Validity of random matrix theories for many-particle systems

J. B. FRENCH, S. S. M. Wong

Maxime Dixmerias, Giuseppe Del Vecchio Del Vecchio, Cyprien Daix +6 more

Since its initial use by Wigner in describing heavy atom spectra, random matrix theory (RMT) has found applications and revealed universal laws in diverse physical contexts, from quantum field theory to disordered systems and wireless communications. A particularly compelling application arises in the mathematical structure of the many-body wavefunction of non-interacting Fermi gases, which underpins a complex spatial organization driven by Pauli’s exclusion principle. However, experimental validation of the counting statistics predicted in such systems has remained elusive. Here, we probe at the single-atom level ultracold atomic Fermi gases made of two interacting spin states, obtaining direct access to their counting statistics in situ. We show that, while the system is strongly attractive, each spin component is described to a high accuracy by RMT predictions based on Fredholm determinants. Our results constitute an experimental validation of the Fermi-sphere point process through the lens of RMT, and establish its relevance for strongly-interacting systems. Random matrix theory (RMT) is used to describe non-interacting ultracold Fermi gases. Here, the authors directly probe the full counting statistics in a 2D system of 6Li atoms with two interacting spin states, by means of a continuum quantum gas microscopy technique. The measured number and spacing statistics agree well with predictions from RMT, also in the strongly interacting regime.

Léa Ferreira dos Santos, Dimitri Kusnezov, Philippe Jacquod

Stephen J. Gustafson, Israel Michael Sigal

Agneta F. Cederström

Random Matrix Theory considers symmetries in the QM energy matrix. The differential geometric implications of matrix groups and their relation to many particle quantum mechanical manifolds is explored. The RMT reduction of phase space is described by co-adjoint orbits on Lie matrix groups. Disorder is identified as a statistical mixture of entangled states. The system' s density matrix eigenvalues remain invariant. The spectrum of eigenvalues in an asymmetric quartic matrix potential is calculated in the unitary ensemble. This measure is interpreted to be indicative of the density of pseudo-particle states in compromised spin systems. The splitting of the spectral density into two or more intervals is taken as some type of phase transition. Finally, the temperature-doping phase diagrams of the ceramic compounds (La,Nd,Eu)2- xSrxCuO4 is partitioned into the three RMT symmetry classes by mapping onto the appearance and disappearance of an NMR/NQR signal.

Stephen J. Gustafson, Israel Michael Sigal