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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - CORFRONMAT (Correlated frontiers of many-body quantum mathematics and condensed matter physics)

Teaser

Summary

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.

Work performed

During the first year and a half of the project we made progress in several directions.

As regards the first part of the project we have studied a certain 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, by us and another group. 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.

In a very recent preprint 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. This is in accordance with the physicists\' expectations, but had never been attacked from a mathematical standpoint.

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.

We also improved the derivation of the average-field model 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 based on a new variant of the quantum de Finetti approach to quantum mean-field limits.

Final results

The derivation of classical field theories from many-body bosonic Gibbs states seems to us 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 ?

Another research direction in the first part of the project is the derivation of a Bose-Hubbard Hamiltonian for interacting bosons in a double-well potential.

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. A first step is to study higher Landau level states in analogy with what we did for lowest Landau level states.

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. Any partial progress would be of significance, but a particular case seems within closer reach: statistics transmutation, turning bosons to fermions or vice-versa via strong correlations.

As for anyons, several important problems remain as to the derivation of effective models from the many-body Hamiltonian. In particular, the case of almost-fermions should be within reach, and help discuss the properties of systems where the almost-bosonic approximation would not be appropriate.