Gauging non-invertible symmetries on the lattice

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.

66 pages, 1 figure, 1 table; v2: minor changes

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• Generalized Landau Paradigm for quantum phases and phase transitions
• (2+1)d Lattice Models and Tensor Networks for Gapped Phases with Categorical Symmetry
• Gauging Modulated Symmetries via Multiple Gauge Symmetry Operators and Adaptive Quantum Circuits
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• Les Houches Lecture Notes on Tensor Networks
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• Higher-order topological phases protected by noninvertible and subsystem symmetries