Chemical Potential in the First Law for Holographic Entanglement Entropy

TL;DR

The paper addresses how the first law of holographic entanglement entropy extends when the boundary theory's degrees of freedom vary by promoting variations in the bulk cosmological constant $\Lambda$. It develops an extended bulk first law using Hamiltonian perturbation theory and Killing potentials, maps these variations to the boundary as a chemical-potential-like term governed by a thermodynamic volume $V$, and demonstrates that for spherical entangling surfaces $V$ is geometrically tied to the area of the minimal surface $A_\Sigma$. The main results include the explicit extended relation $\delta E_\xi = \frac{\delta A_\Sigma}{4G} - \frac{V \delta\Lambda}{8\pi G}$ with $V = -\int_\Sigma da_a \omega^{ab} n_b$ and the interpretation $\mu_{CFT} = -\frac{S_E}{c}$ (or its higher-dimensional analogs), linking bulk geometry to boundary conformal data such as central charges. This work provides a rigorous framework for expressing entanglement-entropy responses to changes in $N_{CFT}$ and suggests directions for general entangling surfaces and higher-curvature bulk theories.

Abstract

Entanglement entropy in conformal field theories is known to satisfy a first law. For spherical entangling surfaces, this has been shown to follow via the AdS/CFT correspondence and the holographic prescription for entanglement entropy from the bulk first law for Killing horizons. The bulk first law can be extended to include variations in the cosmological constant $Λ$, which we established in earlier work. Here we show that this implies an extension of the boundary first law to include varying the number of degrees of freedom of the boundary CFT. The thermodynamic potential conjugate to $Λ$ in the bulk is called the thermodynamic volume and has a simple geometric formula. In the boundary first law it plays the role of a chemical potential. For the bulk minimal surface $Σ$ corresponding to a boundary sphere, the thermodynamic volume is found to be proportional to the area of $Σ$, in agreement with the variation of the known result for entanglement entropy of spheres. The dependence of the CFT chemical potential on the entanglement entropy and number of degrees of freedom is similar to how the thermodynamic chemical potential of an ideal gas depends on entropy and particle number.