# Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians

## TL;DR

This paper develops methods to compute the average entanglement entropy of eigenstates in quadratic fermionic systems, revealing how entanglement scales with subsystem size and system size, and showing eigenstate thermalization in certain limits.

## Contribution

It provides exact bounds for the average entanglement entropy of eigenstates in translationally invariant quadratic fermionic Hamiltonians, highlighting deviations from typical pure states.

## Key findings

- Average entanglement entropy is less than maximal for finite fraction subsystems.
- Eigenstates exhibit maximal entanglement when the subsystem is a vanishing fraction of the system.
- Results demonstrate eigenstate thermalization in the limit of small subsystems.

## Abstract

In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page proved that the average entanglement entropy of subsystems of random pure states is $S_{\rm ave}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm A}^2/{\cal D}$ for $1\ll{\cal D}_{\rm A}\leq\sqrt{\cal D}$, where ${\cal D}_{\rm A}$ and ${\cal D}$ are the Hilbert space dimensions of the subsystem and the system, respectively. Hence, typical pure states are (nearly) maximally entangled. We develop tools to compute the average entanglement entropy $\langle S\rangle$ of all eigenstates of quadratic fermionic Hamiltonians. In particular, we derive exact bounds for the most general translationally invariant models $\ln{\cal D}_{\rm A} - (\ln{\cal D}_{\rm A})^2/\ln{\cal D} \leq \langle S \rangle \leq \ln{\cal D}_{\rm A} - [1/(2\ln2)] (\ln{\cal D}_{\rm A})^2/\ln{\cal D}$. Consequently we prove that: (i) if the subsystem size is a finite fraction of the system size then $\langle S\rangle<\ln{\cal D}_{\rm A}$ in the thermodynamic limit, i.e., the average over eigenstates of the Hamiltonian departs from the result for typical pure states, and (ii) in the limit in which the subsystem size is a vanishing fraction of the system size, the average entanglement entropy is maximal, i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.