Anderson Transitions (0707.4378v1)

Abstract: The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed. The term ``Anderson transition'' is understood in a broad sense, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states. The emphasis is put on recent developments, which include: multifractality of critical wave functions, criticality in the power-law random banded matrix model, symmetry classification of disordered electronic systems, mechanisms of criticality in quasi-one-dimensional and two-dimensional systems and survey of corresponding critical theories, network models, and random Dirac Hamiltonians. Analytical approaches are complemented by advanced numerical simulations.

• The paper reveals that multifractal analysis of critical wave functions uncovers unique scaling exponents at the Anderson transition.
• The paper employs the power-law random banded matrix model to simulate phase transitions from localized to metallic states through numerical and analytical methods.
• The paper classifies disordered systems into distinct symmetry classes, offering insights into universality and critical phenomena in quantum phase transitions.

Overview of "Anderson Transitions" by Ferdinand Evers and Alexander D. Mirlin

The paper by Ferdinand Evers and Alexander D. Mirlin explores the physics of Anderson transitions, focusing on transitions between localized and metallic phases within disordered systems. The review spans a breadth of pertinent developments, particularly highlighting multifractality of wave functions, criticality in power-law banded random matrix models, and the symmetry classification of disordered electronic systems. Through both analytical approaches and numerical simulations, the authors provide a comprehensive paper of the underlying quantum phase transition physics.

Critical Points and Multifractality

One of the focal points of the paper is the exploration of multifractality in critical wave functions. Multifractality captures the fluctuations in quantum states at criticality, characterized by an infinite set of critical exponents. These exponents describe the scaling of wave function amplitudes, which manifest as multifractal fluctuations. Critical wave functions exhibit a scaling behavior distinct from both metallic and insulating phases, illuminating the nuanced characteristics of Anderson transitions. The paper underscores the importance of this multifractal spectrum, which provides crucial insights into the fixed points governing these transitions.

Power-Law Random Banded Matrix Model

The research paper thoroughly examines the power-law random banded matrix (PRBM) model, which serves as a paradigm for studying Anderson criticality. The PRBM model displays critical behavior at the transition from localized to delocalized phases, making it a pivotal tool in understanding Anderson transitions. The model's ability to capture the nuances of 1D systems with long-range hopping characteristics and its adaptability for numerical simulations enhances its utility in theoretical explorations. The paper articulates the significance of this model in linking multifractality to the system's physical properties and establishing a bridge between theoretical predictions and numerical findings.

Symmetry Classes and Universality

A vital aspect discussed is the classification of disordered systems into symmetry classes that dictate their phase behaviors and critical properties. The paper identifies ten distinct symmetry classes, incorporating new chiral and Bogoliubov-de Gennes classes, each associated with unique universality classes of Anderson transitions. This classification elucidates the role of symmetry in determining the nature of critical phenomena, providing a framework within which the universality of such transitions can be assessed. Notably, the exploration of additional symmetries and corresponding universality deviations expands the theoretical understanding of disordered systems.

Numerical and Theoretical Implications

Evers and Mirlin provide meticulous numerical analyses alongside their theoretical explorations, emphasizing the importance of numerical simulations in validating theoretical predictions. They touch on recent computational advancements which enable highly accurate studies of critical phenomena, thereby refining the parameters of criticality and enhancing the predictability of system behavior at transitions. Additionally, the RG methodologies applied to analyze $\sigma$-models are discussed for different dimensions, emphasizing the transitional behavior from metallic to insulating regimes. These methods also highlight how these systems deviate from non-interacting models, providing insights into the role of interactions in real-world materials.

Conclusion and Future Directions

The implications of the research extend beyond the specifics of each model or critical point, weaving a narrative of interconnected phenomena across varying dimensionalities and disorder types. By embedding multifractality, symmetry classifications, and computational analyses into a unified discourse, the authors present future prospects in the theoretical and numerical exploration of Anderson transitions.

Moving forward, the alluded multifractality concepts and symmetrical properties suggest a world of yet unexplored quantum dimensions. As computational methods advance, so will the opportunity to explore higher-dimensional systems and complex disorder characters, potentially uncovering novel phases and transitions. Evers and Mirlin’s paper is thus both a compendium of current knowledge and a springboard for future research into the fascinating domain of disordered quantum systems.