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Vibrational spectrum on an exact fractal lattice

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  • Yu, Kin-Wah

Abstract

The vibrational properties were studied on the two-dimensional Sierpinski gasket. If one allows the mass on each site to move in two orthogonal directions, one needs to introduce a transverse coupling K in addition to the longitudinal couplings K1 and K2. Using the standard decimation procedures, one derives the rescaling properties of the couplings K1, K2 and K. One finds, if K is the geometric mean of K1, K2, then upon rescaling, the new coupling K' remains as the geometric mean of K'1 and K'2. By using the lattice Green's function and its rescaling properties, one is able to obtain the density of vibrational states as well as the correlation between two given sites. According to the correlation function, one finds there exists an energy dependent localization length ξ, such that when |r1−r2|>ξ, the ratios K/K1 and K/K2 go to constants upon rescaling. Also, in the limit of K = 0, the two orthogonal motions are completely decoupled and our results reduce to the case of one degree of freedom.

Suggested Citation

  • Yu, Kin-Wah, 1984. "Vibrational spectrum on an exact fractal lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 128(1), pages 307-317.
    • Handle: RePEc:eee:phsmap:v:128:y:1984:i:1:p:307-317
    DOI: 10.1016/0378-4371(84)90094-3
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