Letters in Mathematical Physics

, Volume 75, Issue 1, pp 25–37 | Cite as

Spontaneous Edge Currents for the Dirac Equation in Two Space Dimensions

  • Michael J. Gruber
  • Marianne LeitnerEmail author


Spontaneous edge currents are known to occur in systems of two space dimensions in a strong magnetic field. The latter creates chirality and determines the direction of the currents. Here we show that an analogous effect occurs in a field-free situation when time reversal symmetry is broken by the mass term of the Dirac equation in two space dimensions. On a half plane, one sees explicitly that the strength of the edge current is proportional to the difference between the chemical potentials at the edge and in the bulk, so that the effect is analogous to the Hall effect, but with an internal potential. The edge conductivity differs from the bulk (Hall) conductivity on the whole plane. This results from the dependence of the edge conductivity on the choice of a selfadjoint extension of the Dirac Hamiltonian. The invariance of the edge conductivity with respect to small perturbations is studied in this example by topological techniques


Dirac operator boundary condition Hall effect spectral flow 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akkermans E., Avron J., Narevich R. Seiler R. (1998). Boundary conditions for bulk and edge states in quantum Hall systems. Eur. Phys. J. B 1(1):117–121CrossRefADSGoogle Scholar
  2. 2.
    Avron J., Seiler R. (1985). Quantization of the Hall conductance for general, multiparticle Schrödinger Hamiltonians. Phys. Rev. Lett. 54(4):259–262CrossRefADSMathSciNetGoogle Scholar
  3. 3.
    Avron J., Seiler R., Shapiro B. (1986). Generic properties of quantum Hall Hamiltonians for finite systems. Nuclear Phys. B 265(FS15):364–374CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Bellissard J., van Elst A., Schulz-Baldes H. (1994). The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35(10):5373–5451zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Elbau P., Graf G. (2002). Equality of bulk and edge Hall conductance revisited. Comm. Math. Phys. 229:415–432zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Fröhlich J., Kerler T. (1991). Universality in quantum Hall systems. Nuclear Phys. B 354:369–417CrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Haldane F. (1988). Model for a quantum Hall effect without Landau levels: condensed-matter realization of the parity anomaly. Phys. Rev. Lett. 61:2015–2018CrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Halperin B.I. (1982). Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential. Phys. Rev. B 25(40):2185–2190CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Hatsugai Y. (1993). Chern number and edge states in the integer quantum Hall-effect. Phys. Rev. Lett. 71(22):3697–3700zbMATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Hatsugai Y. (1993). Edge states in the integer quantum Hall-effect and the Riemann surface of the Bloch function. Phys. Rev. B 48(16):11851–11862CrossRefADSGoogle Scholar
  11. 11.
    Kellendonk J., Richter T., Schulz-Baldes H. (2002). Edge current channels and Chern numbers in the integer quantum Hall effect. Rev. Math. Phys. 14(1):87–119zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kohmoto M. (1985). Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160:343–354CrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Laughlin R.B. (1981). Quantized Hall conductivity in two dimensions. Phys. Phys. B 23(10):5632–5633ADSGoogle Scholar
  14. 14.
    Leitner M. (2004). Zero Field Hall-Effekt für Teilchen mit Spin 1/2, vol 5 of Augsburger Schriften zur Mathematik, Physik und Informatik. Logos-Verlag, BerlinGoogle Scholar
  15. 15.
    Leitner, M.: Cond-mat/0505428 (2005)Google Scholar
  16. 16.
    Ludwig A., Fisher M., Shankar R., Grinstein G. (1994). Integer quantum Hall transition: an alternative approach and exact results. Phys. Rev. B 50:7526–7552CrossRefADSGoogle Scholar
  17. 17.
    Redlich A. (1984). Parity violation and gauge invariance of the effective gauge field action in three dimensions. Phys. Rev. D 29(10):2366–2374CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Reed M., Simon B. (1975). Fourier analysis, self-adjointness, vol II Methods of modern mathematical physics. Academic Press, New YorkGoogle Scholar
  19. 19.
    Reed M., Simon B. (1978). Analysis of operators, vol. IV Methods of modern mathematical physics. Academic Press, New YorkGoogle Scholar
  20. 20.
    Schulz-Baldes H., Kellendonk J., Richter T. (2000). Simultaneous quantization of edge and bulk Hall conductivity. J. Phys. A: Math. Gen. 33:L27–L32zbMATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Semenoff G. (1984). Condensed-matter simulation of a three-dimensional anomaly. Phys. Rev. Lett. 53:2449–2452CrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Thouless D., Kohmoto M., Nightingale M., de Nijs M. (1982). Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49:405–408CrossRefADSGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Institut für Physik, Theoretische Physik IIUniversität AugsburgAugsburgGermany
  2. 2.School of Theoretical PhysicsDublin Institute for Advanced StudiesDublinIreland

Personalised recommendations