Tensor Renormalization Group for interacting quantum fields

TL;DR

This paper introduces a tensor network algorithm based on the Tensor Renormalization Group for calculating the partition function of interacting quantum field theories in two dimensions, incorporating self-interactions via perturbation theory.

Contribution

It extends the TRG method to interacting quantum fields at the field level, including a perturbative treatment of self-interactions, and demonstrates its effectiveness with a $bbb4$ theory.

Findings

Fast convergence with bond dimension.

Accurately captures interaction effects on entanglement.

Effective for perturbative corrections in quantum field theories.

Abstract

We present a new tensor network algorithm for calculating the partition function of interacting quantum field theories in 2 dimensions. It is based on the Tensor Renormalization Group (TRG) protocol, adapted to operate entirely at the level of fields. This strategy was applied in Ref.[1] to the much simpler case of a free boson, obtaining an excellent performance. Here we include an arbitrary self-interaction and treat it in the context of perturbation theory. A real space analogue of the Wilsonian effective action and its expansion in Feynman graphs is proposed. Using a $\lambda \phi^4$ theory for benchmark, we evaluate the order $\lambda$ correction to the free energy. The results show a fast convergence with the bond dimension, implying that our algorithm captures well the effect of interaction on entanglement.