Distinct properties of vortex bound states driven by temperature

We investigate the behavior of vortex bound states in the quantum limit by self-consistently solving the Bogoliubov-de Gennes equation.\revision{We find that the energies of the vortex bound states deviates from the analytical result $E_\mu=\mu\Delta^2/E_F$ with the half-integer angular momentum $\mu$ at very low temperature. Specifically, the energy ratio for the first three orders is more close to $1:2:3$ instead of $1:3:5$ in the extreme quantum limit $T/T_c\ll\Delta/E_F$.} The local density of states reveals a Friedel-like behavior associated with that of the pair potential, which will be smoothed out by thermal effect above the quantum limit. Our studies show that the vortex bound states can exhibit very distinct features in different temperature regimes, which provides a comprehensive understanding and should stimulate more experimental efforts for verifications.