Topological classification of excitations in quadratic bosonic systems

We investigate the topological classification of excitations in quadratic bosonic systems with an excitation band gap. Time-reversal, charge-conjugation, and parity symmetries in bosonic systems are introduced to realize a ten-fold symmetry classification. We find a specific decomposition of the quadratic bosonic Hamiltonian and use it to prove that each quadratic bosonic system is homotopic to a direct sum of two single-particle subsystems. The topological classification table is thus derived via inheriting from that of Atland-Zirnbauer classes and unique topological phases of bosons are predicted. Finally, concrete topological models are proposed to demonstrate the peculiarity of bosonic excitations.