Critical specific heat of systems confined by a wall

We analyze quantitatively the surface (excess) specific heat near criticality for semi-infinite thermodynamic systems of the Ising universality class when they are in contact with a planar wall belonging to the so-called normal surface universality class in spatial dimension d=3 and in the mean-field limit. In order to determine the universal surface amplitudes and the spatially varying scaling functions which govern the behavior of the specific heat we use the local-functional theory. Our analysis pertains to the critical isochore, near two-phase coexistence, and along the critical isotherm if (in magnetic language) the surface and the weak bulk magnetic fields are either collinear or anti-collinear. The universal scaling functions which underly calculations of the specific heat are monotonic in most cases, except along the critical isotherm for anti-collinear bulk and surface magnetic fields. Among other results, we find in the latter case a divergent behavior of the inhomogeneous universal scaling functions associated with the surface specific heat, generated by the interface between the wall and the bulk phase of a macroscopically thick wetting film. We show that the singularity of this profile near the interface in spatial dimension three is governed by a power-law divergence with the exponent 2α/β, while it is of logarithmic nature in the mean-field limit. It turns out that this feature has a strong influence on the values of the universal surface amplitudes, regarding both their magnitude and sign.