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. 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 $\mu$ in the extreme quantum limit. Specifically, the energy ratio for the first three orders is more close to $1:2:3$ instead of $1:3:5$ at extremely low temperature. The local density of states reveals an Friedel-like behavior associated with that of the pair potential in the extreme quantum limit, which will be smoothed out by thermal effect above a certain temperature even the quantum limit condition, namely $T/T_c<\Delta/E_F$ is still satisfied. 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.