A minimal tensor network beyond free fermions

TL;DR

This paper introduces a minimal tensor network model that extends classical-fermionic duality beyond free fermions, revealing complex phases and topological features with potential applications in tensor network analysis.

Contribution

It presents a simple, two-parameter tensor network model that captures interacting fermionic behavior and complex phase structures beyond free fermion theories.

Findings

Exhibits a rich phase diagram with three phases and a tricritical point.

Connects classical spin models, fermionic systems, and loop gas models.

Serves as a minimal reference for studying non-linearities in tensor networks.

Abstract

This work proposes a minimal model extending the duality between classical statistical spin systems and fermionic systems beyond the case of free fermions. A Jordan-Wigner transformation applied to a two-dimensional tensor network maps the partition sum of a classical statistical mechanics model to a Grassmann variable integral, structurally similar to the path integral for interacting fermions in two dimensions. The resulting model is simple, featuring only two parameters: one governing spin-spin interaction (dual to effective hopping strength in the fermionic picture), the other measuring the deviation from the free fermion limit. Nevertheless, it exhibits a rich phase diagram, partially stabilized by elements of topology, and featuring three phases meeting at a tricritical point. Besides the interpretation as a spin and fermionic system, the model is closely related to loop gas and vertex models and can be interpreted as a parity-preserving (non-unitary) circuit. Its minimal construction makes it an ideal reference system for studying non-linearities in tensor networks and deriving results by means of duality.