Holographic order parameter for charge fractionalization

TL;DR

This work introduces a flux-enhanced holographic order parameter for charge fractionalization at finite density by defining a deformed entanglement entropy $S_E^\gamma = \frac{A_\Gamma}{4G_N} + \gamma \Phi_\Gamma$ with $\Phi_\Gamma$ the electric flux through a bulk hypersurface $\Gamma$. By analyzing both RT-minimal surfaces and the IR scaling geometries with hyperscaling violation $\theta$, it distinguishes fully fractionalized, partially fractionalized, deconfined cohesive, and confined cohesive phases through distinct scaling laws: fractionalized cases exhibit a volume-law flux, while cohesive cases show boundary- or sub-volume-law behavior or zero flux in confinement. The authors develop the dynamics of dipole-bearing hypersurfaces, derive conditions for endpoint structures, and discuss how minimization of $S_E^\gamma$ yields different phase-sensitive scaling, including potential bubble solutions in the bulk. They propose two field-theoretic interpretations—the flux as information in the large-$N$ reduced density matrix and as dual to polarized bulk D-brane surface operators carrying a dipole moment—offering a bridge between holographic geometry and boundary fractionalization physics. Overall, the paper provides a concrete, computable diagnostic for fractionalization in holographic matter and outlines avenues to relate bulk flux to boundary density and surface operator dynamics.

Abstract

Nonlocal order parameters for deconfinement, such as the entanglement entropy and Wilson loops, depend on spatial surfaces Σ. These observables are given holographically by the area of a certain bulk spatial surface Γ, ending on Σ. At finite charge density it is natural to consider the electric flux through the bulk surface Γ, 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 Σ. The second is as surface operators dual to polarized bulk `D-branes', carrying an electric dipole moment.

Figures (4)

• Figure 1: Left, minimal bulk hypersurfaces in a confining geometry. At large boundary separations $L$ the surface is disconnected and leads to a boundary law for the entanglement entropy. Right, minimal hypersurfaces in a finite temperature black hole geometry. At large separations $L$ the surface remains connected and leads to a volume law. In both cases, the surfaces for small boundary separations are connected.
• Figure 2: From left to right: (i) all flux emanates from horizon, (ii) a fraction of the flux emanates from a horizon, a fraction is from charged bulk fields, (iii) all flux from bulk fields outside a neutral horizon, (iv) all flux from bulk fields in a confining geometry.
• Figure 3: By Gauss's law, in the absence of bulk charges, the electric flux through the surface $\Gamma$ is equal to the flux at the asymptotic boundary in the region bounded by the surface $\Sigma = \partial \Gamma$. The total asymptotic flux is $\rho \, L \cdot \text{Vol} \left(\Sigma \right)$.
• Figure 4: Typical plots of the function $|D(r)|$ in various parts of parameter space. The horizontal lines correspond to different possible structures of solutions to $|D(r)| = 1/4G_N$. The shaded regions are where a solution to the Euler-Lagrange equation \ref{['eq:xdotsq']} exists. Bubbles are solutions that exist without reaching the boundary.