# Strictly local tensor networks for short-range topological insulators

## TL;DR

This paper demonstrates that for free fermions, strictly local tensor networks can accurately describe thermal averages of gapped Hamiltonians with short-range couplings, overcoming previous no-go theorems for topologically nontrivial states.

## Contribution

It introduces a method to use strictly local tensor networks for thermal states of free fermions with short-range Hamiltonians, including a truncation-reconstruction scheme.

## Key findings

- Tensor networks can describe thermal averages of gapped free fermion systems.
- The scheme applies to any dimensionality without topological obstructions.
- Illustrated on the 2D Haldane model for both topological and trivial phases.

## Abstract

Despite the success in describing a range of quantum many-body states using tensor networks, there is a no-go theorem that rules out strictly local tensor networks as topologically nontrivial groundstates of gapped parent Hamiltonians with short-range (including exponentially decaying) couplings. In this work, we show that for free fermions, strictly local tensor networks may describe nonzero temperature averages with respect to gapped Hamiltonians with exponentially decaying couplings. Parent Hamiltonians in this sense may be constructed for any dimensionality and without any obstructions due to their topology. Conversely, we also show that thermal averages with respect to gapped, strictly short-range free-fermion Hamiltonians can be calculated by tensor networks whose links decay exponentially with distance. We also describe a truncation-reconstruction scheme for such tensor networks that leads to a controlled approximation of exact averages in terms of a sequence of related thermal averages. We illustrate our scheme on the two-dimensional Haldane honeycomb model considering both topological and nontopological phases.