  The wavefunctions of a disordered two-dimensional electron gas at the
quantum-critical Anderson transition are predicted to exhibit multifractal
scaling in their real space amplitude. We experimentally investigate the
appearance of these characteristics in the spatially resolved local density of
states of a two-dimensional mixed surface alloy Bi_xPb_{1-x}/Ag(111), by
combining high-resolution scanning tunneling microscopy with spin and
angle-resolved inverse-photoemission experiments. Our detailed knowledge of the
surface alloy electronic band structure, the exact lattice structure and the
atomically resolved local density of states enables us to construct a realistic
Anderson tight binding model of the mixed surface alloy, and to directly
compare the measured local density of states characteristics with those from
our model calculations. The statistical analyses of these two-dimensional local
density of states maps reveal their log-normal distributions and multifractal
scaling characteristics of the underlying wavefunctions with a finite anomalous
scaling exponent. Finally, our experimental results confirm theoretical
predictions of an exact scaling symmetry for Anderson quantum phase transitions
in the Wigner-Dyson classes.
