Entanglement entropy of highly degenerate states and fractal dimensions

TL;DR

This paper investigates how the entanglement entropy of highly degenerate ground states in quantum systems scales with subsystem size, revealing a connection to the fractal dimension of basis elements and Goldstone bosons.

Contribution

It establishes a universal relation between entanglement entropy divergence and the fractal dimension of basis states in degenerate ground states.

Findings

Entanglement entropy diverges as (d/2) log m for large subsystems.

The fractal dimension d relates to the number of Goldstone bosons.

The results apply broadly to systems with large ground state degeneracy.

Abstract

We consider the bipartite entanglement entropy of ground states of extended quantum systems with a large degeneracy. Often, as when there is a spontaneously broken global Lie group symmetry, basis elements of the lowest-energy space form a natural geometrical structure. For instance, the spins of a spin-1/2 representation, pointing in various directions, form a sphere. We show that for subsystems with a large number m of local degrees of freedom, the entanglement entropy diverges as (d/2) log m, where d is the fractal dimension of the subset of basis elements with nonzero coefficients. We interpret this result by seeing d as the (not necessarily integer) number of zero-energy Goldstone bosons describing the ground state. We suggest that this result holds quite generally for largely degenerate ground states, with potential applications to spin glasses and quenched disorder.