Lowering the temperature of two-dimensional fermionic tensor networks with cluster expansions

TL;DR

This paper extends the cluster expansion method to two-dimensional fermionic tensor networks, enabling efficient approximations of Gibbs states and revealing phase boundaries in fermionic models at finite temperature.

Contribution

It introduces a novel extension of the cluster expansion to 2D fermionic systems and applies it to construct PEPO approximations of Gibbs states.

Findings

Successfully constructed PEPO approximations for 2D fermionic Gibbs states

Benchmarking of truncation schemes for PEPO layer multiplication

Resolved a clear phase boundary at finite temperature in a fermionic model

Abstract

Representing the time-evolution operator as a tensor network constitutes a key ingredient in several algorithms for studying quantum lattice systems at finite temperature or in a non-equilibrium setting. For a Hamiltonian composed of strictly short-ranged interactions, the Suzuki-Trotter decomposition is the main technique for obtaining such a representation. In [B.~Vanhecke, L.~Vanderstraeten and F.~Verstraete, Physical Review A, L020402 (2021)], an alternative strategy, the cluster expansion, was introduced. This approach naturally preserves internal and lattice symmetries and can more easily be extended to higher-order representations or longer-ranged interactions. We extend the cluster expansion to two-dimensional fermionic systems, and employ it to construct projected entangled-pair operator (PEPO) approximations of Gibbs states. We also discuss and benchmark different truncation schemes for multiplying layers of PEPOs together. Applying the resulting framework to a two-dimensional spinless fermion model with attractive interactions, we resolve a clear phase boundary at finite temperature.