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

Area of Science:

• Condensed Matter Physics
• Quantum Mechanics
• Materials Science

Background:

• The electronic local density of states (LDOS) describes electron probability in solids.
• Disordered systems exhibit spatial variations in LDOS, complicating analysis.
• Existing methods may not fully capture the impact of disorder on electron properties.

Purpose of the Study:

• To develop a novel framework for analyzing the local density of states in disordered solids.
• To quantify the impact of disorder on electron energy and position distributions.
• To connect information geometry concepts with experimental measurements like scanning tunneling microscopy.

Main Methods:

• Constructing a real-space Fisher information matrix by treating energy as a random variable.
• Constructing an energy-space Fisher information by treating position as a random variable.
• Applying Cramér-Rao bounds to establish limits on electron energy and position variance.

Main Results:

• The proposed Fisher information matrices effectively quantify LDOS variations due to disorder.
• Cramér-Rao bounds provide fundamental limits on electron energy and position variance.
• New interpretations of the local density of states emerge from this information-theoretic approach.

Conclusions:

• Information geometry offers a powerful lens for understanding electron behavior in disordered materials.
• The formalism bridges theoretical concepts with practical scanning tunneling microscopy measurements.
• This work advances the study of electronic properties in disordered metals and topological insulators.