Periodic Reporting for period 1 - FermiMath (The Mathematics of Interacting Fermions)
Reporting period: 2022-05-01 to 2024-10-31
Quantum mechanics is the fundamental theory of nature, describing the behavior of the smallest entities accessible to physicists. This includes electrons in condensed matter systems like metals and semiconductors: technologically promising phenomena such as superconductivity or topological insulators rely on the quantum properties of systems of a large number of electrons. The complexity of such systems is due to the interaction between electrons creating complicated entangled quantum states. Technology has in recent years achieved finer control of these effects with astonishing precision.
However, the thorough mathematical understanding of these systems is just leaving infancy. One of the most prominent examples, at the basis of many applications and theoretical investigations, is the screening of the interaction of electrons: surprisingly, despite the presence of a strong electrostatic repulsion between electrons in the microscopic Schrödinger equation, on a macroscopic level one observes almost non-interacting particles. Moreover, some properties turn out to be universal, i.e. do not depend on the details of the microscopic equation at all. FermiMath aims to mathematically derive emergent theories to describe these correlation effects in systems of interacting fermionic particles; this way the applicability of emergent theories can be critically evaluated.
The approach of FermiMath is based on the analysis of high-density scaling limits. While the analysis of scaling limits has been very successful for bosonic quantum systems, in fermionic systems it has been restricted to the derivation of mean-field theories. At the core of FermiMath is the application of recently developed mathematically rigorous bosonization methods to describe complex correlated behavior. This is one of the few tools that permit an analysis beyond mean-field or perturbation theory, and we aim to use it to mathematically distinguish the fundamental phases of electron systems responsible for the different behavior of metals, semiconductor, and carbon nanotubes.
However, the thorough mathematical understanding of these systems is just leaving infancy. One of the most prominent examples, at the basis of many applications and theoretical investigations, is the screening of the interaction of electrons: surprisingly, despite the presence of a strong electrostatic repulsion between electrons in the microscopic Schrödinger equation, on a macroscopic level one observes almost non-interacting particles. Moreover, some properties turn out to be universal, i.e. do not depend on the details of the microscopic equation at all. FermiMath aims to mathematically derive emergent theories to describe these correlation effects in systems of interacting fermionic particles; this way the applicability of emergent theories can be critically evaluated.
The approach of FermiMath is based on the analysis of high-density scaling limits. While the analysis of scaling limits has been very successful for bosonic quantum systems, in fermionic systems it has been restricted to the derivation of mean-field theories. At the core of FermiMath is the application of recently developed mathematically rigorous bosonization methods to describe complex correlated behavior. This is one of the few tools that permit an analysis beyond mean-field or perturbation theory, and we aim to use it to mathematically distinguish the fundamental phases of electron systems responsible for the different behavior of metals, semiconductor, and carbon nanotubes.
It has been the central research activity of project FermiMath in this stage to break the limitation of collective bosonization methods from collective physical observables like the ground state energy to the computation of the microscopic fermionic observable quantities. In particular, our attention focused on the fermionic momentum distribution in a correlated state modelling the ground state of an interacting gas of fermionic particles. The detailed study of the ground state's properties is crucial to access the signature jump discontinuity characterizing Fermi liquid theory.
The team of project FermiMath has been working on a rigorous mathematical approach, by means of a so-called self-consistent bootstrap, and thus successfully derived explicit formulas for the stability of the Fermi surface, the universality of the Fermi momentum, and the computation of the renormalization of the discontinuity at the Fermi surface. The team has also cast this approach in a diagrammatic computational scheme that greatly simplifies the derivation of explicit formulas and promises to render the future investigation more accessible. The results in this stage have positively verified older conjectures obtained through perturbation theory. Whereas the traditional perturbative methods suffer from severe divergences, in which physical quantities formally appear as infinite and require complicated and mathematically poorly understood resummation and renormalization, the results obtained by project FermiMath are mathematically rigorous and robust. Thereby a central achievement of the work up to this point has not only been to proof existing conjectures, but we have also created a new and robust mathematical theory as an underpinning for the future parts of the project.
In the second year, a research workshop on rigorous methods for the analysis of collective phenomena has been organized as a Lake Como School of Advanced Studies. This school served to develop connections of established mathematical methods such as renormalization group with the bosonization methods central to FermiMath.
The team of project FermiMath has been working on a rigorous mathematical approach, by means of a so-called self-consistent bootstrap, and thus successfully derived explicit formulas for the stability of the Fermi surface, the universality of the Fermi momentum, and the computation of the renormalization of the discontinuity at the Fermi surface. The team has also cast this approach in a diagrammatic computational scheme that greatly simplifies the derivation of explicit formulas and promises to render the future investigation more accessible. The results in this stage have positively verified older conjectures obtained through perturbation theory. Whereas the traditional perturbative methods suffer from severe divergences, in which physical quantities formally appear as infinite and require complicated and mathematically poorly understood resummation and renormalization, the results obtained by project FermiMath are mathematically rigorous and robust. Thereby a central achievement of the work up to this point has not only been to proof existing conjectures, but we have also created a new and robust mathematical theory as an underpinning for the future parts of the project.
In the second year, a research workshop on rigorous methods for the analysis of collective phenomena has been organized as a Lake Como School of Advanced Studies. This school served to develop connections of established mathematical methods such as renormalization group with the bosonization methods central to FermiMath.
In the first part of the research conducted in project FermiMath, the computation of the number of particles occupying different momentum modes in a trial state using the so-called random phase approximation has revealed critical insights into the physical properties of low-energy states of fermionic systems in condensed matter, such as metals or semiconductors. In particular we proved that this momentum distribution in a naturally constructed quantum trial state exhibits a jump discontinuity (an abrupt change of the number of particles occupying a mode as the energy of the mode varies), signifying the presence of a sharp Fermi surface. Moreover, in the trial state, the Fermi momentum, which marks the boundary of the occupied states in momentum space, does not depend on the interaction potential. In physics, this is known as universality. Fundamentally very different systems (ranging even beyond condensed matter into high-energy physics and cosmology) show in some respects the same physical properties. It is a central goal of modern physics to accordingly classify systems into universality classes where they share the same kind of universal behavior. With FermiMath, we start to mathematically understand this classification for large systems of fermionic quantum particles.
The discovery of a well-defined Fermi surface in the mean-field scaling limit is pivotal in advancing theoretical models of fermion systems. In particular, one of the main impacts of our result is to restrict the possible emergence of superconductivity. On the one hand, this facilitates applications of the bosonization approach and the random phase approximation for physical situations in which superconductivity is expected to play a negligible role. On the other hand, it strongly constraints the applicability of mean-field-like scaling limits in the derivation of superconductivity for other materials; it will require further research to identify models that do actually exhibit Cooper pairing and can thus transport electric current without dissipation or loss of energy in form of heat. While the realization of a mathematically rigorous theory of superconductivity from the fundamental laws of physics falls beyond the aims of the research of project FermiMath, it is part of its impact of having clarified already central restrictions and crucial factors in realizing superconductivity.
The research results of the project have already resonated strongly in mathematical physics; new applications of bosonization methods are starting to be employed and developed in other research groups. We expect that after a frenzy of activity in research on Bose-Einstein condensates over the last approximately twenty years, FermiMath is starting to stimulate a similar level of activity of mathematical research on fermionic quantum systems.
The discovery of a well-defined Fermi surface in the mean-field scaling limit is pivotal in advancing theoretical models of fermion systems. In particular, one of the main impacts of our result is to restrict the possible emergence of superconductivity. On the one hand, this facilitates applications of the bosonization approach and the random phase approximation for physical situations in which superconductivity is expected to play a negligible role. On the other hand, it strongly constraints the applicability of mean-field-like scaling limits in the derivation of superconductivity for other materials; it will require further research to identify models that do actually exhibit Cooper pairing and can thus transport electric current without dissipation or loss of energy in form of heat. While the realization of a mathematically rigorous theory of superconductivity from the fundamental laws of physics falls beyond the aims of the research of project FermiMath, it is part of its impact of having clarified already central restrictions and crucial factors in realizing superconductivity.
The research results of the project have already resonated strongly in mathematical physics; new applications of bosonization methods are starting to be employed and developed in other research groups. We expect that after a frenzy of activity in research on Bose-Einstein condensates over the last approximately twenty years, FermiMath is starting to stimulate a similar level of activity of mathematical research on fermionic quantum systems.