Marginal Anisotropy in Layered Aperiodic Ising Systems

Abstract

Two-dimensional layered aperiodic Ising systems are studied in the extreme anisotropic limit where they correspond to quantum Ising chains in a transverse field. The modulation of the couplings follows an aperiodic sequence generated through substitution. According to Luck's criterion, such a perturbation becomes marginal when the wandering exponent of the sequence vanishes. Three marginal sequences are considered: the period-doubling, paper-folding and three-folding sequences. They correspond to bulk perturbations for which the critical temperature is shifted. The surface magnetization is obtained exactly for the three sequences. The scaling dimensions of the local magnetization on both surfaces vary continuously with the modulation factor. The low-energy excitations of the quantum chains are found to scale as L^z with the size L of the system. This is the behaviour expected for a strongly anisotropic system, where z is the ratio of the exponents of the correlation lengths in the two directions. The anisotropy exponent z is equal to the sum of the scaling dimensions of the local magnetization on the two surfaces. The anisotropic scaling behaviour is verified numerically for other surface and bulk critical properties as well.