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 π -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 × (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.
We study a one-dimensional model of interacting bosons coupled to a dynamical
Z2 field, the
Z2 Bose-Hubbard model, and analyze the interplay between spontaneous symmetry breaking and topological symmetry protection. In a previous work, we showed how this model exhibits a spontaneous breaking of the translational symmetry through a bosonic Peierls transition. Here we find how, at half filling, the resulting phase also displays topological features that coexist with the presence of long-range order and yields a topological bond order wave. Using both analytical and numerical methods, we describe the properties of this phase, showing that it cannot be adiabatically connected to a topological phase of noninteracting bosons, and thus constitutes an instance of an interaction-induced symmetry-breaking topological insulator.
In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The lattice spin systems are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by
Z2A or
Z2B projective symmetry group. We find that the
Z2A translation symmetric topological orders can still be divided into 16 subclasses corresponding to 16 translation-symmetry protected topological orders. We introduced four
Z2 topological indices ${\ensuremath{\zeta}}_{\mathbf{k}}=0,1$ at
k=(0,0), $(0,\ensuremath{\pi})$, $(\ensuremath{\pi},0)$, and $(\ensuremath{\pi},\ensuremath{\pi})$ to characterize those 16 topological orders on square lattice. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected
Z2A topological orders for a system on three-dimensional cubic lattice.