Generating Lattice Non-invertible Symmetries

TL;DR

The paper develops a systematic method to generate lattice non-invertible symmetries from a seed KW duality by acting on different sites or by sandwiching unitary gates, uncovering a new dipole KW symmetry tied to Z_N dipole symmetry. It analyzes this symmetry through its fusion algebra, topological defects, and anomaly, revealing that its behavior depends on N and can be anomalous (e.g., N=3). A key advance is the construction of generalized Kennedy-Tasaki transformations KT_k that map between dipole SSB and dipole SPT phases, organizing a duality web of gapped phases. Together, these results enrich the landscape of lattice generalized symmetries, offering tools for classifying non-invertible structures and guiding future explorations in higher dimensions and multipole symmetries.

Abstract

Lattice non-invertible symmetries have rich fusion structures and play important roles in understanding various exotic topological phases. In this paper, we explore methods to generate new lattice non-invertible transformations/symmetries from a given non-invertible seed transformation/symmetry. The new lattice non-invertible symmetry is constructed by composing the seed transformations on different sites or sandwiching a unitary transformation between the transformations on the same sites. In addition to known non-invertible symmetries with fusion algebras of Tambara-Yamagami $\mathbb Z_N\times\mathbb Z_N$ type, we obtain a new non-invertible symmetry in models with $\mathbb Z_N$ dipole symmetries. We name the latter the dipole Kramers-Wannier symmetry because it arises from gauging the dipole symmetry. We further study the dipole Kramers-Wannier symmetry in depth, including its topological defect, its anomaly and its associated generalized Kennedy-Tasaki transformation.

Figures (5)

• Figure 1: Duality web of the dipole Ising model and the XZ model.
• Figure 2: (1) Half-gauging creates a topological interface/defect $\hat{\mathsf{D}}$ located at link $(1,2)$; (2) movement operator $U_{\hat{\mathsf{D}}}^2=CZ_{2,3}(CZ_{2,l}^{\dagger})^2(U^{H}_{2})^{\dagger}S_{2,l}$ moves the interface/defect from $(1,2)$ to $(2,3)$; (3) Fusion operator $\lambda_{\hat{\mathsf{D}}\otimes \hat{\mathsf{D}}}^{(1,2,3)}=CZ_{3,l_2}^{\dagger}CZ_{2,l_1}^{\dagger}(CZ_{2,l_2})^2S_{2,l_2}$ fuses the defects at $(1,2)$ and $(2,3)$.
• Figure 3: The phase diagram of XZ model \ref{['eq:quan torus Hal']} (left) and the dipole Ising model (right) where dashed lines are first order phases transition and solid lines are continuous phase transitions. The red dots $\theta=0.25\pi, 1.25\pi$ (right) are critical points where the dipole Ising model has the anomalous non-invertible symmetry.
• Figure 4: Transformations between the trivial phase, the SSB phase and the SPT phase with level $k$. KT transformation $T_D^k \hat{\mathsf{D}} T_D^{-k}$ is a duality map between the SSB phase and SPT phase with level $k$.
• Figure 5: Duality web between various gapped phases. Unless specified, the SSB phase breaks all $G=\mathbb Z_N^Q\times \mathbb Z_N^D$ symmetry and the SPT phase is protected $G$ symmetry. $k_1,n_1$ are coprime with $N$ and $n_1k_1=-1~\text{mod}~N$. $k_2$ is not coprime with $N$ and $k_2n_2=-\text{gcd}(k_2,N)~\text{mod}~N$. $G_1=\mathbb Z_{N/\text{gcd}(k_2,N)}^Q\times \mathbb Z_{N/\text{gcd}(k_2,N)}^D$ and $G_2=\mathbb Z_{\text{gcd}(k_2,N)}^Q\times \mathbb Z_{\text{gcd}(k_2,N)}^D$.