Crossover from Goldstone to critical fluctuations: Casimir forces in confined O$(\mathbf{n})$ symmetric systems

We study the crossover between thermodynamic Casimir forces arising from long-range fluctuations due to Goldstone modes and those arising from critical fluctuations. Both types of forces exist in the low-temperature phase of O$(n)$ symmetric systems for $n>1$ in a $d$-dimensional $L_{\parallel }^{d-1}\times L$ slab geometry with a finite aspect ratio $\rho =L\mathrm{/}{L}_{\parallel }$. Our finite-size renormalization-group treatment for periodic boundary conditions describes the entire crossover from the Goldstone regime with a nonvanishing constant tail of the finite-size scaling function far below $T_c$ up to the region far above $T_c$ including the critical regime with a minimum of the scaling function slightly below $T_c$. Our analytic result for $\rho \ll 1$ agrees well with Monte Carlo data for the three-dimensional XY model. A quantitative prediction is given for the crossover of systems in the Heisenberg universality class.