Global symmetries of quantum lattice models under non-invertible dualities

TL;DR

This paper investigates how global symmetries transform under non-invertible dualities realized by gauging discrete groups in $(1+1)$-D quantum lattice models. It develops a sector-aware framework based on the sandwiched construction and a ring of double cosets $\mathbb{Z}[G \backslash \mathsf{S} / G]$ to capture dual symmetry content across twist and symmetry sectors, and tests the ideas with explicit XXZ-based dualities to Ising zig-zag, 3-state AFM, and integrable Rydberg ladder models. The results reveal rich, sector-dependent symmetry structures (including cosine-type and hidden $U(1)$ algebras) and demonstrate that the full dual Hilbert space can exhibit $\mathrm{Rep}(G)$ symmetries in addition to sector-specific rings, generalizing known gauging outcomes. The work connects lattice realizations with categorical/field-theory expectations, showing how non-invertible dualities can generate nontrivial, computable symmetry algebras and pointing to future work on weakly symmetric dualities and broader rigorous formulations. The findings have implications for organizing and predicting symmetry content in dual quantum many-body systems and for understanding how non-invertible symmetries emerge from lattice gauging procedures.

Abstract

Non-invertible dualities/symmetries have become an important tool in the study of quantum field theories and quantum lattice models in recent years. One of the most studied examples is non-invertible dualities obtained by gauging a discrete group. When the physical system has more global symmetries than the gauged symmetry, it has not been thoroughly investigated how those global symmetries transform under non-invertible duality. In this paper, we study the change of global symmetries under non-invertible duality of gauging a discrete group $G$ in the context of (1+1)-dimensional quantum lattice models. We obtain the global symmetries of the dual model by focusing on different Hilbert space sectors determined by the $\mathrm{Rep}(G)$ symmetry. We provide general conjectures of global symmetries of the dual model forming an algebraic ring of the double cosets. We present concrete examples of the XXZ models and the duals, providing strong evidence for the conjectures.

Figures (4)

• Figure 1: The mapping between the Hilbert spaces of twisted XXZ models and the Izz model. In the Hilbert space of twisted XXZ models, the $\mathbb Z_2$ odd sectors (orange) are kernels of the duality operators, while the $\mathbb Z_2$ even sectors (blue) are mapped to the Hilbert space of untwisted Izz model (red). We use the same notation in figures of other examples later in Sec. \ref{['sec:examples']}.
• Figure 2: Four models with the dualities between them, where the symmetries gauged are shown inside the brackets. The arrow from the Izz model to the IRL model represents a gauging of the Frobenius subalgebra $\mathcal{A}$ of the fusion-category $\mathrm{Rep}(S_3)$ symmetry, while other lines represent gaugings of finite discrete groups. The diagonal arrow from the XXZ model to the IRL model requires the gauging of the non-Abelian $S_3$ symmetry.
• Figure 3: The mapping between the Hilbert spaces of twisted XXZ models and the 3-state AFM model.
• Figure 4: The mapping between the Hilbert spaces of twisted XXZ models and the IRL model.