  It has recently been proposed that the entanglement entropy can be an order
parameter of confinement/deconfinement transitions. To find a clear evidence,
we introduce a new quantity called the geometric entropy, which is related to
the entanglement entropy via a double Wick rotation. We analyze the geometric
entropy and manifestly show that its value becomes discontinuous at the
Hagedorn temperature both in the free N =4 super Yang-Mills and in its
supergravity dual.
