Theory of weak symmetry breaking of translations in Z2 topologically ordered states and its relation to topological superconductivity from an exact lattice Z2 charge-flux attachment

We study Z(2) topologically ordered states enriched by translational symmetry by employing a recently developed two-dimensional (2D) bosonization approach that implements an exact 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 quasiparticles, in the sense that its action maps them into a different superselection sector. We demonstrate that this phenomenon occurs when the fermionic spinons form a weak topological superconductor in the form of a 2D stack of one-dimensional Kitaev wires, leading to the amusing property that there is no local operator that can transport the p-flux quasiparticle 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 Z(2) topologically ordered states. Moreover, by extending the 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 Z x (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 preserve translational symmetry.

Automated summary

This study explores Z(2) topologically ordered states enriched by translational symmetry using a novel bosonization approach, demonstrating the occurrence of weak symmetry breaking when fermionic spinons form a weak topological superconductor. The research also delves into peculiar features such as ground-state degeneracy and dangling boundary Majorana modes in certain Z(2) topologically ordered states. Additionally, the extension of Z(2) charge-flux attachment to open lattices and cylinders leads to the construction of exact solvable models describing dispersive Majorana gapless boundary modes. The stability of the Z x (Z(2))3 classification of 2D BdG Hamiltonians enriched by translational symmetry against interactions and self-averaging disorder is also discussed.