Level spacing statistics of bidimensional Fermi liquids: II. Landau fixed point and quantum chaos

TL;DR

This paper explores quantum chaos in two-dimensional Fermi liquids by analyzing level spacing statistics, showing a transition from Poissonian to G.O.E. distributions with energy and temperature, and examining effects of a momentum cutoff.

Contribution

It provides a detailed analysis of level statistics in 2D Fermi liquids, highlighting the impact of the momentum cutoff and temperature on quantum chaos signatures, extending previous 1D studies.

Findings

Poissonian level statistics at low energies in 1D with cutoff

G.O.E. statistics at higher energies in 1D with cutoff

Poissonian level statistics in 2D Fermi liquids with certain conditions

Abstract

We investigate the presence of quantum chaos in the spectrum of the bidimensional Fermi liquid by means of analytical and numerical methods. This model is integrable in a certain limit by bosonization of the Fermi surface. We study the effect on the level statisticsof the momentum cutoff $\Lambda$ present in the bidimensional bosonization procedure. We first analyse the level spacing statistics in the $\Lambda$-restricted Hilbert space in one dimension. With $g_2$ and $g_4$ interactions, the level statistics are found to be Poissonian at low energies, and G.O.E. at higher energies, for a given cut-off $\Lambda$. In order to study this cross-over, a finite temperature is introduced as a way of focussing, for a large inverse temperature $\beta$, on the low energy many-body states, and driving the statistics from G.O.E. to Poissonian. As far as two dimensions are concerned, we diagonalize the Fermi liquid Hamiltonian with a small number of orbitals. The level spacing statistics are found to be Poissonian in the $\Lambda$-restricted Hilbert space, provided the diagonal elements are of the same order of magnitude as the off-diagonal matrix elements of the Hamiltonian.