Critical Casimir forces for ${\cal O}(n)$ systems with long-range interaction in the spherical limit

We present exact results on the behavior of the thermodynamic Casimir force and the excess free energy in the framework of the $d$-dimensional spherical model with a power law long-range interaction decaying at large distances $r$ as $r^{-d-\sigma}$, where $\sigma<d<2\sigma$ and $0<\sigma\leq2$. For a film geometry and under periodic boundary conditions we consider the behavior of these quantities near the bulk critical temperature $T_c$, as well as for $T>T_c$ and $T<T_c$. The universal finite-size scaling function governing the behavior of the force in the critical region is derived and its asymptotics are investigated. While in the critical and under critical region the force is of the order of $L^{-d}$, for $T>T_c$ it decays as $L^{-d-\sigma}$, where $L$ is the thickness of the film. We consider both the case of a finite system that has no phase transition of its own, when $d-1<\sigma$, as well as the case with $d-1>\sigma$, when one observes a dimensional crossover from $d$ to a $d-1$ dimensional critical behavior. The behavior of the force along the phase coexistence line for a magnetic field H=0 and $T<T_c$ is also derived. We have proven analytically that the excess free energy is always negative and monotonically increasing function of $T$ and $H$. For the Casimir force we have demonstrated that for any $\sigma \ge 1$ it is everywhere negative, i.e. an attraction between the surfaces bounding the system is to be observed. At $T=T_c$ the force is an increasing function of $T$ for $\sigma>1$ and a decreasing one for $\sigma<1$. For any $d$ and $\sigma$ the minimum of the force at $T=T_c$ is always achieved at some $H\ne 0$.