Abstract
We investigate one-dimensional lattice systems with (symmetric) nearest neighbor transfer ratesW n, n+1 which are independently distributed according to a probability densityρ(w). For two general classes ofρ(w), we rigorously determine the asymptotic behavior of the relevant single site Green function 〈\(\tilde P\) 0(ω)〉 nearω=0, and obtain exact results for the long time decay of the initial probability amplitude and for the low energy density of states. A scaling hypothesis, accurately confirmed by computer simulations, is used to relate the low frequency hopping conductivityσ(ω) uniquely to 〈\(\tilde P\) 0(−iω)〉, and we conjecture that the resulting asymptotic behavior forσ(ω) is also exact. The critical exponents associated with the various asymptotic laws depend onρ(w) and show a crossover from universal to non-universal behavior. Comparison is made with the results of several approximate treatments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bernasconi, J., Alexander, S., Orbach, R.: Phys. Rev. Lett.41, 185 (1978)
Bernasconi, J., Beyeler, H.U., Strässler, S., Alexander, S.: Phys. Rev. Lett.42, 819 (1979)
Alexander, S., Bernasconi, J.: J. Phys. C12, L1 (1979)
See, e.g.: Elliott, R.J., Krumhansl, J.A., Leath, P.L.: Rev. Mod. Phys.46, 465 (1974)
Scher, H., Lax, M.: Phys. Rev. B7, 4491 (1973)
Domb, C., Maradudin, A.A., Montroll, E.W., Weiss, G.H.: Phys. Rev.115, 24 (1959) Rice, M.J., Bernasconi, J.: J. Phys. F3, 55 (1973) Alexander, S., Bernasconi, J., Orbach, R.: Phys. Rev. B17, 4311 (1978)
Theodorou, G., Cohen, M.H.: Phys. Rev. B19, 1561 (1979)
Böttger, H., Bryksin, V.V., Yashin, G.Yu.: J. Phys. C12, 2797 (1979)
Fox, C.: Trans. Amer. Math. Soc.98, 395 (1961)
Gupta, K.C., Jain, U.C.: Proc. Nat. Inst. Sci. India A36, 594 (1966) Raina, R.K., Koul, C.L.: Proc. Amer. Math. Soc.73, 188 (1979)
Doetsch, G.: Handbuch der Laplace-Transformation, pp. 485–504. Basel-Stuttgart: Birkhäuser 1971
Kirkpatrick, S.: Rev. Mod. Phys.45, 574 (1973)
Richards, P.M., Renken, R.L.: to be published
Bernasconi, J., Schneider, W.R., Wyss, W.: to be published
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. New York: Dover 1965
Miller, K.S.: Linear Difference Equations. New York-Amsterdam: Benjamin 1968
Braaksma, B.L.J.: Compos. Math.15, 239 (1963)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bernasconi, J., Schneider, W.R. & Wyss, W. Diffusion and hopping conductivity in disordered one-dimensional lattice systems. Z Physik B 37, 175–184 (1980). https://doi.org/10.1007/BF01365374
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01365374