Communications in Mathematical Physics

, Volume 365, Issue 2, pp 773–796 | Cite as

Persistence of Exponential Decay and Spectral Gaps for Interacting Fermions

  • Wojciech De Roeck
  • Manfred SalmhoferEmail author


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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  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 (2017)Google Scholar
  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)ADSMathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bravyi S., Hastings M.B., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)ADSMathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)Google Scholar
  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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gil, M.I.: Operator Functions and Localization of Spectra. Springer, (2003)Google Scholar
  12. 12.
    Giuliani A., Mastropietro V., Porta M.: Universality of the Hall conductivity in interacting electron systems. Commun. Math. Phys. 349, 1107–1161 (2016)ADSMathSciNetCrossRefzbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle 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)ADSCrossRefGoogle 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)ADSCrossRefGoogle 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)ADSCrossRefGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Michalakis S., Zwolak J.P.: Stability of frustration-free hamiltonians. Commun. Math. Phys. 322, 277–302 (2013)ADSMathSciNetCrossRefzbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Salmhofer M.: Renormalization: An Introduction. Springer Verlag, Heidelberg (1998)zbMATHGoogle Scholar
  25. 25.
    Salmhofer M.: Clustering of fermionic truncated expectation values via functional integration. J. Stat. Phys. 134, 941–952 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Salmhofer M., Wieczerkowski C.: Positivity and convergence in fermionic quantum field theory. J. Stat. Phys. 99, 557–586 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yarotsky D.A.: Ground states in relatively bounded quantum perturbations of classical lattice systems. Commun. Math. Phys. 261, 799–819 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Instituut voor Theoretische FysicaK. U. LeuvenLeuvenBelgium
  2. 2.Institut für Theoretische PhysikUniversität HeidelbergHeidelbergGermany

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