Skip to main content
SpringerLink
Log in

Lifshitz singularities in random harmonic chains: Periodic amplitudes near the band edge and near special frequencies

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We give a complete description of the scaling behavior of the integrated density of states of random harmonic chains with random masses near the band edgeω max and near special frequenciesω s. There are four different situations:ω ↑ω max,ω ↓ω s,ω ↑ω s (critical case),ω ↑ω s (general case). Our analytic results have the form of infinite sums involving Fourier coefficients of the scaling behavior of the Dyson-Schmidt functionat the special frequency or the band edge. Binary mass distributions are considered in detail in the limit of a small fractionp of light masses. Our predictions are compared with extensive numerical data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. M. Lifshitz,Adv. Phys. 13:483 (1964).

    Google Scholar 

  2. C. de Calan, J. M. Luck, Th. M. Nieuwenhuizen, and D. Petritis,J. Phys. A 18:501 (1985).

    Google Scholar 

  3. Th. M. Nieuwenhuizen and J. M. Luck,J. Stat. Phys. 41:745 (1985).

    Google Scholar 

  4. D. Bessis, J. S. Geronimo, and P. Moussa,J. Stat. Phys. 34:75 (1984); B. Derrida, C. Itzykson, and J. M. Luck,Commun. Math. Phys. 94:115 (1984).

    Google Scholar 

  5. Th. M. Nieuwenhuizen, J. M. Luck, J. Canisius, J. L. van Hemmen, and W. J. Ventevogel,J. Stat. Phys. 45:395 (1986).

    Google Scholar 

  6. M. Endrullis and H. Englisch,Commun. Math. Phys. 108:591 (1987).

    Google Scholar 

  7. F. J. Dyson,Phys. Rev. 92:1331 (1953).

    Google Scholar 

  8. H. Schmidt,Phys. Rev. 105:425 (1957).

    Google Scholar 

  9. C. Domb, A. A. Maradudin, E. W. Montroll, and G. W. Weiss,Phys. Rev. 115:24 (1959).

    Google Scholar 

  10. Th. M. Nieuwenhuizen and J. M. Luck, to appear inPhysica A (1987).

  11. Th. M. Nieuwenhuizen,Physica 125A:197 (1984).

    Google Scholar 

  12. Th. M. Nieuwenhuizen,Phys. Lett. 103A:333 (1984).

    Google Scholar 

  13. H. Matsuda,Prog. Theor. Phys. 31:161 (1964).

    Google Scholar 

  14. J. Hori, inSpectral Properties of Disordered Lattices, D. ter Haar, ed. (Pergamon, Oxford, 1968).

    Google Scholar 

  15. B. I. Halperin,Phys. Rev. 139:A104 (1965).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik A, RWTH Aachen, 5100, Aachen, West Germany

    Th. M. Nieuwenhuizen

  2. Service de Physique Théorique, CEN Saclay, 91191, Gif-sur-Yvette, France

    J. M. Luck

Authors
  1. Th. M. Nieuwenhuizen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. M. Luck
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Nieuwenhuizen, T.M., Luck, J.M. Lifshitz singularities in random harmonic chains: Periodic amplitudes near the band edge and near special frequencies. J Stat Phys 48, 393–424 (1987). https://doi.org/10.1007/BF01019680

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01019680

Key words

Search

Navigation