Gauging non-invertible symmetries on the lattice

TL;DR

This work develops a concrete lattice framework for gauging finite non-invertible symmetries in 1+1d quantum systems, centering on the Rep(D8) fusion category and its non-invertible Kennedy–Tasaki generator. It provides explicit lattice data—defining symmetry operators, defects, movement and fusion operators, and F-symbols—and introduces a non-maximal gauging using the algebra object A=1⊕η⊕D, realized by dynamical gauge qubits per link with Gauss’s laws. A key highlight is the gauging map implemented by cosine non-invertible operators, enabling a precise lattice realization of gauged Rep(D8) theories and revealing enhanced fusion-category symmetries such as Rep(D16) in certain limits. The results establish a general, diagrammatic procedure to gauge finite non-invertible symmetries on lattices, with potential wide applicability to other fusion categories and non-invertible symmetries in quantum many-body systems.

Abstract

We provide a general prescription for gauging finite non-invertible symmetries in 1+1d lattice Hamiltonian systems. Our primary example is the Rep(D$_8$) fusion category generated by the Kennedy-Tasaki transformation, which is the simplest anomaly-free non-invertible symmetry on a spin chain of qubits. We explicitly compute its lattice F-symbols and illustrate our prescription for a particular (non-maximal) gauging of this symmetry. In our gauging procedure, we introduce two qubits around each link, playing the role of "gauge fields" for the non-invertible symmetry, and impose novel Gauss's laws. Similar to the Kramers-Wannier transformation for gauging an ordinary $\mathbb{Z}_2$, our gauging can be summarized by a gauging map, which is part of a larger, continuous non-invertible cosine symmetry.

Figures (1)

• Figure 1: The phase diagram of the Hamiltonian \ref{['Ha']}. For $\lambda= -h_0/h_1\in (-1,1]$, it flows to the $c=1$ orbifold CFT with radius $R$ given in \ref{['radius']}. For $\lambda>1$ it is gapped with one ground state.