Classification of Thouless pumps with non-invertible symmetries and implications for Floquet phases

TL;DR

The paper develops a comprehensive framework for Thouless pumps in 1D gapped phases with non-invertible fusion-category symmetries $\mathcal{D}$, establishing a one-to-one correspondence between homotopy classes of adiabatic cycles and invertible defects generated by spatially truncated pump operators, and proving that pumps are classified by the group of $\mathcal{D}$-autoequivalences. Leveraging module-category theory, it constructs explicit lattice models for Vec$_G$, Rep$(G)$, and Rep$(H)$ that realize the predicted autoequivalence groups and pump actions, and demonstrates how these pumps extend to Floquet drives. The Floquet analysis, aided by the Onsager algebra, yields analytic phase diagrams with winding-number invariants that distinguish dynamical phases and edge behaviors in both group-like and non-invertible settings. Overall, the work reveals a deep connection between distinct Thouless pumps and distinct families of Floquet phases, and outlines paths to generalize the framework to higher dimensions and broader symmetry classes.

Abstract

We study symmetry preserving adiabatic and Floquet dynamics of one-dimensional systems. Using quasiadiabatic evolution, we establish a correspondence between adiabatic cycles and invertible defects generated by spatially truncated Thouless pump operators. Employing the classification of gapped phases by module categories, we show that the Thouless pumps are classified by the group of autoequivalences of the module category. We then explicitly construct Thouless pump operators for minimal lattice models with $\text{Vec}_G$, Rep($G$), and Rep($H$) symmetries, and show how the Thouless pump operators have the group structure of autoequivalences. The Thouless pump operators, together with Hamiltonians with gapped ground states, are then used to construct Floquet drives. An analytic solution for the Floquet phase diagram characterized by winding numbers is constructed when the Floquet drives obey an Onsager algebra. Our approach points the way to a general connection between distinct Thouless pumps and distinct families of Floquet phases.

Figures (4)

• Figure 1: The labels $a_i,b_{i+1/2}\in\mathcal{D}$, $m_i\in\mathcal{M}$. $\alpha_{i+1/2}$ and $\mu_{i+1/2}$ denote the basis vectors in the fusion spaces $V^{b_{i+1/2} a_{i+1}}_{a_{i}}$ and $\text{Hom}_{\mathcal{M}}(b_{i+1/2}\otimes m_{i+1},m_{i})$, $\alpha_{i+1/2}=1,\cdots,N^{b_{i+1/2} a_{i+1}}_{a_{i}}$. When $\mathcal{M}$ is a fiber functor over $\mathcal{D}$, $\mu_{i+1/2}=1,\cdots,dim(b_{i+1/2})$. The symmetry $\mathcal{D}$ of the model is defined via fusing a line $A\in \mathcal{D}$ from the bottom. The Thouless pump operators are defined via fusing a line $\gamma\in\Gamma$ from the top, which commute with both the Hamiltonian and the symmetry.
• Figure 2: The Vec$_G$ anyonic chain model can be interpreted as a quasi-1d system defined on the sandwich. A topological order $D(G)$ lives in the bulk, and a representation $\rho$ corresponds to an anyon $e_{\rho}$. Truncating the $U_{\rm TP}$ operator creates the anyon $e_{\rho}$ at the two endpoints, which cannot be condensed on the dynamical boundary for an SPT phase. Thus, the adiabatic cycle pumps a charge $e_{\rho}$ along the quasi-1d system.
• Figure 3: The $\text{Rep}(G)$ anyonic chain model can be interpreted as a quasi-1d system defined on the sandwich. A topological order $D(G)$ lives in the bulk, and a representation $\rho$ corresponds to an anyon $e_{\rho}$. Truncating the $U_{\rm TP}$ operator creates fluxes at the two endpoints, which cannot be condensed on the dynamical boundary for an SPT phase. Thus, the adiabatic cycle pumps a flux $g\in G$ along the quasi-1d system.
• Figure 4: The winding number of the single particle band in the parameter space of the $\mathbb{Z}_2$ Floquet problem. On the boundary of each colored region, the single particle charge is not gapped in the Brillouin zone. The generalization of this phase structure to binary Floquet systems with $\mathbb{Z}_N$ symmetry can be obtained by extending the parameter ranges to $[-N\pi,N\pi)$.