Simulating thermal density operators with cluster expansions and tensor networks

TL;DR

This paper introduces an efficient tensor-network-based cluster expansion method to approximate thermal density operators in quantum spin systems, enabling accurate analysis of phase transitions at various temperatures.

Contribution

The authors develop a novel cluster tensor network operator approach for representing thermal states, improving approximation accuracy for large time steps and low temperatures.

Findings

Accurate approximation of thermal states in 2D quantum spin systems.

Successful identification of the phase transition in the transverse-field Ising model.

Good critical point estimates at low temperatures with increased cluster order.

Abstract

We provide an efficient approximation for the exponential of a local operator in quantum spin systems using tensor-network representations of a cluster expansion. We benchmark this cluster tensor network operator (cluster TNO) for one-dimensional systems, and show that the approximation works well for large real- or imaginary-time steps. We use this formalism for representing the thermal density operator of a two-dimensional quantum spin system at a certain temperature as a single cluster TNO, which we can then contract by standard contraction methods for two-dimensional tensor networks. We apply this approach to the thermal phase transition of the transverse-field Ising model on the square lattice, and we find through a scaling analysis that the cluster-TNO approximation gives rise to a continuous phase transition in the correct universality class; by increasing the order of the cluster expansion we find good values of the critical point up to surprisingly low temperatures.