Fermionic Matrix Product States and One-Dimensional Topological Phases

TL;DR

The paper develops a comprehensive fermionic MPS (fMPS) formalism using super vector spaces to describe interacting 1D fermionic systems and their symmetry-protected topological (SPT) phases. It identifies two irreducible fMPS classes (even and odd) distinguished by the presence of Majorana edge modes, and shows how their entanglement spectra reflect this structure. It then derives time-reversal (Z8) and general on-site symmetry classifications via virtual symmetry actions and cohomology, including anti-unitary cases, and demonstrates how stacking rules yield group structures consistent with Clifford-algebra representations. The work extends to reflection symmetry (via partial reflection) and discusses implications for higher-dimensional fermionic tensor networks and the broader landscape of interacting 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 $\mathbb{Z}_2$ 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 $\mathbb{Z}_8$ 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.

Figures (1)

• Figure 1: Stability of the injectivity property under contraction of tensors. The figure represents $\mathcal{C}(\mathsf{A}_2^{(-1)}\otimes_{\mathfrak{g}} \mathsf{A}_2) =D \mathds{1}\otimes_{\mathfrak{g}} \mathds{1}$, with $D$ the bond dimension. The black dot represents the matrix $\sum_\alpha (-1)^{\alpha}|\alpha)(\alpha|$. The arrows in the diagrammatic notation denote which indices correspond to bra's and which indices correspond to kets in the corresponding super vector spaces.