Probing disorder-induced Fisher information matrix and Cram\'{e}r-Rao bound by STM

Abstract

The electronic local density of states 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 local density of states caused by the disorder. The corresponding Cram\'{e}r-Rao bounds in these two scenarios set a limit on the energy variance and the position variance of electrons, respectively, pointing to new interpretations of STM measurements. Our formalism thus bring the notion of information geometry into STM measurements, as demonstrated explicitly by lattice models of metals and topological insulators.