(2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry

TL;DR

This paper develops a systematic lattice model and tensor network framework for classifying and constructing all gapped phases with fusion 2-categorical symmetries in 2+1 dimensions, based on continuum classifications.

Contribution

It provides a comprehensive lattice construction and tensor network description for gapped phases with fusion 2-category symmetries derived from group symmetries and anomalies.

Findings

Constructed commuting projector Hamiltonians for these phases

Connected continuum classification to lattice data

Presented explicit tensor network ground states

Abstract

Gapped phases in 2+1 dimensional quantum field theories with fusion 2-categorical symmetries were recently classified and characterized using the Symmetry Topological Field Theory (SymTFT) approach arXiv:2408.05266, arXiv:2502.20440. In this paper, we provide a systematic lattice model construction for all such gapped phases. Specifically, we consider "all-boson type" fusion 2-category symmetries, all of which are obtainable from 0-form symmetry groups $G$ (possibly with an 't Hooft anomaly) via generalized gauging--that is, by stacking with an $H$-symmetric TFT and gauging a subgroup $H$. The continuum classification directly informs the lattice data, such as the generalized gauging that determines the symmetry category, and the data that specifies the gapped phase. We construct commuting projector Hamiltonians and ground states applicable to any non-chiral gapped phase with such symmetries. We also describe the ground states in terms of tensor networks. In light of the length of the paper, we include a self-contained summary section presenting the main results and examples.