Tensor networks for non-invertible symmetries in 3+1d and beyond
Pranay Gorantla, Shu-Heng Shao, Nathanan Tantivasadakarn
TL;DR
The paper develops a ZX-calculus–based tensor-network framework for non-invertible symmetries arising from KW and Wegner dualities in low- and higher-dimensional lattice models. It constructs explicit non-invertible duality operators as ZX diagrams (and PEPOs in 3+1d), derives their fusion algebras with condensation operators, and analyzes how these dualities interact with lattice/spatial symmetries. A central result is the diffusion of D^2 into a translation-tacted condensation, $\mathsf D^2=\mathsf C\,T_{1,1,1}$, and the emergence of a nine-fold torus ground-state degeneracy at the self-dual point in the 3+1d theory, controlled by a Lieb–Schultz–Mattis–type constraint. The framework is extended to generalized TFIMs on graphs, yielding a family of non-invertible dualities $\mathsf D_{\rho}$ associated with reversing automorphisms $\rho$, with operator algebras $\mathsf D_\rho^2=T_{\rho^2}\mathsf C$ and $\mathsf D_\rho\mathsf D_{\rho'}=T_{\rho\circ\rho'}\mathsf C$, and rich examples (Ising, Ashkin–Teller, three-spin Ising, plaquette Ising) that reveal when dualities mix with spatial symmetries. These results offer a unifying diagrammatic view of non-invertible symmetries across dimensions and graph-based architectures, with implications for phase structure and SPT-like sectors protected by such symmetries.
Abstract
Tensor networks provide a natural language for non-invertible symmetries in general Hamiltonian lattice models. We use ZX-diagrams, which are tensor network presentations of quantum circuits, to define a non-invertible operator implementing the Wegner duality in 3+1d lattice $\mathbb{Z}_2$ gauge theory. The non-invertible algebra, which mixes with lattice translations, can be efficiently computed using ZX-calculus. We further deform the $\mathbb{Z}_2$ gauge theory while preserving the duality and find a model with nine exactly degenerate ground states on a torus, consistent with the Lieb-Schultz-Mattis-type constraint imposed by the symmetry. Finally, we provide a ZX-diagram presentation of the non-invertible duality operators (including non-invertible parity/reflection symmetries) of generalized Ising models based on graphs, encompassing the 1+1d Ising model, the three-spin Ising model, the Ashkin-Teller model, and the 2+1d plaquette Ising model. The mixing (or lack thereof) with spatial symmetries is understood from a unifying perspective based on graph theory.