Chemical Potential in the First Law for Holographic Entanglement Entropy

TL;DR

This paper extends the holographic first law of entanglement entropy to include variations in the cosmological constant, linking it to a chemical potential for degrees of freedom in the boundary CFT, with geometric and thermodynamic implications.

Contribution

It demonstrates that the boundary first law of entanglement entropy can be extended to include a chemical potential for degrees of freedom, derived from the bulk thermodynamic volume.

Findings

Thermodynamic volume is proportional to the entangling surface area.

The boundary chemical potential depends on entanglement entropy and degrees of freedom.

The bulk and boundary laws are consistent with each other in the extended framework.

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 $\Lambda$, 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 $\Lambda$ 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 $\Sigma$ corresponding to a boundary sphere, the thermodynamic volume is found to be proportional to the area of $\Sigma$, 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.