Dirty Hyperbolic Dirac Liquids Exhibit Stable Massless Excitations up to Sites with Vanishing Density of States

The behaviour of electrons in materials with unusual geometric structures is attracting considerable attention, and recent research explores how imperfections influence these systems. Christopher A. Leong, Daniel J. Salib, and Bitan Roy, all from Lehigh University, investigated the properties of electrons confined to two-dimensional hyperbolic lattices, materials possessing a curved geometry. Their work demonstrates that these electrons, behaving as massless particles similar to those described by relativity, remain surprisingly stable even with the presence of impurities. The team reveals a comprehensive picture of how disorder transforms the material from an insulator to a metallic state, and ultimately to a completely localized state, a progression dictated solely by the curvature of the underlying hyperbolic space and offering new insights into the behaviour of relativistic electrons in curved environments.

Impurity Effects on Electronic Density of States

The team investigated how impurities affect the electronic density of states within a hyperbolic material, employing a sophisticated computational method to accurately determine the local density of states in systems containing weak pointlike impurities. Calculations involved modelling a three-dimensional lattice and introducing randomly distributed impurities with a small electric charge, and by analysing systems up to 64³ lattice sites, they obtained statistically significant data. The results reveal a vanishing average density of states at zero energy, indicating the absence of available states due to the impurities, while the typical density of states reflects how a single electron experiences the disordered environment.

Hyperbolic Lattices Resist Disorder and Stay Metallic

This research explores the behavior of electrons in disordered materials structured as hyperbolic lattices, unique crystal structures with negative curvature. The study focuses on how imperfections, or disorder, influence electronic properties, potentially leading to electron trapping and transitions between different quantum states. The team investigates the conditions under which these materials remain conductive, or metallic, versus becoming insulating, and demonstrates that introducing disorder into hyperbolic lattices can drive a transition from a metallic to an insulating state, dependent on the strength of the disorder and the lattice’s properties. The research confirms that Anderson localization occurs in disordered hyperbolic lattices, investigating how far an electron can travel before becoming localized, and suggests that the metal-insulator transition is a quantum phase transition, exhibiting unusual behavior at the transition point. The unique geometry of hyperbolic lattices plays a crucial role, affecting the density of states and electron localization.

Hyperbolic Geometry Drives Electronic Phase Transitions

This research demonstrates the surprising stability of massless, relativistic quasiparticles within a two-dimensional hyperbolic lattice, even when subjected to weak disorder. The team discovered that these quasiparticles maintain their characteristic vanishing density of states at zero energy, indicating a robust, conducting state. However, increasing the level of disorder induces a transition to an insulating state, revealing a complex interplay between material geometry and electronic behavior. The key finding lies in the origin of these transitions, which are shown to be driven by the unique negative curvature inherent to the hyperbolic lattice structure, rather than by the disorder itself.

This contrasts sharply with similar materials lacking this curvature, where even weak disorder immediately leads to insulation. The researchers established a phase diagram detailing these transitions and confirmed the robustness of the initial conducting state against minimal disruption, extending the understanding of this phenomenon. Future work could explore the potential for manipulating these hyperbolic lattices to engineer materials with tailored electronic properties, potentially leading to novel devices.