Abstract: Due to the constraints introduced by boundaries for the long range fluctuations in a massless field, a long-range fluctuation-induced effective force between boundaries is induced, and it is called the Casimir force. The classical Casimir force refers to the interaction between two conducting plates immersed in a vacuum, and it is originated from the quantum fluctuations of the electromagnetic field. In this work, we study the critical Casimir force for the Ising universality class in statistical mechanics. Specifically, we develop an effective method, namely, the Euler-Maclaurin summation method, to calculate the critical Casimir force between two parallel plates at the critical point of the Ising universality class. Our results demonstrate that, in this geometric set-up, the critical Casimir force behaves as a long-range attractive force with a scaling exponent of -3 with respect to the boundary distance.

Key words: critical Casimir force, Ising universality class, parallel plates confinement, Euler-Maclaurin summation