Relevant Aperiodic Modulation in the 2 d Ising Model

We consider the surface critical behaviour of a semi-infinite two-dimensional layered Ising model, where the couplings perpendicular to the surface follow the aperiodic Rudin-Shapiro sequence. The model has unusual critical properties: depending on the strength of the modulation, at the bulk critical point the surface magnetization is either discontinuous or vanishes with an essential singularity. The critical surface magnetization as well as the form of the essential singularity are calculated exactly.