Decoding quantum criticalities from fermionic/parafermionic topological states.

In the Fock representation, we propose the generalized matrix product states to describe one-dimensional topological phases of fermions/parafermions. The defining feature of these topological phases is the presence of Majorana/parafermion zero modes localized at the edges. It is shown that the single-block bipartite entanglement spectrum and its entanglement Hamiltonian are described by the effective coupling between two edge quasiparticles. Furthermore, we demonstrate that sublattice bulk bipartition can create an extensive number of edge quasiparticles in the reduced subsystem, and the symmetric couplings between the nearest neighbor edge quasiparticles lead to the critical entanglement spectra, characterizing the topological phase transitions from the fermionic/parafermionic topological phases to its adjacent trivial phase. The corresponding entanglement Hamiltonians for the critical entanglement spectra can also be derived.