Critical properties of an aperiodic model for interacting polymers

Abstract

We investigate the effects of aperiodic interactions on the critical behaviour of an interacting two-polymer model on hierarchical lattices (equivalent to the Migadal–Kadanoff approximation for the model on Bravais lattices), via renormalization-group and transfer-matrix calculations. The exact renormalization-group recursion relations always present a symmetric fixed point, associated with the critical behaviour of the underlying uniform model. If the aperiodic interactions, defined by substitution rules, lead to relevant geometric fluctuations, this fixed point becomes fully unstable, giving rise to novel attractors of different nature. We present an explicit example in which this new attractor is a two-cycle attractor, with critical indices different from the uniform model. In the case of the four-letter Rudin–Shapiro substitution rule, we find a surprising closed curve whose points are attractors of period two, associated with a marginal operator. Nevertheless, a scaling analysis indicates that this attractor may lead to a new critical universality class. In order to provide an independent confirmation of the scaling results, we turn to a direct thermodynamic calculation of the specific-heat exponent. The thermodynamic free energy is obtained from a transfer-matrix formalism, which had been previously introduced for spin systems, and is now extended to the two-polymer model with aperiodic interactions.