We investigate the spectrum of quasiparticle excitations in the core of isolated pancake vortices in clean layered superconductors. Analysis of the spectral current density shows that both the circular current around the vortex center as well as any transport current through the vortex core is carried by localized states bound to the core by Andreev scattering. Hence the physical properties of the core are governed in clean high-κ superconductors (e.g. the cuprate superconductors) by the

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... bound states, and not by normal electrons as it is the case for traditional (dirty) high-κ superconductors.