${\mathbb{Z}}_N$ generalizations of three-dimensional stabilizer codes

Abstract

In this work, we generalize several three-dimensional Z2 stabilizer models--including the X-cube model, the three-dimensional toric code, and Haah's code--to their ZN counterparts. Under periodic boundary conditions, we analyze their ground state degeneracies and topological excitations, and uncover behaviors that strongly depend on system size. For the X-cube model, we identify excitations with mobility restricted under local operations but relaxed under nonlocal ones derived from global topology. These excitations, previously confined to open boundaries in the Z2 model, now appear even under periodic boundaries. In the toric code, we observe nontrivial braiding between string and point excitations despite the absence of ground state degeneracy, indicating long-range entanglement independent of topological degeneracy. Again, this effect extends from open to periodic boundaries in the generalized models. For Haah's code, we find new excitations--fracton tripoles and monopoles--that remain globally constrained, along with a relaxation of immobility giving rise to lineons and planons. These results reveal new forms of topological order and suggest a broader framework for understanding fracton phases beyond the conventional Z2 setting.

36 pages, 25 figures

References in corpus (44)

• Non-Abelian Anyons and Topological Quantum Computation
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Quantum Orders and Symmetric Spin Liquids
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Local stabilizer codes in three dimensions without string logical operators
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Quantum Glassiness
• Fractons
• A New Kind of Topological Quantum Order: A Dimensional Hierarchy of Quasiparticles Built from Stationary Excitations
• Subdimensional Particle Structure of Higher Rank U(1) Spin Liquids
• Fracton Phases of Matter
• A classification of symmetry enriched topological phases with exactly solvable models
• Exotic topological order in fractal spin liquids
• Classifying fractionalization: symmetry classification of gapped Z2 spin liquids in two dimensions
• Topological order in an exactly solvable 3D spin model
• Choreographed entangle dances: topological states of quantum matter
• Theory of defects in Abelian topological states
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• String and Membrane condensation on 3D lattices
• Fractonic Matter in Symmetry-Enriched U(1) Gauge Theory
• Bulk-boundary correspondence in (3+1)-dimensional topological phases
• Breakdown of a perturbed Z_N topological phase
• Fracton fusion and statistics
• Translation-symmetry protected topological orders on lattice
• Gauging fractons: immobile non-Abelian quasiparticles, fractals, and position-dependent degeneracies
• Sorting topological stabilizer models in three dimensions
• Position-Dependent Excitations and UV/IR Mixing in the ${\mathbb{Z}}_N$ Rank-2 Toric Code and its Low-Energy Effective Field Theory
• Rank-2 Toric Code in Two Dimensions
• Subsystem symmetry enriched topological order in three dimensions
• Ground state degeneracy on torus in a family of ${\mathbb{Z}}_N$ toric code
• Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in $(3+1)$-Dimensions
• Quantum robustness of fracton phases
• Gauging modulated symmetries: Kramers-Wannier dualities and non-invertible reflections
• Effective Fractonic Behavior in a Two-Dimensional Exactly Solvable Spin Liquid
• Defects in the 3-dimensional toric code model form a braided fusion 2-category
• Foliated BF theories and Multipole symmetries
• Effective Field Theory of Dipolar Braiding Statistics in Two Dimensions
• Quantum robustness and phase transitions of the 3D Toric Code in a field
• Competing topological orders in three dimensions: X-cube versus toric code
• Spontaneous symmetry breaking without ground state degeneracy in generalized $N$-state clock model
• Anomaly inflow for dipole symmetry and higher form foliated field theories
• No Strings Attached: Boundaries and Defects in the Cubic Code
• Unveiling UV/IR Mixing via Symmetry Defects: A View from Topological Entanglement Entropy