why entanglement entropy is important in topological phases？ When mentioning interacting topological phases people always talk about entanglement entropy. 
why it is important?
what is its physical meaning?
 A: Entanglement entropy (EE) is a measure of how much entanglement exists between regions of a system. Very loosely, you could think of the EE as telling you how many entangled pairs of particles exist in two spatial regions, usually in the ground state of your system. For a system seperated into two sections, $A$ and $\bar{A}$, the EE is given by:
$$S_A = -Tr(\rho_A\log\rho_A)$$
Where $\rho_A$ is the reduced density matrix of the system, where we have traced out (ignored) contributions from $\bar{A}$: $$\rho_A = {Tr}_{\small\bar{A}}(\rho)$$
Topological entanglement entropy  (TEE) characterises the amount of long-range entanglement that exists in a (gapped) system, which is directly proportional to the number of states in your system with some topological order. The topological order of your system tells you how close to a critical point you are, since phase transitions must occur in order for the topological order to change. Thus, by calculating the TEE in your system, you find out something about the topological phase of your system and how close to criticality it is.
For a 2+1-d topological system, Kitaev and Preskill showed that the EE is of the form $$S_A = \alpha L - \gamma ~+ ...$$
Where $L$ is the (large) length of the boundary of the entangling region (for example a circle encompassing $A$). The first term characterises short distance entanglement arising near the boundary $L$ whereas the term $-\gamma$ is the TEE.
In addition, you can vary the Hamiltonian of your system however you wish and as long as you don't hit a critical point, the TEE will be invariant. This is how phase transitions and the TEE are related, since after a phase transition, you would expect to find a new value for the TEE.
