  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.

Mesoscale and Nanoscale Physics;Quantum Physics