Bulk-edge Correspondence Proves for Finite Two-Dimensional Ergodic Disordered Systems

The behaviour of electrons in disordered materials presents a long-standing challenge in condensed matter physics, and recent work by Habib Ammari and Jiayu Qiu rigorously proves a key connection between the electronic properties deep inside and at the edges of these systems. They demonstrate the ‘bulk-edge correspondence’ for finite two-dimensional materials with disorder, meaning that characteristics of electrons within the material directly relate to those at its boundaries. This research focuses on systems exhibiting the Aizenman-Molchanov mobility gap, and establishes that a quantity describing edge behaviour converges to a corresponding bulk property as the material size increases, providing a fundamental understanding of how disorder impacts electron flow. The findings represent a significant advance in understanding anomalous Hall physics and offer a robust theoretical framework for designing and predicting the behaviour of disordered electronic systems.

The Hall conductance, a well-studied topological number, receives additional contributions from localized states caused by disorder. Conversely, the edge index, which describes the averaged angular momentum of waves at a sample’s edge, is uniquely defined for finite systems. This research demonstrates that as the sample size increases indefinitely, the edge index converges to the bulk index with high probability, rigorously establishing the bulk-edge correspondence principle for disordered systems. The existence of the Aizenman-Molchanov mobility gap is confirmed using a geometric decoupling method.

Disorder, Topology And Mathematical Rigour

This compilation of references demonstrates a comprehensive understanding of topological insulators, disordered systems, and related mathematical concepts, spanning decades of research from the quantum Hall effect to topological photonics and non-Hermitian physics. It reflects an interdisciplinary approach, drawing from physics, mathematics, materials science, and engineering, and covers a broad range of topics including topological insulators, the quantum Hall effect, Anderson localization, and the bulk-edge correspondence.

Disordered Topology And Bulk-Edge Correspondence Proof

Scientists have rigorously proven the bulk-edge correspondence for finite two-dimensional systems with random disorder, a significant step forward in understanding topological materials. The research focuses on systems with disordered on-site potentials, providing a framework for analyzing materials where randomness plays a crucial role. Researchers defined both bulk and edge indices within the Aizenman-Molchanov mobility gap, demonstrating their well-defined nature in this context. The bulk index comprises the Hall conductance, which recovers the Chern number in ordered systems, and an additional contribution from localized states caused by disorder.

Experiments revealed that the edge index accurately characterizes the averaged angular momentum of waves at the sample edge, proving its well-defined nature through statistical analysis. This breakthrough delivers a rigorous proof that, as the sample size increases, the edge index converges to the bulk index, firmly establishing the bulk-edge correspondence principle for disordered systems, extending previous results limited to ordered systems to encompass more realistic scenarios with disorder. Measurements confirm the existence of the Aizenman-Molchanov mobility gap using the geometric decoupling method, validating the underlying assumptions. For completeness, all assumptions were checked using a prototypical Hamiltonian for quantum anomalous Hall physics, further solidifying the theoretical framework. This work provides a robust foundation for understanding and designing topological materials with enhanced robustness against disorder, opening new avenues for advanced technologies.

Disorder Preserves Bulk-Edge Correspondence in 2D Systems

Scientists have rigorously demonstrated the bulk-edge correspondence principle for disordered two-dimensional systems, a significant advancement in understanding how electronic properties emerge in complex materials. Their work establishes a precise connection between the bulk electronic properties and the edge states that appear at its boundaries, even when disorder is present. The team proved that, as the size of the disordered system increases, the properties of these edge states converge to those predicted by the bulk properties, confirming a fundamental relationship previously understood only in simpler, ordered systems. This achievement involved the introduction and careful analysis of both bulk and edge indices, mathematical quantities that characterize the electronic behavior of the material.

By establishing a rigorous mathematical link between these indices, the researchers provided a solid foundation for predicting and controlling the behavior of electrons in disordered materials, which is crucial for developing new technologies based on these materials. The study confirms the existence of a specific energy range, known as the Aizenman-Molchanov mobility gap, where this correspondence holds true, and provides a means to calculate its properties. The authors acknowledge that their results rely on certain assumptions regarding the distribution of the random potential within the disordered material, and that further work is needed to explore the generality of these findings. They also highlight the need for future research to investigate the behavior of these systems in three dimensions, which presents significant mathematical challenges. Nevertheless, this work represents a substantial step forward in the theoretical understanding of disordered systems and provides a framework for exploring their potential applications in materials science and condensed matter physics.