Vortex charges in high-temperature superconductor (HTS) are studied by solving the Bogoliubov–de Gennes equations based on a model Hamiltonian with antiferromagnetic (AF) and $d$-wave orders in the presence of the long-range Coulomb repulsion. For a sufficient strength of the AF order, the negative vortex charge is found. A sign change between negative and positive may occur by tuning the long-range Coulomb repulsion strength or the doping parameter. Recent NMR experiments are hopefully understood. We show that the charged vortex can induce a spin-orbit coupling that is important for superconductors with a short coherence length and a large value of the energy gap over the Fermi-level ratio. Fractional flux quanta are possible for HTS.

• Received 9 July 2008

DOI:https://doi.org/10.1103/PhysRevB.78.064505