Topological Phases in Extreme Aperiodic and Correlated Regimes: A View from Noncommutative Geometry
Abstract
The relatively new field of topological phases of matter is extremely active among both mathematics and physics communities. When probed with external fields, these systems display linear and non-linear responses characterized by coefficients that take quantized values. The prototypical example is the quantum Hall effect, where a 2-dimensional electron gas subjected to a perpendicular magnetic field responds to an electric field with a charge current flowing perpendicularly to the applied electric field. The current and electric field enter a linear relation and the constant of proportionality, the Hall conductance, displays quantized values, both integer and fractional, robust against various external stress factors applied to the samples. Topological phases of matter also display robust and predictable spectral features when defects are introduced, such as edges, corners or dislocations. Furthermore, the local excitations of these systems, dubbed anyons, can behave entirely different from their fermionic or bosonic building blocks. For example, they can carry fractions of the electron charge and display non-abelian statistics. Topological phases of matter are well understood in the regimes of (R1) strong aperiodicity and weak interaction and (R2) weak aperiodicity and strong interaction, but they are poorly understood in the regime of (R3) strong aperiodicity and strong interaction. Noncommutative Geometry (NCG) was successfully applied in the regime R1, where it produced some of the finest results available, such as explaining the quantization and stability of the response coefficients against strong disorder or the metallic character of a sampleÕs edges even when the latter are highly contaminated. The PI plans to make progress with the regime R3 by employing NCG and by building on a PIÕs recent result where the topological algebra associated to the dynamics of interacting fermions was computed. Following the research model emerged from the regime R1, the PI plans to study the K-theories of the topological algebra mentioned above, as well as its cyclic cohomologies and the pairings between them. One goal is to generate index theorems that can be pushed over certain Sobolev spaces that cover the regime R3. If successful, this will supply the first rigorous explanation of the fractional quantization of the Hall conductance observed in laboratory experiments, as well as a natural framework to study the corresponding highly correlated topological phases of matter. Another goal is to establish a rigorous bulk-defect correspondence principle in regime R3, which captures the robust spectral features emerging in highly correlated topological systems when geometric defects are introduced into lattices. Furthermore, the PI plans to make an explicit connection between the NCG and unitary modular tensor categories (UMTC) separate characterizations of highly correlated topological systems, with the primary goal of a high and systematic throughput of microscopic models for various UMTC data. This will be based on new paradigms for taking thermodynamic limits, enabled by the structure of the newly computed topological algebra mentioned above. These microscopic models could facilitate the laboratory engineering of the phases and could shed light on the physical responses of anyons to various external factors. The proposed research is highly connected to the concentrated quest of a broad scientific community towards a laboratory proof of concept of a topological quantum computer. Specifically, the outcomes of the research can broaden the class of highly correlated systems that can serve as engines for these elusive machines, as well as teach us how to operate these machines more optimally and efficiently. The proposed research will very likely open new research directions in pure mathematics and physics and will strengthen the scientific exchanges between these two communities.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Apr 19, 2023
- Source ID
- W911NF2310127
Entities
People
- Emil Prodan
Organizations
- Army Contracting Command
- United States Army
- Yeshiva University