Density profiles in confined critical systems and conformal invariance

The conformal-invariance approach is extended to two-dimensional critical systems with mixed ab boundary conditions. The profiles 〈φ(r)${\mathrm{〉}}_{\mathit{a}\mathit{b}}$ of the magnetization and the energy density in Ising and Q-state Potts strips with +- and fixed-free boundary conditions are derived. The profile of an arbritrary operator degenerate at level 2 is expressed in terms of hypergeometric functions. The universal amplitudes ${\mathit{A}}_{\mathit{a}\mathit{b}}$ ${\mathit{B}}_{\mathit{a}\mathit{b}}$ of the Casimir force and the Fisher–de Gennes distant-wall correction to the profile satisfy ${\mathit{B}}_{\mathit{a}\mathit{b}}$=-4π${\mathit{x}}_{\mathrm{\phi }}$ ${\mathit{c}}^{\mathrm{-}1}$ ${\mathit{A}}_{\mathit{a}\mathit{b}}$. Here ${\mathit{x}}_{\mathrm{\phi }}$ is the scaling dimension of φ(r) and c is the central charge.