Entanglement Renormalization for Weakly Interacting Fields

TL;DR

This paper develops a method to construct local continuous MERA circuits for weakly interacting quantum field theories, linking entanglement renormalization with Wilsonian RG, and providing tools for non-perturbative analysis.

Contribution

It introduces quantum circuit perturbation theory to systematically build unitaries for weakly interacting fields and establishes a correspondence between cMERA and Wilsonian RG in $\

Findings

Constructed a local cMERA circuit for $\

Demonstrated equivalence between perturbative Wilsonian RG and cMERA in $\

Suggested numerical ansatzes for non-perturbative cMERA regimes

Abstract

We adapt the techniques of entanglement renormalization tensor networks to weakly interacting quantum field theories in the continuum. A key tool is "quantum circuit perturbation theory," which enables us to systematically construct unitaries that map between wavefunctionals which are Gaussian with arbitrary perturbative corrections. As an application, we construct a local, continuous MERA (cMERA) circuit that maps an unentangled scale-invariant state to the ground state of $\varphi^4$ theory to 1-loop. Our local cMERA circuit corresponds exactly to 1-loop Wilsonian RG on the spatial momentum modes. In other words, we establish that perturbative Wilsonian RG on spatial momentum modes can be equivalently recast as a local cMERA circuit in $\varphi^4$ theory, and argue that this correspondence holds more generally. Our analysis also suggests useful numerical ansatzes for cMERA in the non-perturbative regime.