Abstract
We prove the validity of linear response theory at zero temperature for perturbations of gapped Hamiltonians describing interacting fermions on a lattice. As an essential innovation, our result requires the spectral gap assumption only for the unperturbed Hamiltonian and applies to a large class of perturbations that close the spectral gap. Moreover, we prove formulas also for higher order response coefficients. Our justification of linear response theory is based on a novel extension of the adiabatic theorem to situations where a time-dependent perturbation closes the gap. According to the standard version of the adiabatic theorem, when the perturbation is switched on adiabatically and as long as the gap does not close, the initial ground state evolves into the ground state of the perturbed operator. The new adiabatic theorem states that for perturbations that are either slowly varying potentials or small quasi-local operators, once the perturbation closes the gap, the adiabatic evolution follows non-equilibrium almost-stationary states (NEASS) that we construct explicitly.
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Acknowledgements
I am grateful to Giovanna Marcelli, Domenico Monaco, and Gianluca Panati for their involvement in a closely related joint project. I would like to thank Horia Cornean, Vojkan Jaksic, Jürg Fröhlich, and Marcello Porta for very valuable discussions and comments. This work was supported by the German Science Foundation within the Research Training Group 1838.
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Teufel, S. Non-equilibrium Almost-Stationary States and Linear Response for Gapped Quantum Systems. Commun. Math. Phys. 373, 621–653 (2020). https://doi.org/10.1007/s00220-019-03407-6
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DOI: https://doi.org/10.1007/s00220-019-03407-6