  Wave functions of low-energy quasiparticle subgap states in d-wave
superconducting rings, threaded by Aharonov-Bohm magnetic flux, are found
analytically. The respective energies are closest to the midgap position at
small magnetic fluxes and deviate from the Fermi surface due to the Doppler
shift, produced by the supercurrent. The Doppler-shifted zero-energy states
result in a paramagnetic response of the ring at small fluxes. The states exist
only for even angular momenta of the center of mass of Cooper pairs, in
agreement with recent numerical studies of the problem. This macroscopic
quantum effect in d-wave rings results in broken h/2e periodicity, retaining
only the h/e periodic behavior of the supercurrent with varying magnetic flux.
