Holographic Charged Renyi Entropies

TL;DR

This work generalizes Rényi entropies to include a conserved charge by inserting a Wilson line around the entangling surface, defining $S_n(\mu)$ and its imaginary-chemical-potential counterpart $\tilde{S}_n(\mu_E)$. Using the replica trick and a spherical entangling surface, the authors map the problem to a thermal ensemble on $\mathbb{R}\times\mathbb{H}^{d-1}$ and compute holographic and free-field results, relating charged twists to energy and current densities via conformal data. In holography, charged topological black holes yield the entropy functions, with explicit expressions for the conformal weight $h_n(\mu)$ and magnetic response $k_n(\mu)$ of the generalized twist operators, and a discussion of thermodynamic stability and possible phase transitions. The paper confirms consistency across CFT methods and gravity, and discusses extensions to higher-derivative gravities, rotating ensembles, and fixed-charge ensembles, highlighting the rich structure of charge-resolved entanglement.

Abstract

We construct a new class of entanglement measures by extending the usual definition of Renyi entropy to include a chemical potential. These charged Renyi entropies measure the degree of entanglement in different charge sectors of the theory and are given by Euclidean path integrals with the insertion of a Wilson line encircling the entangling surface. We compute these entropies for a spherical entangling surface in CFT's with holographic duals, where they are related to entropies of charged black holes with hyperbolic horizons. We also compute charged Renyi entropies in free field theories.

Figures (9)

• Figure 1: Charged Rényi entropy with $n=3$ for a two-dimensional free fermion as a function of $\mu_\textrm{\tiny E}$.
• Figure 2: The $d=3$ charged Rényi entropy (normalized by (a) $S_1(0)$ and (b) $S_1(\mu)$) as a function of $\mu$. The curves correspond to (from top to bottom) $n$=1,2,3,4,10
• Figure 3: The $d=4$ charged Rényi entropy (normalized by (a) $S_1(0)$ and (b) $S_1(\mu)$) as a function of $\mu$. The curves correspond to (from top to bottom) $n$=1,2,3,4,10
• Figure 4: The charged Rényi entropy (normalized by $S_1(0)$) in $d=3$ shown function of $n$ in panel (a). In panel (b), we show $\frac{n-1}{n}S_n(\mu)$ as a function of $n$. Note that the slope of the curves is negative in panel (a) and positive in panel (b). In both cases, the curves correspond to (from top to bottom) $\frac{\mu \ell_*}{2 \pi L}=1.0,0.8,0.6,0.4,0.2$ and $0.0$.
• Figure 5: The $d=4$ charged Rényi entropy (normalized by (a) $\tilde{S}_1(0)$ and (b) $\tilde{S}_1(\mu_\textrm{\tiny E})$) as a function of the imaginary chemical potential $\mu_\textrm{\tiny E}$. The curves correspond to (from top to bottom) $n$=1,2,3,4,10