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.
