Higher Order Bosonic Topological Phases in Spin Models

Motivated by the recent discovery of the higher order topological phases in the fermionic systems we propose a natural extension of these phases to bosonic systems. We discuss two bosonic models for a second-order topological phase protected by a global Z2 x Z2 symmetry. One model is built from layers of an exactly solvable cluster model for a one-dimensional Z2 x Z2 topological phase. The other is built from more conventional spin-couplings (XY or Heisenberg) and repeats the structure of the quadrupole model. These models host gapped, but topologically protected, edges, as well as protected corner modes that fall into a projective representation of the symmetry. Using Jordan-Wigner transformations we show that our models are both related to a bilayer of free Majorana fermions that form a fermionic second-order topological phase. In fact, the XY model was shown to be in exact correspondence with the fermionic Quadrupole model. We also discuss possible extension to 3D bosonic models for 2nd and 3rd order topological phases.