  The critical behavior of the order-disorder phase transition in the buckled
dimer structure of the Si(001) surface is investigated both theoretically by
means of first-principles calculations and experimentally by spot profile
analysis low-energy electron diffraction (SPA-LEED). We use density functional
theory (DFT) with three different functionals commonly used for Si to determine
the coupling constants of an effective lattice Hamiltonian describing the dimer
interactions. Experimentally, the phase transition from the low-temperature
$c(4 {\times} 2)$- to the high-temperature $p(2 {\times} 1)$-reconstructed
surface is followed through the intensity and width of the superstructure spots
within the temperature range 78-400 K. Near the critical temperature $T_c =
190.6\,\mathrm{K}$, we observe universal critical behavior of spot intensities
and correlation lengths which falls into the universality class of the
two-dimensional (2D) Ising model. From the ratio of correlation lengths along
and across the dimer rows we determine effective nearest-neighbor couplings of
an anisotropic 2D Ising model, $J_\parallel = (-24.9 \pm 0.9_\mathrm{stat} \pm
1.3_\mathrm{sys})\,\mathrm{meV}$ and $J_\perp = (-0.8 \pm
0.1_\mathrm{stat})\,\mathrm{meV}$. We find that the experimentally determined
coupling constants of the Ising model can be reconciled with those of the more
complex lattice Hamiltonian from DFT when the critical behavior is of primary
interest. The anisotropy of the interactions derived from the experimental data
via the 2D Ising model is best matched by DFT calculations using the PBEsol
functional. The trends in the calculated anisotropy are consistent with the
surface stress anisotropy predicted by the DFT functionals, pointing towards
the role of surface stress reduction as a driving force for establishing the
$c(4 {\times} 2)$-reconstructed ground state.
