Fermionic matrix product states and one-dimensional topological phases

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 fM...

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Bibliographic Details
Published in:Physical review. B 2017-02, Vol.95 (7), Article 075108
Main Authors: Bultinck, Nick, Williamson, Dominic J., Haegeman, Jutho, Verstraete, Frank
Format: Article
Language:English
Subjects:
Entanglement
Fermions
Formalism
Invariants
Partitions (mathematics)
Periodic variations
Phases
Symmetry
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Summary: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 twofold 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 onsite unitary and antiunitary 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.
ISSN:2469-9950
2469-9969
DOI:10.1103/PhysRevB.95.075108