Holographic entanglement entropy of a $1+1$ dimensional $p$-wave superconductor

We examine the behavior of entanglement entropy of a subsystem $A$ in a fully backreacted holographic model of a $1+1$ dimensional $p$ wave superconductor across the phase transition. For a given temperature, the system goes to a superconducting phase beyond a critical value of the charge density. The entanglement entropy, considered as a function of the charge density at a given temperature, has a cusp at the critical point. In addition, we find that there are three different behaviors in the condensed phase, depending on the subsystem size. For a subsystem size $l$ smaller than a critical size $l_{c1}$, entanglement entropy continues to increase as a function of the charge density as we cross the phase transition. When $l$ lies between $l_{c1}$ and another critical size $l_{c2}$ the entanglement entropy displays a non-monotonic behavior, while for $l > l_{c2}$ it decreases monotonically. At large charge densities entanglement entropy appears to saturate. The non-monotonic behavior leads to a novel phase diagram for this system.