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We study $\mathbb{Z}_2$ topologically ordered states enriched bytranslational symmetry by employing a recently developed 2D bosonizationapproach that implements an exact $\mathbb{Z}_2$ charge-flux attachment in thelattice. Such states can display `weak symmetry breaking' of translations, inwhich both the Hamiltonian and ground state remain fully translationalinvariant but the symmetry is `broken' by its anyon quasi-particles, in thesense that its action maps them into a different super-selection sector. Wedemonstrate that this phenomenon occurs when the fermionic spinons form a weaktopological superconductor in the form of a 2D stack of 1D Kitaev wires,leading to the amusing property that there is no local operator that cantransport the $\pi$-flux quasi-particle across a single Kitaev wire of fermonicspinons without paying an energy gap in spite of the vacuum remaining fullytranslational invariant. We explain why this phenomenon occurs hand-in-handwith other previously identified peculiar features such as ground statedegeneracy dependence on the size of the torus and the appearance of danglingboundary Majorana modes in certain $\mathbb{Z}_2$ topologically ordered states.Moreover, by extending the $\mathbb{Z}_2$ charge-flux attachment to openlattices and cylinders, we construct a plethora of exactly solvable modelsproviding an exact description of their dispersive Majorana gapless boundarymodes. We also review the $\mathbb{Z}\times (\mathbb{Z}_2)^3$ classification of2D BdG Hamiltonians (Class D) enriched by translational symmetry and providearguments on its robust stability against interactions and self-averagingdisorder that preserves translational symmetry.

physical review research

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