Algebraic Fusion in a (2+1)-dimensional Lattice Model with Generalized Symmetries

TL;DR

This paper develops an algebraic framework to derive fusion rules of topological defects in higher-dimensional lattice models with generalized symmetries, exemplified by a (2+1)-D quantum Ising plaquette model, revealing non-invertible duality operators and their constrained fusion algebra.

Contribution

It introduces a systematic algebraic approach to understand fusion rules of non-invertible symmetries in higher-dimensional lattice systems, with explicit constructions and verifications.

Findings

Fusion algebra of non-invertible defects established.

Explicit representation of duality operators as quantum circuits.

Verification of partial isometry property for non-invertible transformations.

Abstract

The notion of quantum symmetry has recently been extended to include reduced-dimensional transformations and algebraic structures beyond groups. Such generalized symmetries lead to exotic phases of matter and excitations that defy Landau's original paradigm. Here, we develop an algebraic framework for systematically deriving the fusion rules of topological defects in higher-dimensional lattice systems with non-invertible generalized symmetries, and focus on a (2+1)-dimensional quantum Ising plaquette model as a concrete illustration. We show that bond-algebraic automorphisms, when combined with the so-called half-gauging procedure, reveal the structure of the non-invertible duality symmetry operators, which can be explicitly represented as a sequential quantum circuit. The resulting duality defects are constrained by the model's rigid higher symmetries (lower-dimensional subsystem symmetries), leading to restricted mobility. We establish the fusion algebra of these defects. Finally, in constructing the non-invertible duality transformation, we explicitly verify that it acts as a partial isometry on the physical Hilbert space, thereby satisfying a recent generalization of Wigner's theorem applicable to non-invertible symmetries.