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.
Highlights
The toric code (TC) [1] is a simple example of an exactly solvable model of Z2 topologically ordered states [2,3]
By enforcing a local symmetry which freezes the motion of isolated e and m particles, but allows the fluctuations of their fermionic bound state, the ε particle, the underlying spin Hilbert separates into subspaces of ε fermions coupled to nondynamical background gauge fields
We have elucidated this construction in geometries with fully open boundaries and cylinders, and extended it to the torus. This formalism allows to construct a relatively simple unifying picture of a series of amusing properties of Z2 topologically ordered states enriched by translational symmetry [28,29,30,32,33], including their anomalous ground-state degeneracy (GSD) dependence on the size of the torus and the appearance of dangling Majorana modes at the boundaries of open lattices even in states whose bulk topological order is identical to the toric code
Summary
The toric code (TC) [1] is a simple example of an exactly solvable model of Z2 topologically ordered states [2,3]. On the other hand, when a finite density of the bosonic anyons are added to the TC vacuum but instead they form an “atomic insulator” state in which they are localized at sites without spontaneously breaking their parity symmetry, the resulting state is still Z2 topologically ordered, it can display a projective symmetry implementation of the translation group [6,7]. If e particles are kept dynamically immobile, these constructions can be viewed as a form of Z2 charge-flux attachment implementing a type of local two-dimensional (2D) Jordan-Wigner transformation [4,5,8,9,10,11,12,13,14,15,16,17] In this case, and in contrast to the bosonic case, any local fermion Hamiltonian always respects parity. Several technical aspects and alternative derivations are presented in Appendices A–G
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