Probing disorder-induced Fisher information matrix and Cramér-Rao bound by STM

The electronic local density of states (LDOS) of solids, if normalized correctly, represents the probability density that the electron at a specific position has a particular energy. Because this probability density can vary in space in disordered systems, we propose that one can either treat the energy as a random variable and position as an external parameter to construct a real space Fisher information matrix, or treat the position as a random variable and energy as an external parameter to construct an energy space Fisher information, both quantify the variation of LDOS caused by the disorder. The corresponding Cramér-Rao bounds in these two scenarios set a limit on the energy variance and the position variance of electrons in any disordered solids, pointing to new interpretations of LDOS. Our formalism thus brings the notion of information geometry into scanning tunneling microscopy measurements, as demonstrated explicitly by lattice models of metals and topological insulators.