Advertisement

Communications in Mathematical Physics

, Volume 349, Issue 3, pp 1107–1161 | Cite as

Universality of the Hall Conductivity in Interacting Electron Systems

  • Alessandro Giuliani
  • Vieri Mastropietro
  • Marcello PortaEmail author
Article

Abstract

We prove the quantization of the Hall conductivity for general weakly interacting gapped fermionic systems on two-dimensional periodic lattices. The proof is based on fermionic cluster expansion techniques combined with lattice Ward identities, and on a reconstruction theorem that allows us to compute the Kubo conductivity as the analytic continuation of its imaginary time counterpart.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agazzi A., Eckmann J.-P., Graf G.M.: The colored Hofstadter butterfly for the honeycomb lattice. J. Stat. Phys. 156, 417–426 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aizenman M., Graf G.M.: Localization bounds for an electron gas. J. Phys. A: Math. Gen. 31, 6783 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Araki H.: Mathematical Theory of Quantum Fields. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
  4. 4.
    Avron J.E., Seiler R., Simon B.: Homotopy and quantization in condensed matter physics. Phys. Rev. Lett. 51, 51 (1983)ADSCrossRefGoogle Scholar
  5. 5.
    Avron, J., Seiler, R.: Why is the Hall conductance quantized? In: Open Problems in Mathematical Physics. Available at http://web.math.princeton.edu/~aizenman/OpenProblems.iamp
  6. 6.
    Avron J.E., Seiler R., Simon B.: Charge deficiency, charge transport and comparison of dimensions. Commun. Math. Phys. 159, 399–422 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Battle G.A., Federbush P.: A note on cluster expansions, tree graph identities, extra 1/N! factors!!!. Lett. Math. Phys. 8, 55–57 (1984)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Bellissard, J., van Els, A., Schulz-Baldes, H.: The non-commutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373 (1994)Google Scholar
  9. 9.
    Benfatto G., Mastropietro V.: On the density-density critical indices in interacting Fermi systems. Commun. Math. Phys. 231, 97–134 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Benfatto G., Mastropietro V.: Ward identities and chiral anomaly in the Luttinger liquid. Commun. Math. Phys. 258, 609–655 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Benfatto G., Mastropietro V.: Universality relations in non-solvable quantum spin chains. J. Stat. Phys. 138, 1084–1108 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Benfatto G., Falco P., Mastropietro V.: Universal relations for non solvable statistical models. Phys. Rev. Lett. 104, 075701 (2010)ADSCrossRefGoogle Scholar
  13. 13.
    Benfatto G., Falco P., Mastropietro V.: Universality of one-dimensional Fermi systems, I. Response functions and critical exponents. Commun. Math. Phys. 330, 153–215 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Benfatto G., Falco P., Mastropietro V.: Universality of one-dimensional Fermi systems, II. The Luttinger liquid structure. Commun. Math. Phys. 330, 217–282 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Benfatto G., Gallavotti G., Procacci A., Scoppola B.: Beta function and Schwinger functions for a many fermions system in one dimension. Anomaly of the Fermi surface. Commun. Math. Phys. 160, 93–171 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bieri S., Fröhlich J.: Physical principles underlying the quantum Hall effect. Compt. Rend. Phys. 12, 332–346 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    Bishop M., Nachtergaele B., Young A.: Spectral gap and edge excitations of d-dimensional PVBS models on half-spaces. J. Stat. Phys. 162(6), 1485–1521 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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
  19. 19.
    Bravyi S., Hastings M.B., Michalakis S.: Topological quantum order: stability under local perturbations. J. Math. Phys. 51, 093512 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bru, J.B., de S. Pedra,W.A.: Lieb-Robinson Bounds forMulti-Commutators andApplications to Response Theory. Springer Briefs in Mathematical Physics, vol. 13. Springer (2016)Google Scholar
  21. 21.
    Bru, J.B., de S. Pedra, W.A.: Universal bounds for large determinants from non-commutative Hölder inequalities in fermionic constructive quantum field theory. Preprint mp_arc 16-16 Google Scholar
  22. 22.
    Brydges, D.C.: A short course on cluster expansions. In: Phénomènes critiques, systèmes aléatoires, théories de jauge (Les Houches, 1984), pp. 129–183. North-Holland, Amsterdam (1986)Google Scholar
  23. 23.
    Brydges D.C., Federbush P.: A new form of the Mayer expansion in classical statistical mechanics. J. Math. Phys. 19, 2064–2067 (1978)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Coleman S., Hill B.: No more corrections to the topological mass term in QED3. Phys. Lett. B. 159, 184 (1985)ADSCrossRefGoogle Scholar
  25. 25.
    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 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Datta N., Fernández R., Fröhlich J., Rey-Bellet L.: Low-temperature phase diagrams of quantum lattice systems. II. Convergent perturbation expansions and stability in systems with infinite degeneracy. Helv. Phys. Acta. 69, 752 (1996)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Fröhlich J., Kerler T.: Universality in quantum Hall systems. Nucl. Phys. B. 354, 369–417 (1991)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Fröhlich J., Studer U.M.: Gauge invariance and current algebra in nonrelativistic many-body theory. Rev. Mod. Phys. 65, 733 (1993)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Fröhlich, J., Studer, U.M., Thiran, E.: Quantum Theory of Large Systems of Non-relativistic Matter. Les Houches Lectures 1994, Elsevier, New York (1995). arXiv:cond-mat/9508062
  30. 30.
    Fröhlich J., Zee A.: Large scale physics of the quantum Hall fluid. Nucl. Phys. B. 364, 517–540 (1991)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Gallavotti G.: Renormalization group and ultraviolet stability for scalar fields via renormalization group methods. Rev. Mod. Phys. 57, 471–562 (1985)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Gallavotti G., Nicolò F.: Renormalization theory for four dimensional scalar fields, Part I. Commun. Math. Phys. 100, 545–590 (1985)ADSCrossRefGoogle Scholar
  33. 33.
    Gallavotti G., Nicolò F.: Renormalization theory for four dimensional scalar fields, Part II. Commun. Math. Phys. 101, 471–562 (1985)CrossRefGoogle Scholar
  34. 34.
    Gentile G., Mastropietro V.: Renormalization group for one-dimensional fermions. A review on mathematical results. Phys. Rep. 352(4), 273–437 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Giuliani, A.: The ground state construction of the two-dimensional Hubbard model on the honeycomb lattice. In: Quantum Theory from Small to Large Scales. Lecture Notes of the Les Houches Summer School, vol. 95 (August 2010)Google Scholar
  36. 36.
    Giuliani A., Mastropietro V.: The 2D Hubbard model on the honeycomb lattice. Commun. Math. Phys. 293, 301–346 (2010)ADSCrossRefzbMATHGoogle Scholar
  37. 37.
    Giuliani A., Mastropietro V., Porta M.: Universality of conductivity in interacting graphene. Commun. Math. Phys. 311, 317–355 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Giuliani A., Mastropietro V., Porta M.: Absence of interaction corrections in the optical conductivity of graphene. Phys. Rev. B 83, 195401 (2011)ADSCrossRefGoogle Scholar
  39. 39.
    Haldane F.D.M.: Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett. 61, 2015 (1988)ADSCrossRefGoogle Scholar
  40. 40.
    Hastings M.B., Michalakis S.: Quantization of Hall conductance for interacting electrons on a torus. Commun. Math. Phys. 334, 433–471 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Hofstadter D.R.: Energy levels and wavefunctions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249 (1976)ADSCrossRefGoogle Scholar
  42. 42.
    Ishikawa K., Matsuyama T.: Magnetic field induced multi-component QED3 and quantum Hall effect. Z. Phys C. 33, 41–45 (1986)ADSCrossRefGoogle Scholar
  43. 43.
    Jotzu G. et al.: Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014)ADSCrossRefGoogle Scholar
  44. 44.
    Katsura H., Koma T.: The \({\mathbb{Z}_{2}}\) index of disordered topological insulators with time reversal symmetry. J. Math. Phys. 57, 021903 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Kubo R.: Statistical-mechanical theory of irreversible processes, I. J. Phys. Soc. Jpn. 12, 570–586 (1957)ADSCrossRefGoogle Scholar
  46. 46.
    Lieb E.H., Robinson D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    Mahan G.D.: Many-Particle Physics, 3rd edn. Kluwer/Plenum, New York (2010)Google Scholar
  48. 48.
    Mastropietro V.: Non-perturbative Renormalization. World Scientific, Singapore (2008)CrossRefzbMATHGoogle Scholar
  49. 49.
    Michalakis S., Zwolak J.P.: Stability of frustration-free Hamiltonians. Commun. Math. Phys. 322, 277–302 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Nachtergaele B., Ogata Y., Sims R.: Propagation of correlations in quantum lattice systems. J. Stat. Phys. 124, 1–13 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Nachtergaele B., Sims R.: Lieb–Robinson bounds in quantum many-body physics. Contemp. Math. 529, 141–176 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    de S. Pedra W.A., Salmhofer M.: Determinant bounds and the Matsubara UV problem of many-fermion systems. Commun. Math. Phys. 282, 797–818 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Stauber T., Peres N.M.R., Geim A.K.: Optical conductivity of graphene in the visible region of the spectrum. Phys. Rev. B 78, 085432 (2008)ADSCrossRefGoogle Scholar
  54. 54.
    Thouless D.J., Kohmoto M., Nightingale M.P., den Nijs M.: Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405 (1982)ADSCrossRefGoogle Scholar
  55. 55.
    Varney C.N., Sun K., Rigol M., Galitski V.: Topological phase transitions for interacting finite systems. Phys. Rev. B. 84, 241105 (2011)ADSCrossRefGoogle Scholar
  56. 56.
    von Klitzing, K., Dorda, G., Pepper, M.: New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45, 494 (1980)Google Scholar
  57. 57.
    Wen X.G.: Chiral Luttinger liquid and the edge excitations in the fractional quantum Hall states. Phys. Rev. B. 41, 12838–12844 (1990)ADSCrossRefGoogle Scholar
  58. 58.
    Zhang S.-C.: The Chern–Simons–Landau–Ginzburg theory of the fractional quantum Hall effect. Int. J. Mod. Phys. B. 6, 25–58 (1992)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Alessandro Giuliani
    • 1
  • Vieri Mastropietro
    • 2
  • Marcello Porta
    • 3
    Email author
  1. 1.Department of Mathematics and PhysicsUniversity of Roma TreRomeItaly
  2. 2.Department of Mathematics “F. Enriquez”University of MilanoMilanItaly
  3. 3.Mathematics DepartmentUniversity of ZürichZurichSwitzerland

Personalised recommendations