Equivariant Homotopy Theory Advances Classification of Fragile Topological Phases in 2D Chern Insulators

The search for robust quantum states of matter drives innovation in materials science and quantum computing, and recent work focuses on understanding the subtle properties of crystalline materials. Hisham Sati, Urs Schreiber, and colleagues at the Center for Quantum and Topological Systems, New York University Abu Dhabi, investigate the fragile topological phases and potential for topological order within two-dimensional crystalline Chern insulators. Their research applies advanced mathematical techniques to classify these phases, revealing that their behaviour is governed by the geometric properties of the material’s momentum space, a connection previously overlooked in this field. The team demonstrates that any emergent quantum particles within these materials must be confined to specific regions of momentum space, a crucial finding that informs the development of stable and controllable quantum systems for future technologies.

We investigate the topological order arising from fractional quantum Hall states, focusing on their logical order relative to their fractional counterparts. The research highlights that the phases are determined by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta, considering actions of wallpaper point groups, a connection that has not previously been explored in this context. The team argues that any topological order must manifest in the adiabatic monodromy of gapped quantum ground states over the space of band topologies, and computes this monodromy in examples where the group is non-abelian. Results demonstrate that any potential fractional quantum Hall anyons must be localized in momentum space, offering insights relevant to ongoing searches for these states.

Topological Matter, Anyons, and Quantum Computation

Scientists have achieved a comprehensive classification of fragile topological phases in two-dimensional crystalline Chern insulators, utilizing methods of equivariant homotopy theory, a mathematical framework previously underappreciated in condensed matter physics. The work establishes that these phases are defined by the equivariant 2-Cohomotopy of the Brillouin torus, a concept that, despite extensive study of crystalline Chern insulators, had not been previously considered. Researchers demonstrate that the topological phases are labeled by pairs consisting of an integer Chern number and an element of a finite set defining how high-symmetry points map within the system, revealing a potentially richer diversity of phases than previously understood. Further research extends this classification to fractional Chern insulators exhibiting the fractional quantum anomalous Hall effect, investigating the nature of anyonic topological order within these systems.

Data shows that the adiabatic monodromy of gapped ground states, reflecting how states evolve under changes in parameters, is localized in momentum space, meaning any potential anyons are confined to specific momentum values. The team highlights that topological order is fundamentally linked to the linear representations of the fundamental group of the system’s parameter space, suggesting a pathway for identifying and manipulating anyonic states for potential applications in topological computing hardware. These findings provide a theoretical foundation for understanding and detecting anyons in crystalline systems, paving the way for advancements in quantum technologies.

Topological Phases Classified by Equivariant Cohomotopy

Scientists have classified fragile topological phases in two-dimensional crystalline Chern insulators using equivariant homotopy theory. The team demonstrates that these phases are determined by the equivariant 2-Cohomotopy of the Brillouin torus, a concept previously overlooked in the study of these materials, and successfully computed these phases for various crystal symmetries. This work establishes a more refined understanding of topological phases beyond standard classifications. Experiments reveal that for systems with symmetries, the topological phases are characterized by a Chern number divisible by the order of the point group, paired with an element from a finite set representing the mapping of high-symmetry points.

The researchers extended their analysis to consider the topological order of fractional Chern insulators, investigating how topological order manifests in the adiabatic behavior of gapped ground states. Their calculations reveal that any potential anyons arising from these states are localized in momentum space, offering insights into their potential properties and behavior. This finding has implications for understanding the fundamental nature of fractional quantum Hall states and their potential for use in quantum computing.

Equivariant Cohomotopy Classifies Topological Phases

This research presents a novel classification of fragile topological phases in two-dimensional crystalline Chern insulators, building upon the mathematical framework of equivariant homotopy theory. The team demonstrates that these phases are determined by the equivariant 2-Cohomotopy of the Brillouin torus, a concept previously overlooked in the study of these materials. The team computed these phases for various crystal symmetries, establishing a more refined understanding of topological phases. Furthermore, the researchers extended their analysis to consider the topological order of fractional Chern insulators, investigating how topological order manifests in the adiabatic behavior of gapped ground states. Their calculations reveal that any potential anyons arising from these states are localized in momentum space, offering insights into their potential properties and behavior. This finding has implications for understanding the fundamental nature of fractional quantum Hall states and their potential for use in quantum computing.