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Noninvertible symmetry and topological holography for modulated SPT in one dimension

Jintae Kim, Yizhi You, Jung Hoon Han

Abstract

We examine noninvertible symmetry (NIS) in one-dimensional (1D) symmetry-protected topological (SPT) phases protected by dipolar and exponential-charge symmetries, which are two key examples of modulated SPT (MSPT). To set the stage, we first study NIS in the $\mathbb{Z}_N \times \mathbb{Z}_N$ cluster model, extending previous work on the $\mathbb{Z}_2 \times \mathbb{Z}_2$ case. For each symmetry type (charge, dipole, exponential), we explicitly construct the noninvertible Kramers-Wannier (KW) and Kennedy-Tasaki (KT) transformations, revealing dual models with spontaneous symmetry breaking (SSB). The resulting symmetry group structure of the SSB model is rich enough that it allows the identification of other SSB models with the same symmetry. Using these alternative SSB models and KT duality, we generate novel MSPT phases distinct from those associated with the standard decorated domain wall picture, and confirm their distinctiveness by projective symmetry analyses at their interfaces. Additionally, we establish a topological-holographic correspondence by identifying the 2D bulk theories-two coupled layers of toric codes (charge), anisotropic dipolar toric codes (dipole), and exponentially modulated toric codes (exponential)-whose boundaries host the respective 1D MSPT phases.

Noninvertible symmetry and topological holography for modulated SPT in one dimension

Abstract

We examine noninvertible symmetry (NIS) in one-dimensional (1D) symmetry-protected topological (SPT) phases protected by dipolar and exponential-charge symmetries, which are two key examples of modulated SPT (MSPT). To set the stage, we first study NIS in the cluster model, extending previous work on the case. For each symmetry type (charge, dipole, exponential), we explicitly construct the noninvertible Kramers-Wannier (KW) and Kennedy-Tasaki (KT) transformations, revealing dual models with spontaneous symmetry breaking (SSB). The resulting symmetry group structure of the SSB model is rich enough that it allows the identification of other SSB models with the same symmetry. Using these alternative SSB models and KT duality, we generate novel MSPT phases distinct from those associated with the standard decorated domain wall picture, and confirm their distinctiveness by projective symmetry analyses at their interfaces. Additionally, we establish a topological-holographic correspondence by identifying the 2D bulk theories-two coupled layers of toric codes (charge), anisotropic dipolar toric codes (dipole), and exponentially modulated toric codes (exponential)-whose boundaries host the respective 1D MSPT phases.

Paper Structure

This paper contains 30 sections, 234 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Charge symmetry
  3. Gauging and noninvertible charge symmetry
  4. Charge SPT
  5. Kennedy-Tasaki transformation
  6. Other cSPT states
  7. Other cSPT states : $H_{c'}$
  8. Other cSPT states : $H_{c"}$
  9. Edge modes
  10. cSPT from holography
  11. Dipole symmetry
  12. Gauging and noninvertible dipole symmetry
  13. Dipolar SPT
  14. Kennedy-Tasaki transformation
  15. Other dSPT states
  16. ...and 15 more sections

Figures (3)

  • Figure 1: (a) Bulk stabilizers of the anistropic dipolar toric code given in \ref{['Ebisu-model']}. (b) Boundary operators \ref{['e-fixing-bc']} at the bottom rough boundary fixing the $e$-condensing boundary conditions, active boundary operators at the top boundary, vertical Wilson loop operator playing the role of charge operator, and two horizontal Wilson loop operators playing the role of charge and dipole symmetry operators $g_c^{(e)}, g_d^{(e)}$ are depicted. (c) Same as (b), for the smooth bottom boundary realizing the $m$-condensing boundary condition. The top boundary is rough in both (b) and (c).
  • Figure 2: Stabilizers for each layer of eTC. The layers are labeled by 1,2. $r$ and $\tilde{r}$ represent the vertex and plaquette coordinates.
  • Figure 3: (a) Boundary operators at the bottom rough boundary [Eq. \ref{['rough-bottom-H-eTC']}] and the top rough boundary [Eq. \ref{['top']}], serving as the boundary condition and the active degrees of freedom, respectively, of each layer of the exponential toric codes. Two horizontal Wilson loop operators serving as two symmetry operators of the eSPT, $g_1 , g_2$ for layers 1 and 2, are shown along with the vertical Wilson loop operators serving as the charge operators. (b) Similar to (a) for smooth bottom boundary and rough top boundary.