The topological vacuum in exceptional Brillouin zone

The vacuum of a band system, with respect to particles or holes, can become topologically nontrivial when the exceptional points of the non-hermitian Hamiltonian spread over the whole Brillouin zone. The coalescence of the eigenstates eliminates the geometric multiplicity of the Hamiltonian and puts the remaining bands on a topological background. We find several distinct phenomena in these systems: a single band topological model, a two-band Chern insulator in which the topological number of the low band does not compensate to that of the up band, a topologically protected boundary state that is less damping when stronger dissipative forces are subjected and topologically protected in-gap extended states. Our results are presented by the vibrational spectrum on a classical lattice. The active constrains are the key ingredient to collapse the demanding Hilbert subspace.