Holographic entanglement entropy probe on spontaneous symmetry breaking with vector order

We study holographic entanglement entropy in 5-dimensional charged black brane geometry obtained from Einstein-SU(2)Yang-Mills theory defined in asymptotically AdS space. This gravity system undergoes second order phase transition near its critical point, where a spatial component of the Yang-Mills fields appears, which is normalizable mode of the solution. This is known as phase transition between isotropic and anisotropic phases, where in anisotropic phase, SO(3)-isometry(spatial rotation) in bulk geometry is broken down to SO(2) by emergence of the spatial component of Yang-Mills fields, which corresponds to a vector order in dual field theory. We get analytic solutions of holographic entanglement entropies by utilizing the solution of bulk spacetime geometry given in arXiv:1109.4592 , where we consider subsystems defined on AdS boundary of which shapes are wide and thin slabs and a cylinder. It turns out that the entanglement entropies near the critical point shows scaling behavior such that for both of the slabs and cylinder, Δ_εS∼(1-T/T_c)^β and the critical exponent β = 1, where ∆ ε S ≡ S iso − S aniso , and S iso denotes the entanglement entropy in isotropic phase whereas S aniso denotes that in anisotropic phase. We suggest a quantity O 12 ≡ S 1 − S 2 as a new order parameter near the critical point, where S 1 is entanglement entropy when the slab is perpendicular to the direction of the vector order whereas S 2 is that when the slab is parallel to the vector order. O 12 = 0 in isotropic phase but in anisotropic phase, the order parameter becomes non-zero showing the same scaling behavior. Finally, we show that even near the critical point, the first law of entanglement entropy is held. Especially, we find that the entanglement temperature for the cylinder is 𝒯_cy=c_ent/a , where c ent = 0 . 163004 ± 0 . 000001 and a is the radius of the cylinder.