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

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.

Figures (19)

• Figure 1: (a) Xu-Moore model with sites on vertices of a periodic 5 $\times$ 4 lattice. Red diamonds indicate the new sites extending the bond algebra so as to permit the construction of an automorphism. $S$ and $\bar{S}$ are the half-gauging procedure. (b) The original, ungauged lattice sites (black) and the gauged sites on the dual lattice (red diamonds) are separated by the interface forming the duality defect. The Gauss law operator $G_q$ of the gauged Xu-Moore is shaded in purple.
• Figure 2: (a) A pair of fractons fuse into a horizontal $\eta$-defect: $f \times f = \eta^{\text{row}}$. The two fracton membranes combine to form a rigid string. (b) A vertical $\eta$-defect fuses with the duality defect. The $\hat{\eta}$ operator and $\tilde{Z}$ operator pair combine to form the fusion operator in Eq. (\ref{['eqn:EtaDualityFusionOps']}).
• Figure 3: Line Defect Movement and Fusion Steps are schematically indicated. The operator $\mathcal{U}$ on the arrow indicates that the Hamiltonian on the left of the arrow is to be conjugated as $\mathcal{U}()\mathcal{U}^{-1}$ to obtain that on the right.
• Figure S1: Xu-Moore model with periodic boundary conditions enforced by identifying opposite edges
• Figure S2: Xu-Moore model with periodic boundary conditions and extra sites coupled to plaquette terms.