Defect Fusion in Higher-Dimensional Systems with Non-Invertible Symmetries

The conventional Landau paradigm of classifying phases of matter by the associated (anti)unitary symmetries has recently been augmented with the notion of non-unitary, non-invertible symmetries. These are found at the self-dual points of certain lattice models, in particular the 1+1D transverse field Ising model, wherein the topological defects associated with the duality form a fusion algebra (rather than a group). By contrast, much less is known of such non-invertible symmetries in 2+1D. In this work, we develop this formalism for the non-invertible self-duality in the Ising plaquette model in a transverse field [1]. We show that the framework of bond-algebraic automorphisms [2], combined with the so-called partial gauging procedure [3], provides insight into the structure of the duality operator, which we express as a sequential quantum circuit. The associated duality defects are constrained by the rigid subsystem symmetries of the model, resulting in restricted mobility. We study the fusion rules of these duality defects with the defects of the subsystem symmetries. In constructing the non-invertible duality transformation, we verify explicitly that it acts as a “projective” unitary on the physical Hilbert space, thus satisfying the recently formulated generalized Wigner theorem [4].

[1] C. Xu and J. E. Moore, Strong-weak coupling self-duality in the two-dimensional quantum phase transition of p+ ip superconducting arrays, Phys. Rev. Lett. 93, 047003 (2004).

[2] E. Cobanera, G. Ortiz, and Z. Nussinov, The bond-algebraic approach to dualities, Advances in Physics 60, 679 (2011).

[3] N. Seiberg, S. Seifnashri, and S.-H. Shao, Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space, SciPost Phys. 16, 154 (2024).

[4] Gerardo Ortiz et al. Generalized Wigner theorem for non-invertible symmetries arXiv: 2509.25327 (2025).