Crossover between aperiodic and homogeneous semi-infinite critical behaviors in multilayered two-dimensional Ising models

We investigate the surface critical behavior of two-dimensional multilayered aperiodic Ising models in the extreme anisotropic limit. The system under consideration is obtained by piling up two types of layers with respectively $p$ and $q$ spin rows coupled via nearest neighbor interactions $\lambda r$ and $\lambda$, where the succession of layers follows an aperiodic sequence. Far away from the critical regime, the correlation length $\xi_\perp$ is smaller than the first layer width and the system exhibits the usual behavior of an ordinary surface transition. In the other limit, in the neighborhood of the critical point, $\xi_\perp$ diverges and the fluctuations are sensitive to the non-periodic structure of the system so that the critical behavior is governed by a new fixed point. We determine the critical exponent associated to the surface magnetization at the aperiodic critical point and show that the expected crossover between the two regimes is well described by a scaling function. From numerical calculations, the parallel correlation length $\xi_\parallel$ is then found to behave with an anisotropy exponent $z$ which depends on the aperiodic modulation and the layer widths.