Fermionic matrix product states and one-dimensional topological phases

2016/10/31 by Nick Bultinck, Dominic J. Williamson, Jutho Haegeman +1
Mathematics · Physics and Astronomy · #Advanced Condensed Matter Physics #Algebra over a field #Clifford algebra #Fermion #Geometry #Homogeneous space #Invariant (physics) #MAJORANA #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum entanglement #Quantum many-body systems #Quantum mechanics #Theoretical physics #Topological Materials and Phenomena #cond-mat.str-el #quant-ph

paper · pdf · doi:10.1103/physrevb.95.075108

published as Phys. Rev. B 95, 075108 (2017)

openalex publication_date 2017/02/03 · arxiv created 2017/03/01 · arxiv updated 2017/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

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 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.

Citations

• Detection of symmetry-protected topological phases in one dimension
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• Fermionic Matrix Product Operators and Topological Phases of Matter