ö}dinger system. We find that for dense systems, a semi-classical approximation yields qualitatively good results. Namely, we derive a density functional theory of magnetic Thomas-Fermi type, which correctly captures the trends of our numerical results. In particular, we explore the subtle dependence of the ground state with respect to the fraction of magnetic flux units attached to particles.

math.AP↗

arXiv2025

Gyrokinetic limit of the 2D Hartree equation in a large magnetic field

We study the dynamics of two-dimensional interacting fermions submitted to a homogeneous transverse magnetic field. We consider a large magnetic field regime, with the gap between Landau levels set to the same order as that of potential energy contributions. Within the mean-field approximation, i.e. starting from Hartree's equation for the first reduced density matrix, we derive a drift equation for the particle density. We use vortex coherent states and the associated Husimi function to define a semi-classical density almost satisfying the limiting equation. We then deduce convergence of the density of the true Hartree solution by a Dobrushin-type stability estimate.

math.AP↗

arXiv2025

Convergence of states for polaron models in the classical limit

We consider the quasi-classical limit of Nelson-type regularized polaron models describing a particle interacting with a quantized bosonic field. We break translation-invariance by adding an attractive external potential decaying at infinity, acting on the particle. In the strong coupling limit where the field behaves classically we prove that the model's energy quasi-minimizers strongly converge to ground states of the limiting Pekar-like non-linear model. This holds for arbitrarily small external attractive potentials, hence this binding is fully due to the interaction with the bosonic field. We use a new approach to the construction of quasi-classical measures to revisit energy convergence, and a localization method in a concentration-compactness type argument to obtain convergence of states.

math.AP↗

arXiv2024

Thomas-Fermi profile of a fast rotating Bose-Einstein condensate

We study the minimizers of a magnetic 2D non-linear Schrödinger energy functional in a quadratic trapping potential, describing a rotating Bose-Einstein condensate. We derive an effective Thomas-Fermi-like model in the rapidly rotating limit where the centrifugal force compensates the confinement, and available states are restricted to the lowest Landau level. The coupling constant of the effective Thomas-Fermi functional is linked to the emergence of vortex lattices (the Abrikosov problem). We define it via a low density expansion of the energy of the corresponding homogeneous gas in the thermodynamic limit.

math-ph↗

arXiv2024

Dimensional reduction for a system of 2D anyons

Anyons with a statistical phase parameter $α\in(0,2)$ are a kind of quasi-particles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave-guide. More specifically, we study the 2D magnetic-gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the $y$-direction than in the $x$-direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the $x$-direction is given by the impenetrable Tonks-Girardeau Bose gas, which has no dependency on $α$, and no trace left of the long-range interactions of the 2D model.

math.AP↗

arXiv2023

Anyons in a tight wave-guide and the Tonks-Girardeau gas

We consider a many-body system of 2D anyons, free quantum particles with general statistics parameter α\in ]0,2[. In the magnetic gauge picture they are described as bosons attached to Aharonov-Bohm fluxes of intensity 2 παgenerating long-range magnetic forces. A dimensional reduction to 1D is obtained by imposing a strongly anisotropic trapping potential. This freezes the motion in the direction of strong trapping, leading to 1D physics along the weak direction. The latter is governed to leading order by the Tonks-Girardeau model of impenetrable bosons, independently of α.

math.AP↗

arXiv2023

Invariant Gibbs measures for 1D NLS in a trap

We consider the one dimensional cubic nonlinear Schr{ö}dinger equation with trapping potential behaving like |x| s (s > 1) at infinity. We construct Gibbs measures associated to the equation and prove that the Cauchy problem is globally well-posed almost surely on their support. Consequently, the Gibbs measure is indeed invariant under the flow of the equation. We also address the construction and invariance of canonical Gibbs measures, conditioned on the L 2 mass.

math.AP↗

arXiv2023

Quantum statistics transmutation via magnetic flux attachment

We consider a model for two types (bath and tracers) of 2D quantum particles in a perpendicular magnetic field. Interactions are short range and inter-species, and we assume that the bath particles are fermions, all lying in the lowest Landau level of the magnetic field. Heuristic arguments then indicate that, if the tracers are strongly coupled to the bath, they effectively change their quantum statistics, from bosonic to fermionic or vice-versa. We rigorously compute the energy of a natural trial state, indeed exhibiting this phenomenon of statistics transmutation. The proof involves estimates for the characteristic polynomial of the Ginibre ensemble of random matrices.

math-ph↗

arXiv2023

A Lieb-Thirring inequality for extended anyons

We derive a Pauli exclusion principle for extended fermion-based anyons of any positive radius and any non-trivial statistics parameter. That is, we consider 2D fermionic particles coupled to magnetic flux tubes of non-zero radius, and prove a Lieb-Thirring inequality for the corresponding many-body kinetic energy operator. The implied constant is independent of the radius of the flux tubes, and proportional to the statistics parameter.

math-ph↗

arXiv2022

Quantum Hall Phases of Cold Bose Gases

Cold atomic gases of interacting bosons subject to rapid rotation and confined in anharmonic traps can theoretically exhibit analogues of the fractional quantum Hall effect for electrons in strong magnetic fields. In this setting the Coriolis force due to the rotation mimics the Lorentz force on charged particles but artificial gauge fields can also be obtained by coupling the internal structure of the atoms to light fields. The chapter discusses mathematical aspects of transitions to different strongly correlated phases that appear when the parameters of a model Hamiltonian are varied.

math-ph↗

arXiv2022

On the stability of Laughlin's fractional quantum Hall phase

The fractional quantum Hall effect in 2D electron gases submitted to large magnetic fields remains one of the most striking phenomena in condensed matter physics. Historically, the first observed signature is a Hall resistance quantized to the value (2m+1) when the filling factor (electron density divided by magnetic flux quantum density) of a 2D electron gas is in the vicinity of an inverse odd integer 1/(2m +1). This was one of the first observation of fractional quantum numbers. A large part of our basic theoretical understanding of this effect (and descendants) originates from Laughlin's theory of 1983, reviewed here from a mathematical physics perspective. We explain in which sense Laughlin's proposed ground and excited states for the system are rigid/incompressible liquids, and why this is crucial for the explanation of the effect.

cond-mat.mes-hall↗

arXiv2022

Blow-up of 2D attractive Bose-Einstein condensates at the crittical rotational speed

We study the ground states of a 2D focusing non-linear Schrödinger equation with rotation and harmonic trapping. When the strength of the interaction approaches a critical value from below, the system collapses to a profile obtained from the optimizer of a Gagliardo--Nirenberg interpolation inequality. This was established before in the case of fixed rotation frequency. We extend the result to rotation frequencies approaching, or even equal to, the critical frequency at which the centrifugal force compensates the trap. We prove that the blow-up scenario is to leading order unaffected by such a strong deconfinement mechanism. In particular the blow-up profile remains independent of the rotation frequency.

math.AP↗

arXiv2022

The classical Jellium and the Laughlin phase

I discuss results bearing on a variational problem of a new type, inspired by fractional quantum Hall physics. In the latter context, the main result reviewed herein can be spelled as "the phase of independent quasi-holes generated from Laughlin's wave-function is stable against external potentials and weak long-range interactions". The main ingredient of the proof is a connection between fractional quantum Hall wave-functions and statistical mechanics problems that generalize the 2D one-component plasma (jellium model). Universal bounds on the density of such systems, coined "Incompressibility estimates" are obtained via the construction of screening regions for any configuration of points with positive electric charges. The latter regions are patches of constant, negative electric charge density, whose shape is optimized for the total system (points plus patch) not to generate any electric potential in its exterior.

math.AP↗