Simulating thermal density operators with cluster expansions and tensor networks

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 bench-mark 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 ap-ply 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 approx-imation 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.

Automated summary

We propose an efficient approximation method using tensor-network representations to calculate the exponential of a local operator in quantum spin systems. Through benchmarking, we demonstrate its effectiveness for large time steps in one-dimensional systems. We apply this method to represent the thermal density operator of a two-dimensional spin system and successfully obtain a continuous phase transition in the correct universality class.