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Correlated frontiers of many-body quantum mathematics and condensed matter physics

Periodic Reporting for period 3 - CORFRONMAT (Correlated frontiers of many-body quantum mathematics and condensed matter physics)

Reporting period: 2021-01-01 to 2022-06-30

One of the main challenges in condensed matter physics is to understand strongly correlated quantum systems. Our purpose is to approach this issue from the point of view of rigorous mathematical analysis. The goals are two-fold: develop a mathematical framework applicable to physically relevant scenarii, take inspiration from the physics to introduce new topics in mathematics. The scope of the proposal thus goes from physically oriented questions (theoretical description and modelization of physical systems) to analytical ones (rigorous derivation and analysis of reduced models) in several cases where strong correlations play the key role.

In a first part, we aim at developing mathematical methods of general applicability to go beyond mean-field theory in different contexts. Our long term goal is to forge new tools to attack important open problems in the field. Particular emphasis will be put on the structural properties of large quantum states as a general tool.

A second part is concerned with so-called fractional quantum Hall states, host of the fractional quantum Hall effect. Despite the appealing structure of their built-in correlations, their mathematical study is in its infancy. They however constitute an excellent testing ground to develop ideas of possible wider applicability. In particular we introduce and study a new class of many-body variational problems.

In the third part we discuss so-called anyons, exotic quasi-particles thought to emerge as excitations of highly-correlated quantum systems. Their modelization gives rise to rather unusual, strongly interacting, many-body Hamiltonians with a topological content. Mathematical analysis will help us shed light on those, clarifying the characteristic properties that could ultimately be experimentally tested.
We have studied a large temperature (semi-classical) limit of the thermal equilibria of an interacting quantum Bose gas. This limit could previously be tackled only in one space dimension. For the physically relevant case of a gas in 2 or 3 space dimensions we prove that the free-energy and the correlation functions converge to those of a classical field theory. The latter is a singular object which must be renormalized in order to make sense mathematically. The corresponding renormalization at the level of the quantum model is a simple adjustment of the chemical potential of the theory. A decisive ingredient of our proof is a new method to control the high-momentum correlations in the interacting Bose gas.

We made progress on stability properties of fractional quantum Hall states built on the Laughlin wave-function. We studied a variational problem of a new kind, which physically corresponds to asking whether the Laughlin phase is stable against external perturbations and long-range interactions. We prove that this is the case: a minimizer of the variational problem can, for so-called filling factors close to that of the Laughlin function itself, be looked for in the form of uncorrelated quasi-holes generated from the Laughlin function.

Concerning anyons, we conducted a numerical study of an effective model (almost-bosonic average-field functional) derived by us previously. The results are in very good agreement with rigorous theorems we obtained previously. They also shed light on some expectations we had on the structure of the solutions: they spontaneously generate inhomogeneous vortex lattices, a hallmark of the anyonic statistics entering the underlying many-body model.

The improvement of the mathematical techniques to deal with the mean-field limit of quantum mechanics is also a theme of the project. We have borrowed a new tool from quantum information theory (a refined version of the quantum de Finetti theorem) which, combined with approaches we developed previously, permits to extend the range of validity of previous results, in particular for the 2D attractive Bose gas.

We also improved the derivation of the average-field model for almost bosonic anyons from many-body quantum mechanics by including external magnetic fields in the picture and improving on some technical assumptions previously made in the derivation. This is partly based on the aforementioned new variant of the quantum de Finetti approach to quantum mean-field limits.

We have been dealing with the mean-field limit for many bosons in a double-well potential. For all relevant values of the tunneling energy between the two wells, and under a technical assumption of a small coupling constant, we prove that the ground state of the system violates the central limit theorem and hence contains strong interparticle correlations. This is achieved by expanding the full energy to high precision, identifying the contributions of a reduced two-modes (Bose-Hubbard like) Hamiltonian and those of Bogoliubov fluctuations.

Currently we are finishing work on a different mean-field limit for anyons: the almost fermionic one. When the statistical parameter converges to the Fermi end at the same time as the particle number goes to infinity, the problem becomes semi-classical. An important challenge was to prove a sufficiently quantitative version of a Pauli principle on phase space.

An important open problem is the rigorous proof of the emergence of anyons from usual non-relativistic quantum mechanics. We are currently finishing a proof that in a natural trial state for a coupled quantum system (bath/tracers) in high magnetic fields, the tracers change their quantum statistics from bosonic to fermionic by coupling to the bath (statistics transmutation). This is a proof of concept for the more general case where the tracer particles can become anyons.
The derivation of classical field theories from many-body bosonic Gibbs states seems a promising direction. It would be highly desirable to extend the results we have obtained in several directions: consider a thermodynamic limit jointly with the semi-classical limit, consider canonical (instead of grand-canonical) ensembles, allow for more realistic (short-range, singular) inter-particle interactions ... In the physics literature, classical fields were used to study in details the Bose-Einstein phase transition. Could our results help shed some light on this topic, whose mathematical side is a major open problem ?

Concerning fractional quantum Hall states, all mathematical studies so far have focused on the simplest case: states in the lowest Landau level. A desirable direction is the extension to promising variants, the so-called fractional Chern insulators.

At the juncture of the two last parts of the project, the emergence of fractional statistics in correlated condensed matter systems is still an important open problem. We would like to make some progress following our recent study of statistic transmutation.

Regarding effective models for anyons we currently have a satisfying understanding of the almost-bosonic and almost fermionic limits, although some problems remain open. A more challenging direction concerns the cases in between where the statistics parameter stays fixed. A rough idea is to attack this as a particular mean-field limit for fermions under strong self-consistent magnetic fields. Given that even the case of fixed inhomogeneous magnetic fields is open, this is a natural first step before tackling the full anyonic problem.
Vortex lattices in a trapped almost-bosonic anyon gas