Area Law for Gapless States from Local Entanglement Thermodynamics

TL;DR

This paper establishes an area law bound for the entanglement entropy of gapless quantum states using local entanglement thermodynamics, linking it to the hyper-scaling violation exponent and thermodynamic data.

Contribution

It introduces a thermodynamics-based approach to bound entanglement entropy in gapless states, relating the hyper-scaling violation exponent to area law violations.

Findings

Systems with $ heta < d-1$ obey the area law.

Systems with $ heta = d-1$ can violate the area law logarithmically.

Violating the area law significantly requires many low energy states.

Abstract

We demonstrate an area law bound on the ground state entanglement entropy of a wide class of gapless quantum states of matter using a strategy called local entanglement thermodynamics. The bound depends only on thermodynamic data, actually a single exponent, the hyper-scaling violation exponent $\theta$. All systems in $d$ spatial dimensions obeying our scaling assumptions and with $\theta < d-1$ obey the area law, while systems with $\theta = d-1$ can violate the area law at most logarithmically. We also discuss the case of frustration-free Hamiltonians and show that to violate the area law more than logarithmically these systems must have an unusually large number of low energy states. Finally, we make contact with the recently proposed $s$-source framework and argue that $\theta$ and $s$ are related by $s=2^\theta$.