Simplex tensor network renormalization group for boundary theory of 3+1D symTFT

TL;DR

This paper introduces a symmetry-preserving tensor network renormalization method for boundary theories of 3+1D topological field theories, enabling the study of phase transitions and symmetry breaking in 3D symmetric systems.

Contribution

It develops a novel simplex tensor network approach and a numerical RG algorithm for boundary conditions of 3+1D symTFTs, demonstrated on a $\

Findings

Mapped phase transitions using tensor network RG

Detected 0-form and 2-form symmetry breaking

Extended formalism to other discrete groups

Abstract

Following the construction in arXiv:2210.12127, we develop a symmetry-preserving renormalization group (RG) flow for 3D symmetric theories. These theories are expressed as boundary conditions of a symTFT, which in our case is a 3+1D Dijkgraaf-Witten topological theory in the bulk. The boundary is geometrically organized into tetrahedra and represented as a tensor network, which we refer to as the "simplex tensor network" state. Each simplex tensor is assigned indices corresponding to its vertices, edges, and faces. We propose a numerical algorithm to implement RG flows for these boundary conditions, and explicitly demonstrate its application to a $\mathbb{Z}_2$ symmetric theory. By linearly interpolating between three topological fixed-point boundaries, we map the phase transitions characterized by local and non-local order parameters, which respectively detects the breaking of a 0-form and a 2-form symmetry. This formalism is readily extendable to other discrete symmetry groups and, in principle, can be generalized to describe 3D symmetric topological orders.