The surface critical behaviour of the semi-infinite one-dimensional quantum Ising model in a transverse field is studied in the presence of an aperiodic surface extended modulation. The perturbed couplings are distributed according to a generalized Fredholm sequence, leading to a marginal perturbation and varying surface exponents. The surface magnetic exponents are calculated exactly whereas the expression of the surface energy density exponent is conjectured from a finitesize scaling study. The system displays surface order at the bulk critical point, above a critical value of the modulation amplitude. It may be considered as a discrete realization of the Hilhorst-van Leeuwen model. † Permanent address: Department of Theoretical Physics, University of Szeged, H-6720 Szeged, Hungary ‡ Unité de Recherche Associée au CNRS No 155 cond-mat/9411030

Geometrical structures of confining surfaces profoundly influence the adsorption of fluids upon approaching a critical point Tc in their bulk phase diagram, i.e., for t = (T −Tc)/Tc → ±0. Guided by general scaling considerations, we calculate, within mean-field theory, the temperature dependence of the order parameter profile in a wedge with opening angle γ < π and close to a ridge (γ > π) for T ≷ Tc and in the presence of surface fields. For a suitably defined reduced excess adsorption Γ±(γ, t → ±0) ∼ Γ±(γ)|t| β−2ν we compute the universal amplitudes Γ±(γ), which diverge as Γ±(γ → 0) ∼ 1/γ for small opening angles, vary linearly close to γ = π for γ < π, and increase exponentially for γ → 2π. There is evidence that, within mean-field theory, the ratio Γ+(γ)/Γ−(γ) is independent of γ. We also discuss the critical Casimir torque acting on the sides of the wedge as a function of the opening angle and temperature.

We consider a system with randomly layered ferromagnetic bonds (McCoy-Wu model) and study its critical properties in the frame of mean-field theory.In the low-temperature phase there is an average spontaneous magnetization in the system, which vanishes as a power law at the critical point with the critical exponents β ≈ 3.6 and β 1 ≈ 4.1 in the bulk and at the surface of the system, respectively. The singularity of the specific heat is characterized by an exponent α ≈ −3.1. The samples reduced critical temperature t c = T av c − T c has a power law distribution P (t c ) ∼ t ω c and we show that the difference between the values of the critical exponents in the pure and in the random system is just ω ≈ 3.1. Above the critical temperature the thermodynamic quantities behave analytically, thus the system does not exhibit Griffiths singularities.

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