Microscopic Theory Explains Anderson Localization of Electrons in Random Lattices, Enabling Description of Diffusive and Localized Regimes

The behaviour of electrons in disordered materials presents a long-standing challenge in condensed matter physics, with understanding how randomness affects their movement being crucial for developing new materials. Václav Janiš from the Institute of Physics, The Czech Academy of Sciences, and colleagues now present a comprehensive microscopic theory that explains electron behaviour across the entire spectrum of disorder, from free movement to complete localization. This work establishes a systematic framework for modelling electrons in random lattices, accurately describing both metallic states where electrons travel as waves and localized states where they become bound to imperfections within the material. By developing a method to calculate key properties while upholding fundamental physical laws, the team provides a powerful tool for analysing materials and predicting their behaviour at the point where electrons transition from moving freely to becoming trapped, a phenomenon known as Anderson localization.

Electron Localization in Random Lattices Explained

Scientists have developed a detailed theory explaining Anderson localization, the phenomenon where electrons become trapped in disordered materials, hindering electrical conductivity. This research investigates the transition between conducting and insulating states, focusing on how interactions between electrons and the degree of disorder influence this change, providing a more accurate description of electron behaviour in these complex systems. The theory employs a self-consistent approach to account for complex electron interactions, allowing researchers to calculate properties like the density of states and the diffusion coefficient. Results demonstrate that the interplay between disorder and electron interactions significantly influences how far an electron can travel before becoming localized, and alters the amount of disorder needed to completely halt electron movement. This provides a more realistic understanding of electron behaviour in disordered materials, with implications for materials science and condensed matter physics.

Previous research has not fully explained the behaviour of electrons in both conducting and localized states. The researchers present a microscopic theory applicable to both metallic phases, where electrons move freely, and localized phases, where electrons become trapped, approximating complex interactions while respecting fundamental physical laws. This allows for quantitative calculations of electron behaviour in disordered systems.

Mean-Field Theory Reveals Localization Transition Mechanisms

This research investigates Anderson localization, the process by which electrons become trapped in disordered materials, suppressing electrical conductivity. It is a fundamental problem in condensed matter physics with implications for materials science, electronics, and even biological systems. The central challenge is understanding how a material transitions from conducting to insulating behaviour as disorder increases. The team employed a mean-field theoretical approach, simplifying complex electron interactions by approximating them with an average potential. Utilizing a sophisticated mathematical technique, the Parquet approach, they calculated how electrons interact with each other, going beyond simple approximations and ensuring the theoretical framework respected fundamental conservation laws.

The researchers investigated the behaviour of the diffusion pole, a quantity related to how easily electrons move through the material, to understand the transition to localization. They obtained an approximate solution to the Anderson localization problem, likely valid in systems with high spatial dimensions, addressing the challenges of considering infinitely large systems and averaging over the random disorder. The team explicitly addressed the relationship between causality and a mathematical constraint known as the Ward identity, essential for a consistent theory. The results suggest a more complex critical behaviour at the transition to localization than previously thought, extending the analysis to disordered systems arranged as random networks. This work contributes to a deeper understanding of how disorder affects the electronic properties of materials, potentially enabling the design of materials with tailored conductivity. The development of improved theoretical approximations and the exploration of complex critical behaviour could lead to new approaches for studying disordered systems, with connections to other fields like optics and acoustics.

Quantum Bound State Drives Anderson Localization

Scientists have developed a microscopic theory of Anderson localization, building upon previous research to create a framework applicable to both conducting and insulating states of electron behaviour in disordered materials. The team successfully developed a method for approximating complex electron interactions, ensuring these approximations adhere to fundamental physical principles across varying levels of disorder. Through this approach, researchers identified a critical point within the conducting phase, directly linking it to the transition into localized behaviour. The central finding demonstrates that localization arises from a quantum bound state formed between a propagating electron and the ‘hole’ it leaves behind, challenging conventional understanding of electron behaviour in disordered systems.

This research reveals that accurately describing the transition to localization requires considering more than just the energy of individual electrons; the interactions between electrons are crucial. Furthermore, the team identified a new property characterizing these localized bound states, which induces a gap in the functions describing electron interactions, without affecting the energy levels of individual electrons. Future work could focus on refining these approaches and exploring the relationship between microscopic calculations and macroscopic, observable properties of disordered materials, as well as the connection between electrical conductivity, electron-hole correlation, and electron movement in both conducting and insulating phases.