Ground-state Vortex Lattice Structures in d-wave Superconductors

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° to them, i.e., along the gap zero direction. The field scale is seen to be 5 T $\sim(\triangle_0/ta)^2\phi_0$, where $\triangle_0$ is the gap maximum, t is the nearest-neighbor hopping, a is the lattice constant, and $\phi_0$ T) there is a is the flux quantum. At much higher fields ($\sim$28 discontinuous transition to a centered 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