Decoding quantum criticalities from fermionic/parafermionic topological states

Under an appropriate symmetric bulk bipartition in a one-dimensional symmetry protected topological phase with the Affleck-Kennedy-Lieb-Tasaki matrix product state wave function for the odd integer spin chains, a bulk critical entanglement spectrum can be obtained, describing the excitation spectrum of the critical point separating the topological phase from the trivial phase with the same symmetry. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry breaking phase transitions. Recently, the framework of matrix product states for topological phases with Majorana fermions/parafermions has been established. Here we first generalize these fixed-point matrix product states with the zero correlation length to the more generic ground-state wave functions with a finite correlation length for the general one-dimensional interacting Majorana fermion/parafermion systems. Then we employ the previous method to decode quantum criticality from the interacting Majorana fermion/parafermion matrix product states. The obtained quantum critical spectra are described by the conformal field theories with central charge $c\leq 1$, characterizing the quantum critical theories separating the fermionic/parafermionic topological phases from the trivial phases with the same symmetry.