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Spontaneous Edge Currents for the Dirac Equation in Two Space Dimensions

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Abstract

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

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References

  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):117–121

    Article  ADS  Google Scholar 

  2. Avron J., Seiler R. (1985). Quantization of the Hall conductance for general, multiparticle Schrödinger Hamiltonians. Phys. Rev. Lett. 54(4):259–262

    Article  ADS  MathSciNet  Google Scholar 

  3. Avron J., Seiler R., Shapiro B. (1986). Generic properties of quantum Hall Hamiltonians for finite systems. Nuclear Phys. B 265(FS15):364–374

    Article  ADS  MathSciNet  Google Scholar 

  4. Bellissard J., van Elst A., Schulz-Baldes H. (1994). The noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35(10):5373–5451

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Elbau P., Graf G. (2002). Equality of bulk and edge Hall conductance revisited. Comm. Math. Phys. 229:415–432

    Article  MATH  ADS  MathSciNet  Google Scholar 

  6. Fröhlich J., Kerler T. (1991). Universality in quantum Hall systems. Nuclear Phys. B 354:369–417

    Article  ADS  MathSciNet  Google Scholar 

  7. Haldane F. (1988). Model for a quantum Hall effect without Landau levels: condensed-matter realization of the parity anomaly. Phys. Rev. Lett. 61:2015–2018

    Article  ADS  MathSciNet  Google Scholar 

  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):2185–2190

    Article  ADS  MathSciNet  Google Scholar 

  9. Hatsugai Y. (1993). Chern number and edge states in the integer quantum Hall-effect. Phys. Rev. Lett. 71(22):3697–3700

    Article  MATH  ADS  MathSciNet  Google Scholar 

  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):11851–11862

    Article  ADS  Google Scholar 

  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):87–119

    Article  MATH  MathSciNet  Google Scholar 

  12. Kohmoto M. (1985). Topological invariant and the quantization of the Hall conductance. Ann. Phys. 160:343–354

    Article  ADS  MathSciNet  Google Scholar 

  13. Laughlin R.B. (1981). Quantized Hall conductivity in two dimensions. Phys. Phys. B 23(10):5632–5633

    ADS  Google Scholar 

  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, Berlin

    Google Scholar 

  15. Leitner, M.: Cond-mat/0505428 (2005)

  16. Ludwig A., Fisher M., Shankar R., Grinstein G. (1994). Integer quantum Hall transition: an alternative approach and exact results. Phys. Rev. B 50:7526–7552

    Article  ADS  Google Scholar 

  17. Redlich A. (1984). Parity violation and gauge invariance of the effective gauge field action in three dimensions. Phys. Rev. D 29(10):2366–2374

    Article  ADS  MathSciNet  Google Scholar 

  18. Reed M., Simon B. (1975). Fourier analysis, self-adjointness, vol II Methods of modern mathematical physics. Academic Press, New York

    Google Scholar 

  19. Reed M., Simon B. (1978). Analysis of operators, vol. IV Methods of modern mathematical physics. Academic Press, New York

    Google Scholar 

  20. Schulz-Baldes H., Kellendonk J., Richter T. (2000). Simultaneous quantization of edge and bulk Hall conductivity. J. Phys. A: Math. Gen. 33:L27–L32

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. Semenoff G. (1984). Condensed-matter simulation of a three-dimensional anomaly. Phys. Rev. Lett. 53:2449–2452

    Article  ADS  MathSciNet  Google Scholar 

  22. Thouless D., Kohmoto M., Nightingale M., de Nijs M. (1982). Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49:405–408

    Article  ADS  Google Scholar 

Download references

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Correspondence to Marianne Leitner.

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Mathematics Subject Classification (2000). 81Q10, 58J32

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Gruber, M.J., Leitner, M. Spontaneous Edge Currents for the Dirac Equation in Two Space Dimensions. Lett Math Phys 75, 25–37 (2006). https://doi.org/10.1007/s11005-005-0036-4

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  • DOI: https://doi.org/10.1007/s11005-005-0036-4

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