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

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Abstract

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.

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Acknowledgements

We wish to thank Nick Crawford for many helpful discussions. Furthermore, we are thankful to the DFG (German Science Foundation) and the Belgian Interuniversity Attraction Pole (P07/18 Dygest) for financial support.

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  1. Instituut voor Theoretische Fysica, KU Leuven, Leuven, Belgium

    Wojciech De Roeck & Marius Schütz

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  1. Wojciech De Roeck
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  2. Marius Schütz
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Correspondence to Marius Schütz.

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Roeck, W.D., Schütz, M. An exponentially local spectral flow for possibly non-self-adjoint perturbations of non-interacting quantum spins, inspired by KAM theory. Lett Math Phys 107, 505–532 (2017). https://doi.org/10.1007/s11005-016-0913-z

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  • DOI: https://doi.org/10.1007/s11005-016-0913-z

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