Random matrices and trapped fermions – RaMaTraF

This is a project involving theorists in statistical and condensed matter physics. Based on our recent breakthrough in applying random matrix theory (RMT)
to the physics of trapped cold fermions, our project aims (i) to develop new theoretical methods to describe the statics and dynamics of cold atoms in traps (ii) to explore the connections between fermions and a wide variety of recently studied models in statistical physics (iii) to assess, together with experimentalists, how our theoretical predictions on the universal features at the edge of the trapped Fermi gas may be compared with experiments. Recent progresses in quantum gas microscopy and trap design make this project realistic and timely. We aim to establish new connections and initiate a synergy between two wide and active communities (cold atoms and RMT), which presently have almost no overlap. This ambitious goal will be facilitated by the organisation of a workshop and a school.

Even in the absence of interactions, fermions display non-trivial collective quantum many body effects purely due to the Pauli exclusion principle. Recent developments on Fermi gas microscopes now allow one to probe these quantum correlations, in the non-interacting limit, by direct imaging of the fermion clouds with single atom resolution. The confining trap induces an edge to the Fermi gas, where the density is small and, consequently, quantum and thermal fluctuations are large. Theoretical understanding of the patio-temporal correlations in this edge regime is a challenging problem. Indeed traditional methods, such as the local density approximation, are only valid in the bulk, i.e., near the trap center, but fail to describe the edge properties.

On the theory side, we have recently demonstrated that RMT provides powerful analytical tools to describe the trapped Fermi gas, including
the strong fluctuations at the edge. For instance, one striking prediction of RMT is that the fluctuations of the position of the rightmost fermion in a 1d harmonic trap at T=0 are governed by the universal Tracy-Widom (TW) distribution, emerging in numerous other areas including interface growth. Since thermal fluctuations cannot be ignored in cold atom experiments, it is crucial to compute edge observables at finite temperature, and in higher dimensions -- a major goal of this project. This will be achieved by building on our recent progress, which will also allow us to calculate several other static and dynamic edge observables, in all dimensions, that can be measured in experiments. Given the deep connections of free fermions to a wide variety of other problems, our results will be useful in many other fields.

We also aim to propose experiments on cold atoms, where these results could be tested. Recent advances in theory, design of Fermi gas microscopes, and the ability to tune the atomic interactions and to tailor arbitrary trapping potentials, provide exciting new possibilities at this interface between theoretical physics and experimental atomic physics. We plan to interact with cold atom groups, such as the quantum gas microscope group of I. Bloch in Munich, or the groups of C. Salomon at LKB-ENS to examine the possibility of testing our predictions, gaining understanding of what types of observables would be useful to compute in the theory.

The consortium is composed of three theory teams (LPTMS, LPTENS, LOMA) -- who have extensively collaborated in the past. They are leading international experts in statistical and condensed matter physics. Given their complementary expertise they are in a unique position to make this project a success, with important impact. We request the recruitement of a postdoc in theory for two years, localised for one year at LPTMS and one year at ENS to increase synergy between the teams.