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Frequency Dependent Conductivity of the Fibonacci-Chain

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Abstract

A real-space-renormalization method for the frequency dependent conductivity of the periodic approximants of the Fibonacci chain is developed. This scheme is based on the known 2×2 transfer matrices and additional 5×5 matrices which allow an efficient numerical evaluation of the Kubo formula. Numerical results are presented.

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REFERENCES

  1. For an introduction see, e.g., C. Janot: Quasicrystals: A Primer, Clarendon Press (1994); F. Axel and D. Gratias, eds., Beyond Quasicrystals, Springer (1995).

  2. E.g., F. Hippert und D. Gratias, eds., Lectures on Quasicrystals, Les Éditions de Physique, Les Ulis (France) (1994); Z. M. Stadnik, ed., Physical Properties of Quasicrystals, Springer Series in Solid State Sciences 126, Springer (1999).

  3. T. Fujiwara, Phys. Rev. B 40, 942 (1989); J. Hafner and M. Krajčί, Phys. Rev. B 47, 11 (1993); T. Fujiwara, S. Yamamoto, and G. T. de Laissardière, Phys. Rev. Lett. 71, 4166 (1993); D. Mayou, C. Berger, F. Cyrot-Lackmann, T. Klein, and P. Lanco, Phys. Rev. Lett. 70, 3915 (1993).

    Google Scholar 

  4. E. S. Zijlstra and T. Janssen, Phys. Rev. B 61, 3377 (2000); E. S. Zijlstra and T. Janssen, Europhys. Lett. 52, 578 (2000).

    Google Scholar 

  5. B. Iochum and D. Testard, J. Stat. Phys. 65, 715 (1991); W. Salejda and P. Szyszuk, Physica A 252, 547 (1998); B. Sutherland and M. Kohmoto, Phys. Rev. B 36, 5877 (1987).

    Google Scholar 

  6. S. I. Ben-Abraham and A. Joseph, Proceedings of the 5th Int. Conf. on Quasicrystals, C. Janot and R. Mosseri, eds., World Scientific (1995), p. 621; A. Lahiri, Phys. Rev. B 53, 3702 (1996); S. Roche and D. Mayou, Phys. Rev. Lett. 79, 2518 (1997).

  7. D. Walther and R. v. Baltz, Phys. Rev. B 55, 8852 (1997).

    Google Scholar 

  8. Leonardo Pisano (1170–1250), (named Fibonacci, from filius Bonacci, Liber abbaci (1202). For a historical review see: http://www-groups.dcs.stand.ac.uk/~history/Mathematicians/Fibonacci.html

  9. E.g., C. Brezinski, Padé-Type Approximation and General Orthogonal Polynominals., Birkhäuser-Verlag (1980).

  10. H. Weyl, Ann. Math. 68, 220 (1910); E. C. Titchmarsh: “Eigenfunction Expansions Associated with Second-Order Differential Equations”, Oxford, University Press (1946).

    Google Scholar 

  11. Natl. Bur. Stand. Appl. Math. Ser. No 55, edited by M. Abramowitz and I. A. Stegun, (U.S. GPO, Washington, DC) Handbook of Mathematical Functions, Dover (1968).

    Google Scholar 

  12. E.g., A. Douglas Stone und Aaron Szafer, IBM J. Dev. 32, 384 (1988); H. U. Baranger und A. D. Stone, Phys. Rev. B40, 8169 (1989).

    Google Scholar 

  13. M. Kohmoto, L.P. Kadanoff, and C. Tang, Phys. Rev. Lett. 50, 1870 (1983); S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, and E. Siggia, Phys. Rev. Lett. 50, 1873 (1983).

    Google Scholar 

  14. See the Nielsen-Graph, Fig. 5, of R. v. Baltz, Phys. Rev. B 55, 8852 (1997) Ref. 7. Note, the sequence of substitutions L, R is reversed with respect to the NTs L, R. Here, the path which connects α 0 and σ (k)1 = β (k)1 provides the sequence of NTs of the RG.

    Google Scholar 

  15. D. H. Bailey, Multiprecision Translation and Execution of Fortran Programs, ACM Transactions on Mathematical Software, 19,no. 3, Sept. 1993, p. 288–319.

    Google Scholar 

  16. D. Walther, Dissertation, Universität Karlsruhe, Nov. 2001.

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Walther, D., Baltz, R.v. Frequency Dependent Conductivity of the Fibonacci-Chain. Journal of Low Temperature Physics 126, 1211–1220 (2002). https://doi.org/10.1023/A:1013831716744

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