Letters in Mathematical Physics

, Volume 105, Issue 8, pp 1119–1133 | Cite as

Upper Bounds on the Charge Susceptibility of Many-Electron Systems Coupled to the Quantized Radiation Field



We extend the Kubo–Kishi theorem concerning the charge susceptibility of the Hubbard model in the following way: (i) The electron–photon interaction is taken into account. (ii) Not only on-site but also general Coulomb repulsions are considered.


Hubbard model quantized radiation field Euclidean-Bose field charge susceptibility 

Mathematics subject classification

47N50 46N55 60G15 82B10 82D40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arai A.: Path integral representation of the index of Kahler-Dirac operators on an infinite-dimensional manifold. J. Funct. Anal. 82, 330–369 (1989)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Arai A.: Trace formulas, a Golden-Thompson inequality and classical limit in boson Fock space. J. Funct. Anal. 136, 510–547 (1996)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dollard, J.D., Friedman, C.N.: Product Integration with Application to Differential Equations. Encyclopedia of mathematics and its applications, vol. 10, 1979, Addison-Wesley Publishing CompanyGoogle Scholar
  4. 4.
    Dyson F.J., Lieb E.H., Simon B.: Phase transitions in quantum spin systems with isotropic and nonisotropic interactions. J. Stat. Phys. 18, 335–383 (1978)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Giuliani A., Mastropietro V., Porta M.: Lattice quantum electrodynamics for graphene. Ann. Phys. 327, 461–511 (2012)MathSciNetADSCrossRefMATHGoogle Scholar
  6. 6.
    Giuliani A., Mastropietro V., Porta M.: Universality of conductivity in interacting graphene. Commun. Math. Phys. 311, 317–355 (2012)MathSciNetADSCrossRefMATHGoogle Scholar
  7. 7.
    Gross L.: On the formula of Mathews and Salam. J. Funct. Anal. 25, 162–209 (1977)CrossRefGoogle Scholar
  8. 8.
    Gutzwiller M.: Effect of correlation on the ferromagnetism of transition metals. Phys. Rev. Lett. 10, 159–162 (1963)ADSCrossRefGoogle Scholar
  9. 9.
    Hoegh-Krohn R.: Relativistic quantum statistical mechanics in two-dimensional space-time. Commun. Math. Phys. 38, 195–224 (1974)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Hubbard J.: Electron correlation in narrow energy bands. Proc. Roy. Soc. (Lond.) A 276, 238–257 (1963)ADSCrossRefGoogle Scholar
  11. 11.
    Kanamori J.: Electron correlation and ferromagnetism of transition metals. Prog. Theor. Phys. 30, 275–289 (1963)ADSCrossRefGoogle Scholar
  12. 12.
    Kubo K., Kishi T.: Rigorous bounds on the susceptibilities of the Hubbard model. Phys. Rev. B 41, 4866–4868 (1990)ADSCrossRefGoogle Scholar
  13. 13.
    Lieb E.H.: Two theorems on the Hubbard model. Phys. Rev. Lett. 62, 1201–1204 (1989)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Lieb E.H.: Flux phase of the half-filled band. Phys. Rev. Lett. 73, 2158–2161 (1994)ADSCrossRefGoogle Scholar
  15. 15.
    Lorinczi, J., Hiroshima, F., Betz, V.: Feynman-Kac-Type Theorems And Gibbs Measures On Path Space. With Applications To Rigorous Quantum Field Theory. de Gruyter Studies in Mathematics, vol. 34. Walter de Gruyter Co., Berlin (2011)Google Scholar
  16. 16.
    Miyao T.: Ground state properties of the SSH model. J. Stat. Phys. 149, 519–550 (2012)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Miyao, T.: Some rigorous results on the Holstein-Hubbard model, arXiv:1402.5202
  18. 18.
    Nagaoka Y.: Ferromagnetism in a narrow, almost half-filled s band. Phys. Rev. 147, 392–405 (1966)ADSCrossRefGoogle Scholar
  19. 19.
    Simon, B.: Trace Ideals and their Applications, 2nd edn. Mathematical Surveys and Monographs, vol. 120. American Mathematical Society, Providence, RI (2005)Google Scholar
  20. 20.
    Tasaki H.: From Nagaoka’s ferromagnetism to flat-band ferromagnetism and beyond : an introduction to ferromagnetism in the Hubbard model. Progr. Theor. Phys. 99, 489–548 (1998)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsHokkaido UniversitySapporoJapan

Personalised recommendations