Entanglement and Quantum phase transition in topological insulators

Presence of entangled states is explicitly shown in Topological insulator (TI) $Bi_2Te_3$. The surface and bulk state are found to have the different structures of entanglement. The surface states live as maximally entangled states in the four-dimensional subspace of total Hilbert space (spin, orbital, space). However, bulk states are entangled in the whole Hilbert space. Bulk states are found to be entangled maximally by controlled injection of electrons with momentum only along the z-direction. Scheme to detect entanglement in a 2-D model using measurement, confirming natural implementation of universal Hadamard with Controlled-NOT gates is explicated.