# Holographic spin networks from tensor network states

## TL;DR

This paper demonstrates how to gauge boundary symmetries in tensor network states, specifically MERA, to produce a holographic bulk state with spin network structure, revealing entanglement and emergent phenomena.

## Contribution

It introduces a method to lift MERA states with boundary symmetries to bulk states with gauge invariance, connecting tensor networks to lattice gauge theories in holography.

## Key findings

- Bulk entanglement depends on the conformal field theory's central charge.
- Numerical results on critical spin chains illustrate the bulk structure.
- Discussion of emergent topological order and symmetries in the bulk.

## Abstract

In the holographic correspondence of quantum gravity, a global onsite symmetry at the boundary generally translates to a local gauge symmetry in the bulk. We describe one way how the global boundary onsite symmetries can be gauged within the formalism of the multi-scale renormalization ansatz (MERA), in light of the ongoing discussion between tensor networks and holography. We describe how to "lift" the MERA representation of the ground state of a generic one dimensional (1D) local Hamiltonian, which has a global onsite symmetry, to a dual quantum state of a 2D "bulk" lattice on which the symmetry appears gauged. The 2D bulk state decomposes in terms of spin network states, which label a basis in the gauge-invariant sector of the bulk lattice. This decomposition is instrumental to obtain expectation values of gauge-invariant observables in the bulk, and also reveals that the bulk state is generally entangled between the gauge and the remaining ("gravitational") bulk degrees of freedom that are not fixed by the symmetry. We present numerical results for ground states of several 1D critical spin chains to illustrate that the bulk entanglement potentially depends on the central charge of the underlying conformal field theory. We also discuss the possibility of emergent topological order in the bulk using a simple example, and also of emergent symmetries in the non-gauge ("gravitational") sector in the bulk. More broadly, our holographic model translates the MERA, a tensor network state, to a superposition of spin network states, as they appear in lattice gauge theories in one higher dimension.