Topological phases with generalized global symmetries

We present simple lattice realizations of symmetry-protected topological phases with q-form global symmetries where charged excitations have q spatial dimensions. Specifically, we construct d space-dimensional models supported on a (d + 1)-colorable graph by using a family of unitary phase gates, known as multiqubit control-Z gates in quantum information community. In our construction, charged excitations of different dimensionality may coexist and form a short-range entangled state which is protected by symmetry operators of different dimensionality. Nontriviality of proposed models, in a sense of quantum circuit complexity, is confirmed by studying protected boundary modes, gauged models, and corresponding gapped domain walls. We also comment on applications of our construction to quantum error-correcting codes, and discuss corresponding fault-tolerant logical gates.