Non-invertible defects in generalized Ising models via strange correlator

TL;DR

This work introduces a versatile strange-correlator framework to realize Kramers–Wannier duality defects for generalized Ising models by gauging subregions and stitching topologically ordered cluster states. It provides explicit KW defect constructions in the 2D Ising model, extends the approach to higher-dimensional chain-complex Ising models, and analyzes defect fusion and Hamiltonian realizations, including an interface between 3D Ising gauge theory and 3D Ising matter. A key result is the fusion rule $\mathcal{D}\times\mathcal{D}=\mathrm{I}+D_{\eta}$ with a contractible defect loop contributing a quantum dimension $\sqrt{2}$, signaling a non-invertible symmetry structure. The framework generalizes to arbitrary dimensions and chain complexes, offering a flexible path to studying KW defects across dimensions and potentially to higher-categorical symmetries and lattice realizations of non-invertible dualities. Potential extensions include $\mathbb{Z}_N$ generalizations, broader fusion categories, and applications to extracting universal data from defect partition functions in higher dimensions.

Abstract

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.

Figures (4)

• Figure 1: Left: $\mathbb{Z}_2$ defect (blue) being moved by transforming $s_{\rm v} \to -s_{\rm v}$. Right: Moving the $\mathbb{Z}_2$ defect across $v$ in presence of $s_{\rm v}$ inside the correlation function gives a $-1$ factor.
• Figure 2: Kramers-Wannier Duality on a periodic square lattice relating Ising Model with coupling $J$ on the left to the Ising Model with coupling $J^*$ on the dual lattice (red) with its $\mathbb{Z}_2$ symmetry (denoted as blue) being gauged, on the right.
• Figure 3: The figure illustrate the cluster state \ref{['eq:2Dclusterstateinterface']}. The black edges denote the collection $\Delta_{\rm e}^{\rm A^c}$ and the red edges denote the collection $\Delta_{\rm e}^{\rm A}$. The blue edges denote the cluster entangler between vertices and edges or plaquette centers and edges. The green edges denote the cluster entangler between vertices and the plaquette centers. Samples of entanglement patterns on the black edges and red edges away from the interface and near the interface region are shown in the figure. The whole square lattice is on a torus as indicated by the arrows on the boundary edges.
• Figure 4: A relation on a periodic square lattice (torus) between the Ising model with coupling $J$ on the left and Ising model with both couplings $J$ (for interaction on black edges) and $J^*$ (for interaction on red edges) separated by an interface (green edges) on the right. The $\mathbb{Z}_2$ symmetry is being gauged in the region with coupling $J^*$ (between the two green interface zig-zag lines).