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We analyze the holographic entanglement entropy in a soliton background withWilson lines and derive a relation analogous to the first law ofthermodynamics. The confinement/deconfinement phase transition occurs due tothe competition of two minimal surfaces. The entropic c function probes theconfinement/deconfinement phase transition. It is sensitive to the degrees offreedom (DOF) smaller than the size of a spatial circle. When the Wilson linebecomes large, the entropic c function becomes non-monotonic as a function ofthe size and does not satisfy the usual c-theorem. We analyze the entanglemententropy for a small subregion and the relation analogous to the first law ofthermodynamics. For the small amount of Wilson lines, the excited amount of theentanglement entropy decreases from the ground state. It reflects thatconfinement decreases degrees of freedom. We finally discuss the second ordercorrection of the holographic entanglement entropy.

Thermodynamical property of entanglement entropy and deconfinement phase transition