An exponentially local spectral flow for possibly non-self-adjoint perturbations of non-interacting quantum spins,inspired by KAM theory

Since its introduction by Hastings (Phys Rev B 69:104431, 2004), the technique of quasi-adiabatic continuation has become a central tool in the discussion and classification of ground-state phases. It connects the ground states of self-adjoint Hamiltonians in the same phase by a unitary quasi-local transformation. This paper takes a step towards extending this result to non-self-adjoint perturbations, though, for technical reason, we restrict ourselves here to weak perturbations of non-interacting spins. The extension to non-self-adjoint perturbation is important for potential applications to Glauber dynamics (and its quantum analogues). In contrast to the standard quasi-adiabatic transformation, the transformation constructed here is exponentially local. Our scheme is inspired by KAM theory, with frustration-free operators playing the role of integrable Hamiltonians.