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 a phase transition when 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 a D0 + D4 system on a circle and see that it scales as N ² at low energy, as in gauge theory with instantons. However, at high energy it transforms to a phase that scales as N ³ , as an M5 brane system. We also present the general form of holographic entanglement entropy of Dp, D _p + D _{p + 4} and M-branes on a circle and see some simple relations among them. Finally, we present an analytic method to prove that they all have phase transitions from connected to disconnected surfaces as one increases the line segment that divides the entangling regions.