Thermodynamical property of entanglement entropy and deconfinement phase transition

We analyze the holographic entanglement entropy in a soliton background with a background gauge field and derive a relation analogous to the first law of thermodynamics. The confinement/deconfinement phase transition occurs due to the competition of two minimal surfaces. The entropic C function probes the confinement/deconfinement phase transition. It is sensitive to the degrees of freedom smaller than the size of a spatial circle. When the background gauge field becomes large, the entropic C function becomes nonmonotonic as a function of the size and does not satisfy the usual c-theorem. We analyze the entanglement entropy for a small subregion and the relation analogous to the first law of thermodynamics. For the small amount of a background gauge field, the excited amount of the entanglement entropy decreases from the ground state. It reflects that confinement decreases degrees of freedom. We finally discuss the second order correction of the holographic entanglement entropy.