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Macroscopic Behavior of Many-Body Quantum Systems

Periodic Reporting for period 2 - MaMBoQ (Macroscopic Behavior of Many-Body Quantum Systems)

Reporting period: 2020-03-01 to 2021-08-31

A fundamental problem in physics is to understand the collective behavior of macroscopic systems from the fundamental laws of motion of the elementary constituents. Due to the large dimensionality of the problem, related to the enormous number of particles involved in macroscopic compounds, in realistic situations it is essentially impossible to extract precise information about the physical evolution from the microscopic dynamics, defined by the Schroedinger equation. For this reason, one is often led to consider effective theories, namely approximate descriptions depending on much less degrees of freedom, which are expected to capture the behavior of the system on a macroscopic scale. These simplified models are usually motivated by heuristic averaging mechanisms, taking place at a mesoscopic/macroscopic scale, whose rigorous justification represents a major challenge for mathematicians. An important problem in mathematical physics is to put on rigorous grounds the validity of such emergent descriptions starting from first principles, and in particular to understand the sensitivity of the macroscopic description on the microscopic structure of the system. In general, different microscopic models might produce completely different microscopic evolutions; in this view, it is remarkable that some important physical properties appear to be universal, that is largely insensitive from the microscopic details, and only dependent on few qualitative properties of the system, such as its symmetries.

The goal of the ERC project MaMBoQ "Macroscopic Behavior of Many-Body Quantum Systems" is to develop new mathematical methods for the derivation of effective theories for many-body quantum systems, and to prove the emergence of universality in models of relevance for condensed matter physics. The project focuses on a broad spectrum of mathematical problems, such as: the description of topological phase transitions in interacting condensed matter systems; the stability or instability of semimetallic phases against disorder; the understanding of edge transport in interacting topological insulators, and the connection with bosonization in the scaling limit; the derivation of effective evolution equations for many-body quantum systems on kinetic time scales; the validity of the bosonization approach for the dynamics and for the equilibrium properties of interacting Fermi gases at high and low density. The methods that we plan to use to attack these problems include the renormalization group (RG), supersymmetric (SUSY) localization, functional analytic methods, semiclassical analysis.
As planned, the project developed along two parallel directions. The first concerns the transport properties of interacting or disordered topological insulators and semimetals, while the second focuses on the derivation of effective theories for interacting Fermi gases in the mean-field regime or in the dilute regime. About the first direction, the project combined rigorous renormalization group methods with other tools, to achieve new results in mathematical condensed matter physics. In particular, the combination with supersymmetry allowed to construct the scaling limit of a hierarchical model for weakly disordered three-dimensional semimetals, and to prove the irrelevance of disorder and the algebraic decay of correlations; the combination with Ward identities allowed to prove the universality of the lattice analogue of the chiral anomaly for interacting Weyl semimetals, and to compute the edge conductance for a generic class of interacting quantum Hall systems. This last result provides a rigorous justification for the predictions about edge transport based on heuristic bosonization methods; moreover, it allows to extend the bulk-edge correspondence for quantum Hall systems from non-interacting models to the realm of weakly interacting systems. Concerning the second research direction, the project succeeded in characterizing the ground state properties of interacting fermionic systems in the mean-field regime beyond the quasi-free approximation, also called Hartree-Fock approximation, by providing a rigorous, nonperturbative computation of the leading contribution to the correlation energy. The same techniques have then been used to derive a norm approximation for the many-body evolution of a class of fermionic states, in terms of the evolution of a suitable quasi-free Bose gas. The progress obtained in this second research direction is based on a rigorous bosonization approach to the study of excitations around the Fermi surface. Similar techniques have also been used to describe correlations in the somewhat opposite regime of dilute Fermi gases.
All the results mentioned above introduce a progress in the state of the art for the corresponding research lines. The methodology that has been developed provides the foundation for the second part of the project. Concerning disordered systems, the project aims at setting up a rigorous renormalization group method for the analysis of the supersymmetric representation of disordered systems with pointlike Fermi surface. So far we developed a method that applies to the hierarchical approximation of three dimensional semimetals; we expect the extension to the full non-hierarchical setting to be within reach, by combining the renormalization group method with cluster expansion techniques for supersymmetric theories, which have also been developed in the first period. Renormalization group also played a key role in the proof of universality of the chiral anomaly in Weyl semimetals, and for the edge conductance of quantum Hall systems. Both results are the first of their kind; in particular, the last result is the culmination of a research line that allowed us to prove the validity of the bulk-edge duality for weakly interacting quantum Hall systems, a longstanding open problem in mathematical physics. In the second part of the project, we plan to extend the renormalization group analysis in order to fully characterize multipoint edge correlation functions for interacting quantum Hall systems, and to prove that a suitable scaling limit of such correlation functions converges to the correlation functions of a quasi-free Bose gas, as predicted by the bosonization approach to edge transport. A complementary set of techniques, of functional analytic type, has been introduced for the analysis of interacting Fermi gases in the mean-field regime. In particular, these nonperturbative methods allowed to rigorously justify the bosonization approach for the study of the low energy excitations of mean-field Fermi gases, and to give a rigorous justification of the random-phase approximation. This is the first rigorous, nonperturbative result about the correlation energy of an interacting, three-dimensional Fermi gas. In the second part of the project we plan to extend these results to study the thermodynamic limit of interacting Fermi gases at high density, and to study the dynamics of interacting Fermi gases on kinetic time scales.