Fermionic topological quantum states as tensor networks

TL;DR

This paper develops a tensor network framework for fermionic topological quantum states, enabling the classification and analysis of topological order in fermionic systems, extending tensor network methods beyond spin and bosonic models.

Contribution

It introduces a fermionic tensor network formalism with axioms of fermionic matrix product operator injectivity, and applies it to fermionic topological models like the fermionic toric code.

Findings

Framework captures topological order in fermionic models

Formalism successfully applied to fermionic twisted quantum double models

Extends tensor network methods to fermionic topological phases

Abstract

Tensor network states, and in particular projected entangled pair states, play an important role in the description of strongly correlated quantum lattice systems. They do not only serve as variational states in numerical simulation methods, but also provide a framework for classifying phases of quantum matter and capture notions of topological order in a stringent and rigorous language. The rapid development in this field for spin models and bosonic systems has not yet been mirrored by an analogous development for fermionic models. In this work, we introduce a framework of tensor networks having a fermionic component capable of capturing notions of topological order. At the heart of the formalism are axioms of fermionic matrix product operator injectivity, stable under concatenation. Building upon that, we formulate a Grassmann number tensor network ansatz for the ground state of fermionic twisted quantum double models. A specific focus is put on the paradigmatic example of the fermionic toric code. This work shows that the program of describing topologically ordered systems using tensor networks carries over to fermionic models.