Abstract
The density of states of two-dimensional electrons in a strong perpendicular magnetic field and white-noise potential is calculated exactly under the provision that only the states of the free electrons in the lowest Landau level are taken into account. It is used that the integral over the coordinates in the plane perpendicular to the magnetic field in a Feynman graph yields the inverse of the number λ of Euler trails through the graph, whereas the weight by which a Feynman graph contributes in this disordered system is λ times that of the corresponding interacting system. Thus the factors λ cancel which allows the reduction of thed dimensional disordered problem to a (d-2) dimensional φ4 interaction problem. The inverse procedure and the equivalence of disordered harmonic systems with interacting systems of superfields is used to give a mapping of interacting systems withU(1) invariance ind dimensions to interacting systems with UPL(1,1) invariance in (d+2) dimensions. The partition function of the new systems is unity so that systems with quenched disorder can be treated by averaging exp(−H) without recourse to the replica trick.
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References
Ando, T., Matsumoto, Y., Uemura, Y.: J. Phys. Soc. Jpn.39, 279 (1975)
von Klitzing, K., Dorda, G., Pepper, M.: Phys. Rev. Lett.45, 494 (1980)
Ando, T.: Anderson localisation. In: Springer Series in Solid-State Sciences. Nagaoka, Y., Fukuyama, H. (eds.) Vol. 39, p. 176, Berlin, Heidelberg, New York: Springer 1982
Thouless, D.J., p. 191,
Imry, Y.:, p. 198,
Ono, Y.: p. 207,
Ando, T.: J. Phys. Soc. Jpn.37, 622 (1974)
Lloyd, P., J. Phys. C2, 1717 (1969)
Euler, L.: Comm. Acad. Sci. Petropolitanae8, 128 (1736); Opera OmniaI-7, 1 (1766)
Kasteleyn, P.W.: In: Graph theory and theoretical physics. Harary, F. (ed.), p. 44. New York, London: Academic Press 1967
Harary, F., Graph theory, p. 204. Reading: Addison-Wesley 1969
Harary, F., Palmer, E.M.: Graphical enumeration. p. 25. New York, London: Academic Press 1973
McKane, A.J.: Phys. Lett.76A, 22 (1980)
Efetov, K.B.: Zh. Eksp. Teor. Fiz.82, 872 (1982); Sov. Phys. JETP55, 514 (1982)
Ziegler, K.: Z. Phys. B-Condensed Matter48, 293 (1982)
Wegner, F.: Z. Phys. B-Condensed Matter49, 297 (1983)
Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities. Cambridge: Cambridge University Press 1934
Kac, M.: Ark. Fys. Semin. Trondheim11, 1 (1968)
Lin, T.F.: J. Math. Phys.11, 1584 (1970)
Edwards, S.F.: In: Proceedings of the Third International Conference on Amorphous Materials 1970. Douglas, R.W., Ellis, B. (eds.). New York: Wiley 1972
Imry, Y., Mas, S.K.: Phys. Rev. Lett.35, 1399 (1975)
Young, A.P.: J. Phys. C10, L257 (1977)
Parisi, G., Sourlas, N.: Phys. Rev. Lett.43, 744 (1979)
Smith, C.A.B., Tutte, W.T.: Am. Math. Monthly48, 233 (1941)
Rittenberg, V., Scheunert, M.: J. Math. Phys.19, 709 (1978)
van Aardenne-Ehrenfest, T., de Bruijn, N.G.: Simon Stevin Wis. Natuurkd. Tijdschr.28, 203 (1951)
Matsubara, T.: Prog. Theor. Phys.14, 351 (1955)
Abrikosov, A.A., Gorkev, L.P., Dzyaloshinski, I.E.: Zh. Eksp. Teor. Fiz.36, 900 (1959); Sov. Phys. JETP9, 636 (1959)
Fradkin, E.S.: Zh. Teor. Fiz.36, 1286 (1959); Sov. Phys. JETP9, 912 (1959)
Abrikosov, A.A., Gorkov, L.P., Dzyaloskinski, I.E.: Methods of quantum field theory in statistical physics. Englewood Cliffs: Prentice Hall 1964
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Supported in part by the DFG through SFB123 “Stochastic Mathematical Models”
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Wegner, F. Exact density of states for lowest Landau level in white noise potential superfield representation for interacting systems. Z. Physik B - Condensed Matter 51, 279–285 (1983). https://doi.org/10.1007/BF01319209
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DOI: https://doi.org/10.1007/BF01319209