Deviation from maximal entanglement for mid-spectrum eigenstates of local Hamiltonians

TL;DR

This paper demonstrates that mid-spectrum eigenstates of local Hamiltonians exhibit entanglement entropy and thermodynamic entropy deviations from maximum and thermodynamic values, highlighting differences from random states.

Contribution

It proves that eigenstates in a microcanonical ensemble with sufficient bandwidth deviate from maximal entanglement and thermodynamic entropy, revealing fundamental differences from random states.

Findings

Eigenstates deviate from maximum entanglement entropy by a positive constant.

Eigenstates deviate from thermodynamic entropy at the same energy by a positive constant.

Results distinguish mid-spectrum eigenstates of local Hamiltonians from random states.

Abstract

In a spin chain governed by a local Hamiltonian, we consider a microcanonical ensemble in the middle of the energy spectrum and a contiguous subsystem whose length is a constant fraction of the system size. We prove that if the bandwidth of the ensemble is greater than a certain constant, then the average entanglement entropy (between the subsystem and the rest of the system) of eigenstates in the ensemble deviates from the maximum entropy by at least a positive constant. This result highlights the difference between the entanglement entropy of mid-spectrum eigenstates of (chaotic) local Hamiltonians and that of random states. We also prove that the former deviates from the thermodynamic entropy at the same energy by at least a positive constant.