Holographic entanglement density for spontaneous symmetry breaking

TL;DR

This paper explores how spontaneous symmetry breaking affects holographic entanglement entropy, revealing universal behaviors and violations of the area theorem, with implications for understanding strongly coupled systems.

Contribution

It classifies conditions for the first law of entanglement entropy and area theorem in systems with spontaneous symmetry breaking, highlighting universal features and differences between weak and strong coupling.

Findings

FLEE is always obeyed regardless of symmetry type.

The area theorem is violated when translational symmetry is weakly broken.

Log contributions from Goldstone modes may be absent in strongly coupled systems.

Abstract

We investigate the properties of the holographic entanglement entropy of the systems in which the $U(1)$ or the translational symmetry is broken \textit{spontaneously}. For this purpose, we define the entanglement density of the strip-subsystems and examine both the first law of entanglement entropy (FLEE) and the area theorem. We classify the conditions that FLEE and/or the area theorem obey and show that such a classification may be useful for characterizing the systems. We also find universalities from both FLEE and the area theorem. In the spontaneous symmetry breaking case, FLEE is always obeyed regardless of the type of symmetry: $U(1)$ or translation. For the translational symmetry, the area theorem is always violated when the symmetry is weakly broken, independent of the symmetry breaking patterns (explicit or spontaneous). We also argue that the $\log$ contribution of the entanglement entropy from the Goldstone mode may not appear in the strongly coupled systems.