the one-dimensional topological superconducting phase of paired fermions, which is characterized by Majorana zero modes at the edges[@Kitaev2001]. Such zero modes actually resemble the ones found in the cores of vortices in two-dimensional topological superconductors[@ReadGreen], and have been shown useful for quantum information processing[@AliceaRPP]. However, it has been argued that one-dimensional fermion systems with interactions and no extra symmetry than the intrinsic fermion parity can only realize two topologically distinct phases[@Fidkowski2011; @Turner2011]. In order to search for a universal quantum computation platform and fully understand topological excitations in strongly interacting electron systems, more exotic parafermion excitations have been investigated in an effectively one-dimensional system[@Fendley2012; @Pollmann2013; @AliceaFendley], which exists at the edges of a two-dimensional fractionalized topological state and cannot be realized in a strictly one-dimensional system[Lindner,Clarke,Cheng]{}.

For various correlated low-dimensional gapped systems, it has been known that matrix product states[@FannesWerner1992] (MPS) and their high dimensional generalizations, tensor network states or projective entangled pair states[@VerstraeteCirac] (PEPS), have been proven increasingly successful. The framework of MPS and PEPS naturally provides an efficient method to classify topologically ordered phases[@Schuch2013; @Haegeman2015], symmetry protected topological phases[Chen-Gu-Wen-2011,Schuch,chen-gu-liu-wen]{}, and the long-range ordered phases with spontaneous symmetry breaking[@Chen-Gu-Wen-2011; @Rispler2015]. However, it is not straight forward to extend the 