Unitary Networks from the Exact Renormalization of Wave Functionals

TL;DR

This paper extends the exact renormalization group formalism to wave functionals of the free $O(N)$ vector model, revealing a unitary, local tensor network structure that offers insights into holography and entanglement in quantum field theories.

Contribution

It introduces a unitary, local tensor network representation of ERG flow for arbitrary states in the free $O(N)$ model, connecting to holographic tensor networks like cMERA.

Findings

ERG flow of states is implemented by local unitary operators.

The ERG tensor network shares structure with cMERA but has key differences.

Provides a continuum tensor network perspective related to holography.

Abstract

The exact renormalization group (ERG) for $O(N)$ vector models (at large $N$) on flat Euclidean space can be interpreted as the bulk dynamics corresponding to a holographically dual higher spin gauge theory on $AdS_{d+1}$. This was established in the sense that at large $N$ the generating functional of correlation functions of single trace operators is reproduced by the on-shell action of the bulk higher spin theory, which is most simply presented in a first-order (phase space) formalism. In this paper, we extend the ERG formalism to the wave functionals of arbitrary states of the $O(N)$ vector model at the free fixed point. We find that the ERG flow of the ground state and a specific class of excited states is implemented by the action of unitary operators which can be chosen to be local. Consequently, the ERG equations provide a continuum notion of a tensor network. We compare this tensor network with the entanglement renormalization networks, MERA, and its continuum version, cMERA, which have appeared recently in holographic contexts. In particular the ERG tensor network appears to share the general structure of cMERA but differs in important ways. We comment on possible holographic implications.