Density of states and dynamic scaling on the Vicsek snowflake fractal

Abstract

Real-space rescaling techniques have been employed to obtain recursion relations for the parameters characterising the dynamics of both the Heisenberg ferromagnet and antiferromagnet on the infinite Vicsek snowflake fractal. The authors have calculated the associated dynamic exponents zF and zA by linearising about the respective fixed points of the ferromagnetic/antiferromagnetic recursion relations and find them to satisfy the relation zA=zF/2. In addition, using a functional integral method both the ferromagnetic and antiferromagnetic density of states rho F( omega ) and rho A( omega ) have been calculated exactly and the resulting spectra were found to satisfy the scaling equation rho alpha ( omega ) approximately bz alpha -df rho ( lambda alpha omega ) near their respective fixed points ( alpha =A,F), df denoting the fractal dimension. Finally, the amplitudes galpha ( omega ) appearing in the solution to the scaling equation rho alpha ( omega ) approximately galpha ( omega ) omega dfz alpha -1/, for the density of states rho alpha ( omega ), were found to be periodic functions of ln( omega ) with period ln( lambda alpha ) where lambda alpha is the eigenvalue associated with ferromagnetic ( alpha =F) or antiferromagnetic ( alpha =A) fixed point.