Full Commuting Projector Hamiltonians of Interacting Symmetry-Protected Topological Phases of Fermions

Using the decorated domain wall procedure, we construct Finite Depth Local Unitaries (FDLUs) that realize Fermionic Symmetry-Protected Topological (SPT) phases. This results in explicit ‘full’ commuting projector Hamiltonians, where ‘full’ implies the fact that the ground state, as well as all excited states of these Hamiltonians, realizes the nontrivial SPT phase. We begin by constructing explicit examples of 1+1D phases protected by symmetry groups $G=\mathbb Z_2^T \times \mathbb Z_2^F$ , which also has a free fermion realization in class BDI, and $G={\mathbb{Z}}_4\times {\mathbb{Z}}_4^F$, which does not. We then turn to 2+1D, and construct the square roots of the Levin-Gu bosonic SPT phase, protected by ${\mathbb{Z}}_2\times {\mathbb{Z}}_2^F$ symmetry, in a concrete model of fermions and spins on the triangular lattice. Edge states and the anomalous symmetry action on them are explicitly derived. Although this phase has a free fermion representation as two copies of $p+ip$ superconductors combined with their $p-ip$ counterparts with a different symmetry charge, the full set of commuting projectors is only realized in the strongly interacting version, which also implies that it admits a many-body localized realization.