Extensive entropy from unitary evolution

TL;DR

This paper proves that generic quantum Hamiltonians with non-degenerate spectra typically generate extensive entanglement entropy, leading to volume-law entanglement in long-time evolution, especially in local and many-body localized systems.

Contribution

It establishes that non-degenerate spectral gaps ensure extensive entropy generation, applicable to most local Hamiltonians, and connects to observed entanglement growth in many-body localized systems.

Findings

Non-degenerate spectrum implies volume-law entanglement.

Almost all local Hamiltonians satisfy the non-degenerate gap condition.

Results explain entanglement behavior in many-body localized systems.

Abstract

In quantum many-body systems, a Hamiltonian is called an ``extensive entropy generator'' if starting from a random product state the entanglement entropy obeys a volume law at long times with overwhelming probability. We prove that (i) any Hamiltonian whose spectrum has non-degenerate gaps is an extensive entropy generator; (ii) in the space of (geometrically) local Hamiltonians, the non-degenerate gap condition is satisfied almost everywhere. Specializing to many-body localized systems, these results imply the observation stated in the title of Bardarson et al. [PRL 109, 017202 (2012)].