Fermionic matrix product states and one-dimensional topological phases

Abstract

We develop the formalism of fermionic matrix product states (fMPS) and show how irreducible fMPS fall in two different classes, related to the different types of simple Z2 graded algebras, which are physically distinguished by the absence or presence of Majorana edge modes. The local structure of fMPS with Majorana edge modes also implies that there is always a two-fold degeneracy in the entanglement spectrum. Using the fMPS formalism we make explicit the correspondence between the Z8 classification of time-reversal invariant spinless superconductors and the modulo 8 periodicity in the representation theory of real Clifford algebras. Studying fMPS with general on-site unitary and anti-unitary symmetries allows us to define invariants that label symmetry-protected phases of interacting fermions. The behavior of these invariants under stacking of fMPS is derived, which reveals the group structure of such interacting phases. We also consider spatial symmetries and show how the invariant phase factor in the partition function of reflection symmetric phases on an unorientable manifold appears in the fMPS framework. CONTENTS

Tensor network approach to two-dimensional bosonization

“…In Appendix A, we show that the algebraic properties of the Majorana operators are indeed satisfied by the tensor representations in Eqs. (16) and (17). Furthermore, using Eq.…”

Fermionic tensor networks for higher-order topological insulators from charge pumping

“…For fermions, we write |Q x = Q x |Ω where Q x is a polynomial of creation operators which acts on the vacuum to create the fiducial state. For fermionic MPSs and PEPSs in one and two dimensions we have [49,56] A…”

Classifying parafermionic gapped phases using matrix product states

“…for l < m. The isomorphism F exchanges two nearby local Fock states, and the whole Fock space is a graded vector space, which is a generalization of super vector space of the fermions 23 . Thus parafermion statistics is encoded into the Fock space by the isomorphism, which becomes crucial for the construction of the MPS wave functions.…”

“…Moreover, according to the holographic principle, the left and right dominant eigenvectors of the transfer operator determine the reduced density matrix in the thermodynamic limit 23,33 . We can study the entanglement spectrum via a bipartition of the parafermionic MPS.…”

“…However, it is not straight forward to extend the MPS representation to the one-dimensional parafermion systems. Recently, the fermionic MPS have been successfully constructed by using the language of super vector space, and all possible topological phases with additional symmetries in terms of Majorana fermions have been classified within the matrix product representation 23,24 . By generalizing the concepts of fermion parity and associated Fock space, the present authors have proposed a general framework to construct the MPS of Z 3 parafermions in the Fock representation, and the corresponding parent Hamiltonians have been also derived 25 .…”