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Noninvertible operators in one, two, and three dimensions via gauging spatially modulated symmetry

Hiromi Ebisu, Bo Han

TL;DR

This work addresses how noninvertible duality defects arise from gauging spatially modulated symmetries in lattice systems. By doubling spin models and applying subsystem or dipole gauging in 1D–3D, it constructs explicit noninvertible operators D and analyzes their fusion rules, revealing 0-form subsystem charges and higher-form operators in 3D (lineon-like excitations) and a hierarchical dipole algebra in 2D with dual structures. The key finding is a unified framework for producing exotic duality defects across dimensions, where D×D closes to a projector-like operator C and DC/CD generate further charged sectors, illustrating deep connections between gauging, mobility constraints, and noninvertible symmetries. The results provide concrete lattice realizations that can inform future classifications, generalizations to non-Abelian or higher multipole symmetries, and potential links to higher category theory, with implications for fracton-like topological order and duality structures.

Abstract

Spatially modulated symmetries have emerged since the discovery of fractons, which characterize unconventional topological phases with mobility-constrained quasiparticle excitations. On the other hand, non-invertible duality defects have attracted substantial attention in communities of high energy and condensed matter physics due to their deep insight into quantum anomalies and exotic phases of matter. However, the connection between these exotic symmetries and defects has not been fully explored. In this paper, we construct concrete lattice models with non-invertible duality defects via gauging spatially modulated symmetries and investigate their exotic fusion rules. Specifically, we construct spin models with subsystem symmetries or dipole symmetries on one, two, and three-dimensional lattices. Gauging subsystem symmetries leads to non-invertible duality defects whose fusion rules involve $0$-form subsystem charges in two dimensions and higher-form operators that correspond to ``lineon'' excitations (excitations which are mobile along one-dimensional line) in three dimensions. Gauging dipole symmetries leads to non-invertible duality defects with dipole algebras that describe a hierarchical structure between global and dipole charges. Notably, the hierarchical structure of the dual dipole charges is inverted compared with the original ones. Our work provides a unified and systematic analytical framework for constructing exotic duality defects by gauging relevant symmetries.

Noninvertible operators in one, two, and three dimensions via gauging spatially modulated symmetry

TL;DR

This work addresses how noninvertible duality defects arise from gauging spatially modulated symmetries in lattice systems. By doubling spin models and applying subsystem or dipole gauging in 1D–3D, it constructs explicit noninvertible operators D and analyzes their fusion rules, revealing 0-form subsystem charges and higher-form operators in 3D (lineon-like excitations) and a hierarchical dipole algebra in 2D with dual structures. The key finding is a unified framework for producing exotic duality defects across dimensions, where D×D closes to a projector-like operator C and DC/CD generate further charged sectors, illustrating deep connections between gauging, mobility constraints, and noninvertible symmetries. The results provide concrete lattice realizations that can inform future classifications, generalizations to non-Abelian or higher multipole symmetries, and potential links to higher category theory, with implications for fracton-like topological order and duality structures.

Abstract

Spatially modulated symmetries have emerged since the discovery of fractons, which characterize unconventional topological phases with mobility-constrained quasiparticle excitations. On the other hand, non-invertible duality defects have attracted substantial attention in communities of high energy and condensed matter physics due to their deep insight into quantum anomalies and exotic phases of matter. However, the connection between these exotic symmetries and defects has not been fully explored. In this paper, we construct concrete lattice models with non-invertible duality defects via gauging spatially modulated symmetries and investigate their exotic fusion rules. Specifically, we construct spin models with subsystem symmetries or dipole symmetries on one, two, and three-dimensional lattices. Gauging subsystem symmetries leads to non-invertible duality defects whose fusion rules involve -form subsystem charges in two dimensions and higher-form operators that correspond to ``lineon'' excitations (excitations which are mobile along one-dimensional line) in three dimensions. Gauging dipole symmetries leads to non-invertible duality defects with dipole algebras that describe a hierarchical structure between global and dipole charges. Notably, the hierarchical structure of the dual dipole charges is inverted compared with the original ones. Our work provides a unified and systematic analytical framework for constructing exotic duality defects by gauging relevant symmetries.
Paper Structure (16 sections, 170 equations, 6 figures)

This paper contains 16 sections, 170 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Conventional Ising chain
  3. Subsystem symmetry
  4. Two dimensions
  5. Three dimensions
  6. Dipole symmetry
  7. One dimension
  8. Two dimensions
  9. Model
  10. Toric code with dipole symmetries
  11. noninvertible operators
  12. $0$-form dual dipole algebra in two dimensions
  13. Outlook
  14. Condensation defects
  15. Subsystem symmetry in three dimensions
  16. ...and 1 more sections

Figures (6)

  • Figure 1: (a) (left) Plaquette Ising term defined in \ref{['plaz']}. (right) The Gauss law, corresponding to \ref{['gauss:pl']}. The blue dots represent original spin degrees of freedom whereas the red squares do $\mathbf{Z}_2$ gauge fields. (b) Example of \ref{['membrane']}. The black dot represents a node with coordinate $\mathbf{r}=(\hat{x},\hat{y}).$
  • Figure 2: (a) Three types of plaquette Ising terms that constitute the Hamiltonian \ref{['3D_Ising']}. (b) The Gauss law \ref{['gs']} is described by the left configuration whereas the three flux operators \ref{['flux']} are depicted by the configurations on the right. The original spin degrees of freedom are depicted as blue dots whereas the gauge fields are indicated by red rectangles. (c) Configuration of an operator given in \ref{['Mr']}.
  • Figure 3: Pictorial understanding of how non-local terms in \ref{['br']} is absorbed in the operator $C_{\eta}$\ref{['eeta']}.
  • Figure 4: (a) Three types of terms defined in \ref{['spin2d2']}. (b) $0$-form dipole symmetries \ref{['algebra']}, forming dipole algebra \ref{['algebra2']} which is schematically portrayed as an "inverse of a triangle" in the bottom. (c) The Gauss law corresponding to \ref{['gauss_44']}. (d) Two flux operators given in \ref{['fluxes']}.
  • Figure 5: (a) Configuration of loops given in \ref{['loops']}. These loops constitute $1$-form dual dipole algebra. (b) When gauging $0$-form dipole symmetry characterized by a dipole algebra (left) yields $1$-form dipole symmetry labeled by dual dipole algebra (right). (c) Configuration of gauge fields defined in \ref{['95']}.
  • ...and 1 more figures