  The Anderson transition of Bogoliubov-de Gennes (BdG) quasiparticles in
superconducting state has been studied theoretically for last three decades.
However, its experimental proof is lacking. In particular, the relationship of
the superconducting order-parameter fluctuations and the Anderson transition of
BdG quasiparticles have not been well understood. Our study, based on scanning
tunneling microscopy measurements, investigates how BdG quasiparticles become
Anderson-localized and delocalized as a function of energy and applied magnetic
field in a quasi-two-dimensional Fe-based superconductor with sufficient
zero-bias BdG quasiparticles. The anomalous multifractal spectra based on the
spatial distributions of the pairing gaps and the coherent peak heights suggest
that superconducting fluctuations play a key role in the delocalization of
in-gap BdG quasiparticles. Our real-space Hartree-Fock-BCS-Anderson simulations
and renormalization group analysis with pairing fluctuations support
quasiparticle localization and suggest that enhanced pairing fluctuations lead
to delocalization of BdG quasiparticles and "weak localization" of
phase-fluctuating Cooper pairs in quasi-two-dimensional disordered
superconductors. The present study proposes that the 10-fold way classification
scheme has to be generalized to take order-parameter fluctuations in actual
quantum matter. Also, it shed light on how ac energy loss due to quasiparticles
at Fermi level can be controlled in a quasi-2d superconductor with sufficient
pairing fluctuation.
