Selection rule for topological amplifiers in Bogoliubov de Gennes systems

Dynamical instability is an inherent feature of a bosonic system described by the Bogoliubov de Geenes (BdG) Hamiltonian. It causes the BdG system to collapse and therefore should be avoided. Recently, there emerged proposals for harnessing this instability for the benefit of creating a topological amplifier characterized with stable bulk bands but unstable edge modes that are populated at an exponentially fast rate. We formulate a theorem for determining the stability of a state with energy sufficiently far from zero, in terms of an unconventional "commutator" between the number conserving part and nonconserving part of the BdG Hamiltonian. We apply the theorem to a generalization of a model by Galilo et al. [Phys. Rev. Lett, 115, 245302(2015)] for creating a topological amplifier in an interacting spin-1 atom system in a honeycomb lattice through a quench process. We use it to illustrate how the vanishing of this \commutator” selects the symmetries that a system has to have so that its bulk states are stable against (weak) pairing interactions. We find that as long as the time reversal symmetry is preserved, our system is capable of acting like a topological atom laser even in the presence of the onsite staggered potential which breaks the inversion symmetry.