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arXiv:2607.13138

Half dualization and non-invertible particle-vortex duality defect on lattice

Gao Z

Published Jul 14, 2026• 0 citations
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Abstract

We introduce half dualization as a general principle to construct non-invertible duality defects. The construction performs a duality transformation only in a subregion of spacetime, leaving an interface between the original theory and its dual. A non-invertible duality defect can be hosted on this interface. Half dualization is applicable whenever a local duality transformation is available, and does not depend on the spacetime dimension. We demonstrate the construction procedure in the (2+1)-dimensional Villainized charge-$n$ XY model on a cubic lattice. Half dualization yields a non-invertible particle-vortex duality defect whose fusion with its orientation reverse produces a (1+1)-dimensional $\mathbb{Z}_n$ gauge theory on the fusion surface. We then apply half dualization to the $\mathbb{Z}_n$-gauged XY model relevant to the 3D XY$^\ast$ transition. In the gauged model, the half dualization interface hosts a gauge covariant duality wall, rather than a genuine gauge invariant duality defect. It flows to an invertible duality defect in infrared when the $\mathbb{Z}_n$ gauge theory is confined or Higgsed.

DOI
10.48550/arxiv.2607.13138

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A bstract In recent years, the concept of global symmetry has generalized considerably. Two dramatic examples of this generalization are the exotic symmetries that govern theories with fractons and non-invertible symmetries, which do not fuse according to a group law. Only recently has the interplay between these two been examined. In this paper, we provide further examples of the interplay in the XY plaquette model, XY cube model, 1+1 d theory with global dipole symmetry, and the 2+1 d Lifshitz theory. They are analogs of the duality symmetries in 2d CTFs and are constructed by first gauging a finite subgroup of the momentum symmetry on half of spacetime and then performing a duality transformation. We analyze the fusion rules of the symmetries and find that they are condensation defects from an analog of higher gauging exotic symmetries. We also address their dependence on the UV cutoff when relevant.
We explore non-invertible symmetries in two-dimensional lattice models with subsystem \mathbb Z_2 ℤ 2 symmetry. We introduce a subsystem \mathbb Z_2 ℤ 2 -gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.
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.
A novel kind of generalized global symmetries is uncovered in a large class of familiar 3+1-dimensional gauge theories, including the free Maxwell theory and Yang-Mills gauge theories. These new symmetries, known as the non-invertible global symmetries, do not have inverses and thus go beyond the traditional paradigm of (anti-)unitary transformations. They are implemented by topological duality defects, generalizing the Kramers-Wannier duality defects in 1+1 dimensions. Remarkably, the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase and hence gives new constraints on renormalization group flows.
Defects associated with non-invertible symmetries have attracted significant attention in recent years. Among them, Kramers-Wannier (KW) duality defects have been investigated in both classical statistical systems and quantum Hamiltonian models. Aasen et al. analyzed duality defects in the 2D Ising model and in statistical models built from fusion categories, while Koide et al. later constructed a duality defect in 4D lattice gauge theory. In this work, we extend these developments by providing a systematic construction of KW duality defects/KW defects for a broad class of models formulated within the chain complex framework. Our construction employs the strange correlator, an overlap between a topologically ordered state and a product state, to realize these KW defects.
Global symmetries can be generalised to transformations generated by topological operators, including cases in which the topological operator does not have an inverse. A family of such topological operators are intimately related to dualities via the procedure of half-space gauging. In this work we discuss the construction of non-invertible defects based on T-duality in two dimensions, generalising the well-known case of the free compact boson to any Non-Linear Sigma Model with Wess-Zumino term which is T-dualisable. This requires that the target space has an isometry with compact orbits that acts without fixed points. Our approach allows us to include target spaces without non-trivial 1-cycles, does not require the NLSM to be conformal, and when it is conformal it does not need to be rational; moreover, it highlights the microscopic origin of the topological terms that are responsible for the non-invertibility of the defect. An interesting class of examples are Wess-Zumino-Witten models, which are self-dual under a discrete gauging of a subgroup of the isometry symmetry and so host a topological defect line with Tambara-Yamagami fusion. Along the way, we discuss how the usual 0-form symmetries match across T-dual models in target spaces without 1-cycles, and how global obstructions can prevent locally conserved currents from giving rise to topological operators.