  Majorana quasiparticles (MQPs) in condensed matter play an important role in
strategies for topological quantum computing but still remain elusive. Vortex
cores of topological superconductors may accommodate MQPs that appear as the
zero-energy vortex bound state (ZVBS). An iron-based superconductor Fe(Se,Te)
possesses a superconducting topological surface state that has been
investigated by scanning tunneling microscopies to detect the ZVBS. However,
the results are still controversial. Here, we performed spectroscopic-imaging
scanning tunneling microscopy with unprecedentedly high energy resolution to
clarify the nature of the vortex bound states in Fe(Se,Te). We found the ZVBS
at 0 $\pm$ 20 $\mu$eV suggesting its MQP origin, and revealed that some
vortices host the ZVBS while others do not. The fraction of vortices hosting
the ZVBS decreases with increasing magnetic field, while chemical and
electronic quenched disorders are apparently unrelated to the ZVBS. These
observations elucidate the conditions to achieve the ZVBS, and may lead to
controlling MQPs.
