Entanglement in many-body eigenstates of quantum-chaotic quadratic Hamiltonians

TL;DR

This paper extends a previously derived formula for the average bipartite entanglement entropy of eigenstates in quadratic Hamiltonians, demonstrating its broad applicability to various models including local and non-particle-number conserving systems, and analyzing second Rényi entropy.

Contribution

It generalizes the analytic expression for entanglement entropy to a wider class of quadratic Hamiltonians, including those with quantum chaos and broken particle-number conservation.

Findings

The formula applies to local Hamiltonians like the 3D Anderson model at weak disorder.

It describes entanglement in models without particle-number conservation, such as the Majorana SYK2.

Analytic and numerical results for the second Rényi entropy support the conjecture for quantum-chaotic quadratic Hamiltonians.

Abstract

In a recent Letter [Phys. Rev. Lett. 125, 180604 (2020)], we introduced a closed-form analytic expression for the average bipartite von Neumann entanglement entropy of many-body eigenstates of random quadratic Hamiltonians. Namely, of Hamiltonians whose single-particle eigenstates have random coefficients in the position basis. A paradigmatic Hamiltonian for which the expression is valid is the quadratic Sachdev-Ye-Kitaev (SYK2) model in its Dirac fermion formulation. Here we show that the applicability of our result is much broader. Most prominently, it is also relevant for local Hamiltonians such as the three-dimensional (3D) Anderson model at weak disorder. Moreover, it describes the average entanglement entropy in Hamiltonians without particle-number conservation, such as the SYK2 model in the Majorana fermion formulation and the 3D Anderson model with additional terms that break particle-number conservation. We extend our analysis to the average bipartite second R\'enyi entanglement entropy of eigenstates of the same quadratic Hamiltonians, which is derived analytically and tested numerically. We conjecture that our results for the entanglement entropies of many-body eigenstates apply to quadratic Hamiltonians whose single-particle eigenstates exhibit quantum chaos, to which we refer as quantum-chaotic quadratic Hamiltonians.