Periodic Reporting for period 5 - MaMBoQ (Macroscopic Behavior of Many-Body Quantum Systems)
Reporting period: 2024-09-01 to 2025-07-31
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 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. 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, 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 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 macroscopic 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.
The goal of the ERC project MaMBoQ 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 macroscopic 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.
The first part of the project 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 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. The approach has also been extended to the case of disordered edge modes, with quasi-periodic potentials. Furthermore, the project succeeded in justifying the expression of many transport coefficients used in physics, by proving the rigorous validity of Kubo formula for interacting, gapless quantum systems, at the basis of the widely used linear response approximation.
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 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. Also, the project succeeded in deriving new analytic methods for the derivation of effective evolution equations for interacting fermions in the thermodynamic limit and at high density. Specifically, it allowed to obtain the first rigorous proof of the Hartree dynamics for extended Fermi gases.
The project opened new research directions beyond the foreseen ones, among which the rigorous analysis of the Z2 gauge theory coupled to fermionic matter in two dimensions, via the combination of reflection positivity, cluster expansion methods, quasi-adiabatic flow and renormalization group tools.
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 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. The approach has also been extended to the case of disordered edge modes, with quasi-periodic potentials. Furthermore, the project succeeded in justifying the expression of many transport coefficients used in physics, by proving the rigorous validity of Kubo formula for interacting, gapless quantum systems, at the basis of the widely used linear response approximation.
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 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. Also, the project succeeded in deriving new analytic methods for the derivation of effective evolution equations for interacting fermions in the thermodynamic limit and at high density. Specifically, it allowed to obtain the first rigorous proof of the Hartree dynamics for extended Fermi gases.
The project opened new research directions beyond the foreseen ones, among which the rigorous analysis of the Z2 gauge theory coupled to fermionic matter in two dimensions, via the combination of reflection positivity, cluster expansion methods, quasi-adiabatic flow and renormalization group tools.
-) Proof of the bulk-edge duality for weakly interacting 2d topological insulators. Renormalization group methods have been used to study the edge side of the duality. In particular, to construct the scaling limit of the boundary theory and to prove the emergence of identities between correlations that have been predicted to hold by non-rigorous bosonization.
-) Development of a rigorous renormalization group framework for the analysis of the supersymmetric representation of disordered systems with pointlike Fermi surface. The method has been applied so far to hierarchical models, and the extension to the full non-hierarchical setting will be the focus of future research.
-) Development of new methods for the analysis of many-body systems exposed to real-time adiabatic perturbations. This allowed to import field theoretic methods for the study of quantum transport; in particular to give the first proof of validity of Kubo formula for the dynamics of gapless interacting systems, achieved in the project for one dimensional Fermi gases.
-) Rigorous bosonization for the ground state properties of non-relativistic Fermi gases. The project allowed to develop a new analytic technique, that has been used to understand non-perturbatively the ground state of mean-field Fermi gases in three dimensions beyond the Hartree-Fock approximation. In particular, we obtained the first rigorous prediction about the correlation energy for mean-field Fermi gases, which reproduced a result obtained in physics via the random phase approximation.
-) Emergence of effective nonlinear dynamics for extended Fermi gases at high density. The project allowed to develop new methods for the derivation of effective evolution equations for interacting Fermi gases in the thermodynamic limit. The technique is centered around the notion of local fluctuations, whose dynamical control allowed to prove the validity of the Hartree dynamics at fixed, high density.
-) Development of a rigorous renormalization group framework for the analysis of the supersymmetric representation of disordered systems with pointlike Fermi surface. The method has been applied so far to hierarchical models, and the extension to the full non-hierarchical setting will be the focus of future research.
-) Development of new methods for the analysis of many-body systems exposed to real-time adiabatic perturbations. This allowed to import field theoretic methods for the study of quantum transport; in particular to give the first proof of validity of Kubo formula for the dynamics of gapless interacting systems, achieved in the project for one dimensional Fermi gases.
-) Rigorous bosonization for the ground state properties of non-relativistic Fermi gases. The project allowed to develop a new analytic technique, that has been used to understand non-perturbatively the ground state of mean-field Fermi gases in three dimensions beyond the Hartree-Fock approximation. In particular, we obtained the first rigorous prediction about the correlation energy for mean-field Fermi gases, which reproduced a result obtained in physics via the random phase approximation.
-) Emergence of effective nonlinear dynamics for extended Fermi gases at high density. The project allowed to develop new methods for the derivation of effective evolution equations for interacting Fermi gases in the thermodynamic limit. The technique is centered around the notion of local fluctuations, whose dynamical control allowed to prove the validity of the Hartree dynamics at fixed, high density.