Connection between low energy effective Hamiltonians and energy level statistics

TL;DR

This paper investigates the energy level statistics of a non-integrable fermionic system, revealing that low energy spectra resemble integrable systems, and proposes using Random Matrix Theory as a tool to analyze low energy effective Hamiltonians.

Contribution

It demonstrates the connection between low energy spectra and integrability in strongly correlated systems using Random Matrix Theory analysis.

Findings

Low energy spectrum follows Poisson distribution, indicating integrability.

Level statistics align with GOE distribution for higher energies.

Proposes RMT as a probe for low energy Hamiltonian properties.

Abstract

We study the level statistics of a non-integrable one dimensional interacting fermionic system characterized by the GOE distribution. We calculate numerically on a finite size system the level spacing distribution $P(s)$ and the Dyson-Mehta $\Delta_3$ correlation. We observe that its low energy spectrum follows rather the Poisson distribution, characteristic of an integrable system, consistent with the fact that the low energy excitations of this system are described by the Luttinger model. We propose this Random Matrix Theory analysis as a probe for the existence and integrability of low energy effective Hamiltonians for strongly correlated systems.