Higher Order Bosonic Topological Phases in Spin Models

We discuss an extension of higher order topological phases to include bosonic systems. We present two spin models for a second-order topological phase protected by a global ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry. One model is built from layers of an exactly solvable cluster model for a one-dimensional ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ topological phase, while the other is built from more conventional spin-couplings (XY or Heisenberg). These models host gapped, but topological, edges, and 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. We also discuss how our models can be extended to three-dimensions to form a third-order topological phase.