Entanglement Entropy of Compactified Branes and Phase Transition

We first calculate the holographic entanglement entropy of M5 branes on a circle and see that it has phase transition during decreasing the compactified radius. In particular, it is shown that the entanglement entropy scales as $N^3$. Next, we investigate the holographic entanglement entropy of D0+D4 system on a circle and see that it scales as $N^2$ at low energy, likes as a gauge theory with instantons. However, at high energy it transforms to a phase which scales as $N^3$, like as M5 branes system. We also present the general form of holographic entanglement entropy of Dp, ${\rm D_p+D_{p+4}}$ and M-branes on a circle and see some simple relations between them. Finally, we present an analytic method to prove that they all have phase transition from connected to disconnected surface during increasing the line segment of length $\ell$ which dividing the space.