  We show in a realistic $d_{x^{2}-y^{2}}$ symmetry gap model for a cuprate
superconductor that the clean vortex lattice has discontinuous structural
transitions (at and near T=0), as a function of the magnetic field $B$ along
the c-axis. The transitions arise from the singular nonlocal and anisotropic
susceptibility of the $d_{x^{2}-y^{2}}$ superconductor to the perturbation
caused by supercurrents associated with vortices. The susceptibility, due to
virtual Dirac quasiparticle-hole excitation, is calculated carefully, and leads
to a ground state transition for the triangular lattice from an orientation
along one of the crystal axis to one at 45$^o$ to them, i.e, along the gap zero
direction. The field scale is seen to be 5 Tesla $ \sim
(\Delta_{0}/ta)^{2}\Phi_{0}$, where $\Delta_{0}$ is the gap maximum, $t$ is the
nearest neighbour hopping, $a$ is the lattice constant, and $\Phi_{0}$ is the
flux quantum. At much higher fields ($\sim 28T$) there is a discontinuous
transition to a centred square structure. The source of the differences from
existing calculations, and experimental observability are discussed, the latter
especially in view of the very small (a few degrees $K$ per vortex) differences
in the ground state energy.
