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

TL;DR

This paper develops an exponentially local spectral flow method inspired by KAM theory to handle weak non-self-adjoint perturbations in quantum spin systems, extending quasi-adiabatic techniques beyond self-adjoint cases.

Contribution

It introduces a novel exponentially local spectral flow for non-self-adjoint perturbations, expanding the applicability of quasi-adiabatic continuation methods.

Findings

Constructs an exponentially local spectral flow for non-self-adjoint perturbations.

Extends quasi-adiabatic continuation to non-self-adjoint, weakly perturbed quantum spins.

Provides potential applications to quantum Glauber dynamics.

Abstract

Since its introduction by Hastings in [10], 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.