The structure of fixed-point tensor network states characterizes patterns of long-range entanglement

TL;DR

This paper explores how the algebraic structure of fixed-point tensor network states encodes long-range entanglement patterns and reveals a holographic relationship with 3D TQFT, linking boundary states to bulk pre-geometry.

Contribution

It establishes a connection between 2D fixed-point tensor networks and 3D TQFT, demonstrating a holographic correspondence and the emergence of pre-geometry from entanglement.

Findings

Tensor network states encode long-range entanglement patterns.

Boundary tensor networks are related to 3D TQFT via holography.

Entanglement leads to emergent pre-geometry in the bulk.

Abstract

The algebraic structure of representation theory naturally arises from 2D fixed-point tensor network states, which conceptually formulates the pattern of long-range entanglement realized in such states. In 3D, the same underlying structure is also shared by Turaev-Viro state-sum topological quantum field theory (TQFT). We show that a 2D fixed-point tensor network state arises naturally on the boundary of the 3D manifold on which the TQFT is defined, and the fact that exactly the same information is needed to construct either the tensor network or the TQFT is made explicit in a form of holography. Furthermore, the entanglement of the fixed-point states leads to an emergence of pre-geometry in the 3D TQFT bulk. We further extend these ideas to the case where an additional global onsite unitary symmetry is imposed on the tensor network states.