Persistence of Exponential Decay and Spectral Gaps for Interacting Fermions

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Abstract

We consider systems of weakly interacting fermions on a lattice. The corresponding free fermionic system is assumed to have a ground state separated by a gap from the rest of the spectrum. We prove that, if both the interaction and the free Hamiltonian are sums of sufficiently rapidly decaying terms, and if the interaction is sufficiently weak, then the interacting system has a spectral gap as well, uniformly in the lattice size. Our approach relies on convergent fermionic perturbation theory, thus providing an alternative method to the one used recently by Hastings (The stability of free Fermi hamiltonians, arXiv:1706.02270, 2017), and extending the result to include non-selfadjoint interaction terms.

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References

  1. 1.

    Aza, N.J.B., Bru, J.B., de Siqueira Pedra, W., Müssnichh, L.C.P.A.M.: Large deviations in weakly interacting fermions I—generating functions as Gaussian Berezin integrals and bounds on large Pfaffians. preprint pdfs.semanticscholar.org (2017)

  2. 2.

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  3. 3.

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

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Borgs C., Koteckỳ R., Ueltschi D.: Low temperature phase diagrams for quantum perturbations of classical spin systems. Commun. Math. Phys. 181, 409–446 (1996)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Bratteli O., Robinson D.W.: Operator Algebras and Quantum Statistical Mechanics 2: Equilibrium States Models in Quantum Statistical Mechanics. Springer-Verlag, Heidelberg (1996)

    MATH  Google Scholar 

  6. 6.

    Bravyi S., Hastings M.B.: A short proof of stability of topological order under local perturbations. Commun. Math. Phys. 307, 609–627 (2011)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Bravyi S., Hastings M.B., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Bru J.-B., Pedra W.: Universal bounds for large determinants fromnon-commutative Hölder inequalities in fermionic constructive quantum field theory. Math. Models Methods Appl. Sci. 27, 1963 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Brydges, D.C.: A short course on cluster expansions. In: Osterwalder, K., Stora, R. (eds.) Critical Phenomena, Random Systems, Gauge Theories (Les Houches 1984) (1986)

  10. 10.

    Datta N., Fernández R., Fröhlich J.: Low-temperature phase diagrams of quantum lattice systems I Stability for quantum perturbations of classical systems with finitely-many ground states. J. Stat. Phys. 84, 455–534 (1996)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Gil, M.I.: Operator Functions and Localization of Spectra. Springer, (2003)

  12. 12.

    Giuliani A., Mastropietro V., Porta M.: Universality of the Hall conductivity in interacting electron systems. Commun. Math. Phys. 349, 1107–1161 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Hastings, M.B.: The stability of free Fermi hamiltonians (2017). arXiv preprint arXiv:1706.02270

  14. 14.

    Hastings M.B., Michalakis S.: Quantization of Hall conductance for interacting electrons on a torus. Commun. Math. Phys. 334, 433–471 (2015)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Hastings M.B., 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)

    ADS  Article  Google Scholar 

  16. 16.

    Hofstadter D.R.: Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239 (1976)

    ADS  Article  Google Scholar 

  17. 17.

    Katsura H., Schuricht D., Takahashi M.: Exact ground states and topological order in interacting Kitaev chains. Phys. Rev. B 92, 115137 (2015)

    ADS  Article  Google Scholar 

  18. 18.

    Mariën M., Audenaert K.M.R., Van Acoleyen K., Verstraete F.: Entanglement rates and the stability of the area law for the entanglement entropy. Commun. Math. Phys. 346, 35–73 (2016)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Michalakis S., Zwolak J.P.: Stability of frustration-free hamiltonians. Commun. Math. Phys. 322, 277–302 (2013)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Monaco, D., Teufel, S.: Adiabatic currents for interacting electrons on a lattice (2017). arXiv preprint arXiv:1707.01852

  21. 21.

    Nachtergaele B.: The spectral gap for some spin chains with discrete symmetry breaking. Commun. Math. Phys. 175, 565–606 (1996)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Nachtergaele, B., Sims, R., Young, A.:Lieb––Robinson bounds, the spectral flow, and stability of the spectral gap for lattice fermion systems (2017). arXiv preprint arXiv:1705.08553

  23. 23.

    PedraW.A. de S., Salmhofer M.: Determinant bounds and the Matsubara UV problem of many-fermion systems. Commun. Math. Phys. 282, 797–818 (2008)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Salmhofer M.: Renormalization: An Introduction. Springer Verlag, Heidelberg (1998)

    MATH  Google Scholar 

  25. 25.

    Salmhofer M.: Clustering of fermionic truncated expectation values via functional integration. J. Stat. Phys. 134, 941–952 (2009)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Salmhofer M., Wieczerkowski C.: Positivity and convergence in fermionic quantum field theory. J. Stat. Phys. 99, 557–586 (2000)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Yarotsky D.A.: Ground states in relatively bounded quantum perturbations of classical lattice systems. Commun. Math. Phys. 261, 799–819 (2006)

    ADS  MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Manfred Salmhofer.

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Communicated by H.-T. Yau

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De Roeck, W., Salmhofer, M. Persistence of Exponential Decay and Spectral Gaps for Interacting Fermions. Commun. Math. Phys. 365, 773–796 (2019). https://doi.org/10.1007/s00220-018-3211-z

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