Entanglement entropy of highly degenerate states and fractal dimensions

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.

5 pages. v2: Small changes, published version

References in corpus (5)

• Area laws in quantum systems: mutual information and correlations
• Entanglement entropy of fermions in any dimension and the Widom conjecture
• Entanglement entropies in free fermion gases for arbitrary dimension
• Bi-partite entanglement entropy in massive two-dimensional quantum field theory
• Permutation operators, entanglement entropy, and the XXZ spin chain in the limit Δ-> -1

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