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 affected by a spatial component of the Yang-Mills fields, which is normalizable mode of the solution. This is known as phase transition between isotropic and anisotropic phases. 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, ${\mathrm{\Delta }}_{\epsilon }S\sim {(1-\frac{T}{T_c})}^{\beta }$ and the critical exponent $\beta =1$, where ${\mathrm{\Delta }}_{\epsilon }S\equiv S^{iso}-S^{aniso}$, and $S^{iso}$ denotes the entanglement entropy in isotropic phase whereas $S^{aniso}$ denotes that in anisotropic phase. We suggest a quantity $O_{12}\equiv S_1-S_2$ 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 hold. Especially, we find that the entanglement temperature for the cylinder is ${\mathcal{T}}_{cy}=\frac{c_{ent}}{a}$, where $c_{ent}=0.163004\pm 0.000001$ and $a$ is the radius of the cylinder.