On Volumes of Subregions in Holography and Complexity

TL;DR

This paper investigates the properties of a volume-based measure in holography, generalizing holographic complexity to subregions, revealing non-monotonic behaviors and discontinuities at phase transitions in various AdS geometries.

Contribution

It derives a formula for the volume of subregions in holography, analyzes its behavior in different geometries, and explores its discontinuities at phase transitions, extending the understanding of holographic complexity.

Findings

Volume exhibits non-monotonic behavior in AdS black holes.

Volume shows finite jumps at entanglement transition points.

Computed volume and action for spherical entangling surfaces in AdS.

Abstract

The volume of the region inside the bulk Ryu-Takayanagi surface is a codimension-one object, and a natural generalization of holographic complexity to the case of subregions in the boundary QFT. We focus on time-independent geometries, and study the properties of this volume in various circumstances. We derive a formula for computing the volume for a strip entangling surface and a general asymptotically AdS bulk geometry. For an AdS black hole geometry, the volume exhibits non-monotonic behaviour as a function of the size of the entangling region (unlike the behaviour of the entanglement entropy in this setup, which is monotonic). For setups in which the holographic entanglement entropy exhibits transitions in the bulk, such as global AdS black hole, geometries dual to confining theories and disjoint entangling surfaces, the corresponding volume exhibits a discontinuous finite jump at the transition point (and so do the volumes of the corresponding entanglement wedges). We compute this volume discontinuity in several examples. Lastly, we compute the codim-zero volume and the bulk action of the entanglement wedge for the case of a sphere entangling surface and pure AdS geometry.