Holographic order parameter for charge fractionalization

TL;DR

This paper introduces a holographic order parameter based on electric flux through a bulk surface, which distinguishes fractionalized phases with charged horizons from cohesive phases with bulk matter, at finite charge density.

Contribution

It proposes a new flux-based order parameter in holography that differentiates fractionalized and cohesive phases at finite density.

Findings

Flux exhibits volume law in fractionalized phases.

Flux is between boundary and volume law in deconfined cohesive phases.

Confined phases have vanishing flux through the surface.

Abstract

Nonlocal order parameters for deconfinement, such as the entanglement entropy and Wilson loops, depend on spatial surfaces \Sigma. These observables are given holographically by the area of a certain bulk spatial surface \Gamma, ending on \Sigma. At finite charge density it is natural to consider the electric flux through the bulk surface \Gamma, in addition to its area. We show that this flux provides a refined order parameter that can distinguish `fractionalized' phases, with charged horizons, from what we term `cohesive' phases, with charged matter in the bulk. Fractionalization leads to a volume law for the flux through the surface, the flux for deconfined but cohesive phases is between a boundary and a volume law, while finite density confined phases have vanishing flux through the surface. We suggest two possible field theoretical interpretations for this order parameter. The first is as information extracted from the large N reduced density matrix associated to \Sigma. The second is as surface operators dual to polarized bulk `D-branes', carrying an electric dipole moment.