Dynamical properties of quasi-crystals: Fibonacci chain and Penrose lattice

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Abstract
The discrete scale invariance of quasi-periodic systems leads to a scaling relationship Omega =L-zf(L) between a characteristic dynamic variable Omega and length L, where f(L) is a periodic function of ln L. The dynamic exponent z can be calculated for diffusion, spin wave and phonon dynamics (where Omega is respectively Gamma , omega , omega 2 where Gamma and omega are characteristic rates or frequencies respectively) by exploiting a crossover argument which results in z=df+t where df is the fractal dimension and t is the length scaling exponent for the conductance. The latter exponents and hence z, are calculated for a Fibonacci chain model ((df,t,z)=(1,1,2)) and, via an exact bond moving technique, for the Penrose lattice ((df,t,z)=(2,0,2)). The technique also provides the thermal exponent of the Heisenberg spin model on the two quasi-crystals ( nu =1, infinity , respectively).

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