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Thermodynamic Casimir forces of film systems in the O$(n)$ universalityclasses with Dirichlet boundary conditions are studied below bulk criticality.Substantial progress is achieved in resolving the long-standing problem ofdescribing analytically the pronounced minimum of the scaling function observedexperimentally in $^4$He films $(n=2)$ by R. Garcia and M.H.W. Chan, Phys. Rev.Lett. ${\bf 83}, 1187 \;(1999)$ and in Monte Carlo simulations for thethree-dimensional Ising model ($n=1$) by O. Vasilyev et al., EPL ${\bf 80},60009 \;(2007)$. Our finite-size renormalization-group approach yieldsexcellent agreement with the depth and the position of the minimum for $n=1$and semiquantitative agreement with the minimum for $n=2$. Our theory alsopredicts a pronounced minimum for the $n=3$ Heisenberg universality class.

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