Pronounced minimum of the thermodynamic Casimir forces of $\mathrm{O}\left(n\right)$ symmetric film systems: Analytic theory

Thermodynamic Casimir forces of film systems in the $\mathrm{O}\left(n\right)$ universality classes with Dirichlet boundary conditions are studied below bulk criticality. Substantial progress is achieved in resolving the long-standing problem of describing analytically the pronounced minimum of the scaling function observed experimentally in ${}^4\mathrm{He}$ films $(n=2)$ by Garcia and Chan [Phys. Rev. Lett. 83, 1187 (1999)] and in Monte Carlo simulations for the three-dimensional Ising model $\left(n=1\right)$ by O. Vasilyev et al. [Europhys. Lett. 80, 60009 (2007)]. Our finite-size renormalization-group approach describes the film systems as the limit of finite-slab systems with vanishing aspect ratio. This yields excellent agreement with the depth and the position of the minimum for $n=1$ and semiquantitative agreement with the minimum for $n=2$. Our theory also predicts a pronounced minimum for the $n=3$ Heisenberg universality class.

Figure 1

Scaling function $X\left(\tilde{x}\right)$ of the Casimir force as a function of $\tilde{x}=t{(L/{\xi }_{0+})}^{1/\nu }$ in three dimensions for $n=1,2,3$. Thick solid lines: RG theory from Eqs. (10) and (11). MC data (i) and (ii) in panel (a) from Ref. [9] for the Ising model with $L=20$. Thin line in panel (b): ${}^4\mathrm{He}$ data from Ref. [5] with ${\xi }_{0+}=1.43 {10}^{-8}$ cm. Dashed line in panel (b): RG improved MF theory from Refs. [8, 10] with a minimum $X_{\text{min}}^{\text{MF}}=-6.92$ at ${\tilde{x}}_{\text{min}}^{\text{MF}}=-9.87$.