Noninvertible symmetry and topological holography for modulated SPT in one dimension

TL;DR

This paper explores noninvertible symmetries in one-dimensional modulated SPT phases, constructing dual models and revealing new phases through topological holography and symmetry analysis.

Contribution

It introduces explicit constructions of noninvertible transformations in 1D MSPT phases and establishes a topological holographic correspondence with 2D bulk theories.

Findings

Constructed noninvertible Kramers-Wannier and Kennedy-Tasaki transformations.

Generated novel MSPT phases beyond decorated domain wall models.

Identified 2D bulk theories with boundary MSPT phases.

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.