Theory of weak symmetry breaking of translations in ${\mathbb{Z}}_2$ topologically ordered states and its relation to topological superconductivity from an exact lattice ${\mathbb{Z}}_2$ charge-flux attachment

Abstract

We study ${\mathbb{Z}}_2$ topologically ordered states enriched by translational symmetry by employing a recently developed 2D bosonization approach that implements an exact ${\mathbb{Z}}_2$ charge-flux attachment in the lattice. Such states can display `weak symmetry breaking' of translations, in which both the Hamiltonian and ground state remain fully translational invariant but the symmetry is `broken' by its anyon quasi-particles, in the sense that its action maps them into a different super-selection sector. We demonstrate that this phenomenon occurs when the fermionic spinons form a weak topological superconductor in the form of a 2D stack of 1D Kitaev wires, leading to the amusing property that there is no local operator that can transport the $\pi$-flux quasi-particle across a single Kitaev wire of fermonic spinons without paying an energy gap in spite of the vacuum remaining fully translational invariant. We explain why this phenomenon occurs hand-in-hand with other previously identified peculiar features such as ground state degeneracy dependence on the size of the torus and the appearance of dangling boundary Majorana modes in certain ${\mathbb{Z}}_2$ topologically ordered states. Moreover, by extending the ${\mathbb{Z}}_2$ charge-flux attachment to open lattices and cylinders, we construct a plethora of exactly solvable models providing an exact description of their dispersive Majorana gapless boundary modes. We also review the $\mathbb{Z}\times ({\mathbb{Z}}_2{)}^3$ classification of 2D BdG Hamiltonians (Class D) enriched by translational symmetry and provide arguments on its robust stability against interactions and self-averaging disorder that preserves translational symmetry.

31 pages, 18 figures

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Cited by in corpus (13)

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• Quantum phases of two-dimensional ${\mathbb{Z}}_2$ gauge theory coupled to single-component fermion matter
• Nature of visons in the perturbed ferromagnetic and antiferromagnetic Kitaev honeycomb models
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Anyon dynamics in field-driven phases of the anisotropic Kitaev model
• (SPT-)LSM theorems from projective non-invertible symmetries
• Berry Phases of Vison Transport in ${\mathbb{Z}}_2$ Topologically Ordered States from Exact Fermion-Flux Lattice Dualities
• Translation symmetry-enriched toric code insulator
• Fractionalization as an alternate to charge ordering in electronic insulators
• Anyon polarons as a window into competing phases of the Kitaev honeycomb model under a Zeeman field
• Wegner's Ising gauge spins versus Kitaev's Majorana partons: Mapping and application to anisotropic confinement in spin-orbital liquids
• Spacetime symmetry-enriched SymTFT: from LSM anomalies to modulated symmetries and beyond
• Jordan-Wigner composite-fermion liquids in 2D quantum spin-ice