Periodic Reporting for period 4 - CORFRONMAT (Correlated frontiers of many-body quantum mathematics and condensed matter physics)
Période du rapport: 2022-07-01 au 2023-12-31
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.
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.
We also worked 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. We also obtained a Lieb-Thirring type inequality for extended anyons, improving our derivation.
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.
We vastly generalized the construction and invariance under the Schrödinger flow of nonlinear Gibbs measures in one space-dimension, allowing a large variety of external potentials, power non-linearities, and dealing with measures conditioned on (renormalized) mass. This paves the way for the derivation of non-linear Gibbs measures conditioned on mass and/or with attractive interactions, from the canonical many-body problem.
We proved that 2D anyons confined in a tight wave-guide are best described in terms of a 1D Girardeau-Tonks model. This means that 2D anyons, irrespective of their statistical parameter, fermionize in 1D. This answers natural theoretical and practical questions and is somewhat suprising in several respects, connected to recent physics literature.
We studied self-binding of polarons in the quasi-classical limit. This was the occasion to revisit the convergence of ground-state energies for such models, and provide tool to improve it to convergence of states.
We considered the semi-classical limit of the Hartree dynamics for fermions in large magnetic fields. We deal with the regime where the appropriate semi-classical phase-space differs from the standard position/momentum one. For large fields the phase-space is the semi-discrete Landau index/position of orbit center, retaining some quantum features. We obtain in the limit a drift equation for the matter density, reflecting the slow motion of cyclotron orbits after the averaging out of the fast cyclotron motion.
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.
We also worked 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. We also obtained a Lieb-Thirring type inequality for extended anyons, improving our derivation.
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.
We vastly generalized the construction and invariance under the Schrödinger flow of nonlinear Gibbs measures in one space-dimension, allowing a large variety of external potentials, power non-linearities, and dealing with measures conditioned on (renormalized) mass. This paves the way for the derivation of non-linear Gibbs measures conditioned on mass and/or with attractive interactions, from the canonical many-body problem.
We proved that 2D anyons confined in a tight wave-guide are best described in terms of a 1D Girardeau-Tonks model. This means that 2D anyons, irrespective of their statistical parameter, fermionize in 1D. This answers natural theoretical and practical questions and is somewhat suprising in several respects, connected to recent physics literature.
We studied self-binding of polarons in the quasi-classical limit. This was the occasion to revisit the convergence of ground-state energies for such models, and provide tool to improve it to convergence of states.
We considered the semi-classical limit of the Hartree dynamics for fermions in large magnetic fields. We deal with the regime where the appropriate semi-classical phase-space differs from the standard position/momentum one. For large fields the phase-space is the semi-discrete Landau index/position of orbit center, retaining some quantum features. We obtain in the limit a drift equation for the matter density, reflecting the slow motion of cyclotron orbits after the averaging out of the fast cyclotron motion.