Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Duality viewpoint of noninvertible symmetry protected topological phases

Weiguang Cao, Masahito Yamazaki, Linhao Li

Abstract

Recent advancements in generalized symmetries have drawn significant attention to gapped phases of matter exhibiting novel symmetries, such as noninvertible symmetries. By leveraging the duality transformations, the classification and construction of gapped phases with noninvertible symmetry can be mapped to those involving conventional group symmetries. We demonstrate this approach by classifying symmetry-protected-topological phases with a broad class of noninvertible symmetries in arbitrary spacetime dimensions. Our results reveal new classifications that extend beyond those based on group symmetries. Additionally, we construct lattice models in $(1+1)D$ and $(2+1)D$ that realize these new phases and explore their anomalous interfaces.

Duality viewpoint of noninvertible symmetry protected topological phases

Abstract

Recent advancements in generalized symmetries have drawn significant attention to gapped phases of matter exhibiting novel symmetries, such as noninvertible symmetries. By leveraging the duality transformations, the classification and construction of gapped phases with noninvertible symmetry can be mapped to those involving conventional group symmetries. We demonstrate this approach by classifying symmetry-protected-topological phases with a broad class of noninvertible symmetries in arbitrary spacetime dimensions. Our results reveal new classifications that extend beyond those based on group symmetries. Additionally, we construct lattice models in and that realize these new phases and explore their anomalous interfaces.

Paper Structure

This paper contains 12 sections, 114 equations, 5 figures, 1 table.

Table of Contents

  1. Unbroken symmetry of $\mathbb{Z}^U_2$-SSB phase with $D_{2n}$ symmetry
  2. Distinct SPTs by one-site lattice translation
  3. Projective representation
  4. Gauging in general dimensions
  5. Anomalies of the interface between stacked SPT phases before gauging
  6. Review of Else-Nayak approach
  7. Interface anomalies of $\mathbb{Z}^{\eta}_4$-SPT with $m=1$
  8. Proof of anomalous symmetry on the interface
  9. The classification based on the conjugacy classes of $\Gamma$-SPTs
  10. One spatial dimension
  11. Two spatial dimensions
  12. Three spatial dimensions

Figures (5)

  • Figure 1: The duality method to study NISPT phases from group SSB phases with examples in the boxes.
  • Figure 2: Effective process (left) and the dictionary of symmetry operators in UV and IR (right).
  • Figure 3: Left: triangular lattice and interface (red) between different NISPTs. Right: seven-body interaction term in \ref{['eq:2Dungauged Hal']}.
  • Figure 4: Local tensors on vertices and links. The physical bonds are in the directions perpendicular to the page. $A$-tensor acts on the vertices and $B$-tensor acts on the edges, which are connected by virtual bonds labeled by $n=\pm 1$. $\prod^6_{i=1}\delta^{n_i}_{\tau_{j}}$ is a projection operator ensuring all the virtual bonds $n_i$ around the vertex $j$ with the same value $n_i=\tau_j=\pm 1$.
  • Figure 5: Local tensors on vertices and links. The physical bonds are in the directions perpendicular to the page. $A$-tensor acts on the vertices and $B$-tensor acts on the edges. $A, B$-tensors are connected by virtual bonds.