Critical Casimir force in slab geometry with finite aspect ratio: analytic calculation above and below $T_c$

Abstract

We present a field-theoretic study of the critical Casimir force of the Ising universality class in a $d$-dimensional $L_{\parallel }^{d-1}\times L$ slab geometry with a finite aspect ratio $\rho =L/L_{\parallel }$ above, at, and below $T_c$. The result of a perturbation approach at fixed dimension $d=3$ is presented that describes the dependence on the aspect ratio in the range $\rho \gtrsim 1/4$. Our analytic result for the Casimir force scaling function for $\rho =1/4$ agrees well with recent Monte Carlo data for the three-dimensional Ising model in slab geometry with periodic boundary conditions above, at, and below $T_c$.

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