Interface roughening in two-dimensional quasicrystals.

Abstract

The equilibrium fluctuations of interfaces in two-dimensional quasiperiodic lattices, such as Penrose tilings, are considered. By a transfer-matrix formulation, this problem is mapped to the one-dimensional Schr\"odinger equation with a quasiperiodic potential; from the scaling properties of the eigenstates near the band edge, one can extract the exponents \ensuremath{\zeta} characterizing the interface roughness. For a cosine potential, we find a true roughening transition in d=2. For a Fibonacci tiling, which approximates the Penrose tiling, we find that \ensuremath{\zeta} is nonuniversal, \ensuremath{\zeta}(1/2, and \ensuremath{\zeta}\ensuremath{\rightarrow}0 continuously as T\ensuremath{\rightarrow}0. .AE