Distorted vortex lattice in a tetrahedral superconductor

Equilibrium shape and orientation of vortex lattice are studied for an $s$-wave tetrahedral superconductor in the vicinity of the upper critical field. The phase diagram, which includes transitions between rhombic and rectangular lattices, is constructed in the parameter space of the Ginzburg-Landau functional. The developed theory is applied to the heavy-fermion superconductor $\mathrm{PrOs_4Sb_{12}}$. In a wide range of parameters the shape of the vortex lattice is only weakly dependent on temperature. The neutron scattering measurements of the vortex lattice in $\mathrm{PrOs_4Sb_{12}}$ can be explained by a peculiarities of the tetrahedral symmetry group and are further supported by analysis of the appropriate band structure calculations.