Selection Rule for Topological Amplifiers in Bogoliubov de Gennes Systems

TL;DR

This paper establishes a criterion for stability in topological amplifiers based on a novel commutator in Bogoliubov de Gennes systems, enabling the design of stable bulk states with unstable edge modes for quantum amplification.

Contribution

It introduces a theorem linking stability to an unconventional commutator, and applies it to a generalized model for topological amplification in spin-1 systems.

Findings

The theorem determines stability of states away from zero energy.

Vanishing of the commutator correlates with symmetry conditions for stability.

Time reversal symmetry enables topological amplification despite symmetry-breaking potentials.

Abstract

Dynamical instability is an inherent feature of bosonic systems described by the Bogoliubov de Geenes (BdG) Hamiltonian. Since it causes the BdG system to collapse, it is generally thought that it should be avoided. Recently, there has been much effort to harness this instability for the benefit of creating a topological amplifier with stable bulk bands but unstable edge modes which can be populated at an exponentially fast rate. We present a theorem for determining the stability of states with energies sufficiently away from zero, in terms of an unconventional commutator between the number conserving part and number nonconserving part of the BdG Hamiltonian. We apply the theorem to a generalization of a model from 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 this model to illustrate how the vanishing of the unconventional commutator selects the symmetries for a system so that its bulk states are stable against (weak) pairing interactions. We find that as long as time reversal symmetry is preserved, our system can act like a topological amplifier, even in the presence of an onsite staggered potential which breaks the inversion symmetry.