Unitary Networks from the Exact Renormalization of Wave Functionals

TL;DR

The work extends the exact renormalization group (ERG) to the wave functionals of states in the free $O(N)$ vector model, revealing that RG flow can be implemented by unitary, (nearly) local operators that act along the holographic coordinate $z$, thereby producing a continuum tensor-network description. The ground state flow is governed by Hermitian generators $oldsymbol{K}(z)$ and $oldsymbol{L}(z)$, while excited states evolve with a bi-local beta function $oldsymbol{eta}$ driving unitary evolution along integral curves $zrac{d}{dz}oldB=oldsymbol{eta}(oldB)$; the resulting network resembles, yet differs from, MERA/cMERA by its momentum-space disentangling and its basis in Wilsonian regulation. The analysis strengthens connections between ERG and holographic duality for higher-spin theories on $AdS_{d+1}$, and frames a concrete continuum tensor-network viewpoint for state renormalization, with potential extensions to multi-trace deformations and real-space entanglement structure. Overall, the paper provides a field-theoretic construction of a tensor-network-like RG for states in a holographically dual free theory, offering new angles on AdS/tensor-network correspondences and quantum information aspects of RG.

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.

Figures (5)

• Figure 1: (a) The ground state wave functional is given by the Euclidean path integral on the lower-half space. (b) Excited states can be constructed by operator insertions in the path integral.
• Figure 2: (a) Operator insertions along the Euclidean branch of the contour prepares a state, while operator insertions in the real time calculate correlation functions in that state. (b) Operator and state norms are given by the path integral over the reflected contour. Evolution by a small imaginary time $\delta$ ensures these norms do not suffer from contact divergences of operators acting at the same spacetime point.
• Figure 3: Along integral curves $\mathcal{B}(z)$ of the beta function, $|\Psi[z;\mathcal{B}(z)]\rangle$ undergoes a unitary flow generated locally by the Hermitian operator $\mathcal{K} = \boldsymbol{K} + \boldsymbol{L}$.
• Figure 4: The darker region indicates entanglement in momentum space. The disentangler acts as a unitary channel that pushes this to lower and lower scales.
• Figure 5: Gluing path integrals with bi-local sources. The shaded regions indicate the support of $B$. After gluing there remains a region of width $2\delta$ along the contour on which $B$ has no support. However the action of a larger background symmetry generates new bi-local sources in this region.