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Non-equilibrium Almost-Stationary States and Linear Response for Gapped Quantum Systems

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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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References

  1. Bachmann S., Bols A., De Roeck W., Fraas M.: Quantization of conductance in gapped interacting systems. Annales Henri Poincaré 19, 695–708 (2018)

    Article  MathSciNet  Google Scholar 

  2. Bachmann S., De Roeck W., Fraas M.: The adiabatic theorem and linear response theory for extended quantum systems. Commun. Math. Phys. 361, 997–1027 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  3. Bachmann S., Michalakis S., Nachtergaele B., Sims R.: Automorphic equivalence within gapped phases of quantum lattice systems. Commun. Math. Phys. 309, 835–871 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  4. Bellissard J., van Elst A., Schulz-Baldes H.: The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373–5451 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  5. Bouclet J., Germinet F., Klein A., Schenker J.: Linear response theory for magnetic Schrödinger operators in disordered media. J. Funct. Anal. 226, 301–372 (2005)

    Article  MathSciNet  Google Scholar 

  6. Bru, J.-B., de Siqueira Pedra, W.: Lieb–Robinson Bounds for Multi-Commutators and Applications to Response Theory. Springer Briefs in Mathematical Physics Vol. 13, Springer (2016)

  7. De Nittis, G., Lein, M.: Linear Response Theory: An Analytic-Algebraic Approach. Springer Briefs in Mathematical Physics Vol. 21, Springer (2017)

  8. de Roeck, W., Salmhofer, M.: Persistence of exponential decay and spectral gaps for interacting fermions. Communications in Mathematical Physics, Online First (2018)

  9. Elgart A., Schlein B.: Adiabatic charge transport and the Kubo formula for Landau-type Hamiltonians. Commun. Pure Appl. Math. 57, 590–615 (2004)

    Article  MathSciNet  Google Scholar 

  10. Graf, G.M.: Aspects of the integer quantum Hall effect. In Proceedings of Symposia in Pure Mathematics 76: 429, American Mathematical Society (2007)

  11. Hastings, M.: The Stability of Free Fermi Hamiltonians. Preprint available at arXiv:1706.02270 (2017)

  12. Hastings M., Wen X.-G.: Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance. Phys. Rev. B 72, 045141 (2005)

    Article  ADS  Google Scholar 

  13. Kato T.: On the convergence of the perturbation method. I. Prog. Theor. Phys. 4, 514–523 (1949)

    Article  ADS  MathSciNet  Google Scholar 

  14. Klein, A., Lenoble, O., Müller, P.: On Mott’s formula for the ac-conductivity in the Anderson model. Ann. Math. 549–577 (2007)

    Article  MathSciNet  Google Scholar 

  15. Kubo R.: Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Japan 12, 570–586 (1957)

    Article  ADS  MathSciNet  Google Scholar 

  16. Laughlin R.: Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398 (1983)

    Article  ADS  Google Scholar 

  17. Lieb E., Robinson D.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)

    Article  ADS  MathSciNet  Google Scholar 

  18. Monaco D., Teufel S.: Adiabatic currents for interacting fermions on a lattice. Rev. Math. Phys. 31, 1950009 (2019)

    Article  MathSciNet  Google Scholar 

  19. Nachtergaele B., Sims R., Young A.: Lieb–Robinson bounds, the spectral flow, and stability of the spectral gap for lattice fermion systems. Math. Problems Quantum Phys. 117, 93 (2018)

    Article  MathSciNet  Google Scholar 

  20. Nenciu G.: On asymptotic perturbation theory for quantum mechanics: almost invariant subspaces and gauge invariant magnetic perturbation theory. J. Math. Phys. 43, 1273–1298 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  21. Panati G., Spohn H., Teufel S.: Space-adiabatic perturbation theory in quantum dynamics. Phys. Rev. Lett. 88, 250405 (2002)

    Article  ADS  Google Scholar 

  22. Panati G., Spohn H., Teufel S.: Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7, 145–204 (2003)

    Article  MathSciNet  Google Scholar 

  23. Panati G., Spohn H., Teufel S.: Effective dynamics for Bloch electrons: Peierls substitution and beyond. Commun. Math. Phys. 242, 547–578 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  24. Simon, B.: Fifteen Problems in Mathematical Physics. Perspectives in Mathematics, Birkhäuser, Basel 423 (1984)

  25. Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics. Lecture Notes in Mathematics 1821, Springer, Berlin (2003)

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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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Correspondence to Stefan Teufel.

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Communicated by M. Salmhofer

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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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