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 Z(2) x Z(2) symmetry. One model is built from layers of an exactly solvable cluster model for a one-dimensional Z(2) x Z(2) topological phase, while the other is built from more conventional spin couplings (XY or Heisenberg). These models host gapped, symmetry-protected topological phases on their edges, and corner modes that fall into a projective representation of the symmetry. Using Jordan-Wigner transformations we show that both our models are 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.