The Division of Materials Research and the Division of Mathematical Sciences contribute funds to this award. It supports theoretical, computational, and mathematical research with an aim to use sophisticated concepts from mathematics to develop a deeper and more precise understanding of topological states of electronic matter in materials and their occurrence.

In thermodynamically stable condensed phases of matter, atoms spontaneously arrange in patterns that determine many of the physical characteristics of the materials. The most studied materials are crystals, where atoms arrange themselves in spatially periodic patterns, like desks in a classroom. Other condensed matter phases such as alloys, amorphous solids, quasicrystals, and liquids display aperiodic atomic structures. In general, the loss of periodicity makes these more difficult to characterize, both theoretically and experimentally. Yet they are of interest because there are many more aperiodic than periodic structures, offering more ways to tailor the physical properties of the material and for the electrons in the material to achieve new states of matter. Besides the naturally occurring condensed matter phases, there is a growing interest in synthetic materials patterned on the microscopic and nanoscopic length scales. These materials enable a more controlled exploration of new properties, new states of matter, and related phenomena enabled by aperiodicity.

The electrons in materials navigate the energy landscape generated by the atomic cores; they are strongly influenced by the pattern in which the atoms are arranged. The electrons also interact with each other through the celebrated electrostatic interaction discovered by Coulomb long ago. In many instances, the effect of the electrostatic interaction can be subsumed in modified atomic potentials. When this is not possible, the electrons are said to be strongly correlated and their dynamics become very difficult to quantify, even with the most powerful computers.

This award supports theoretical and computational research which will explore the dynamics of the electron in conditions of both strong aperiodicity and correlation. The focus will be on detecting those dynamical characteristics which do not change under small modifications of the external conditions and which are said to be topological. The PI has adapted ideas from fields of pure mathematics, such as Operator Algebras, K-theory and Non-Commutative Geometry, into pioneering tools of analysis in materials science. They have been successfully used in the past in the context of strongly disordered topology. In this project, the PI will further extend these tools to generic aperiodic and correlated materials to predict and systematically characterize novel topological phases of electronic matter and related physical effects.

These research activities are ultimately aimed to build accurate and efficient methods and concepts to describe extremely complex electronic systems in materials. It is envisioned that the methods will provide a global view of all possible classes of topological states and the precise conditions that ensure their stability and lead to concrete computer algorithms to locate any synthetically patterned material in this global picture. The award supports graduate student training to become the future experts in applied Non-Commutative Geometry, currently a rare ability in theoretical condensed matter and materials physics. It will also support lectures, pedagogical reviews and textbooks prepared by the PI as an effort to disseminate his ideas among theoretical as well as experimental members of our condensed matter physics community.

Technical Abstract

The Division of Materials Research and the Division of Mathematical Sciences contribute funds to this award. It supports theoretical, computational, and mathematical research with an aim to use sophisticated concepts from mathematics to systematize the search for topological phases and related phenomena in the vast classes of aperiodic correlated materials. Operator Algebras, K-theory and Non-Commutative Geometry will be utilized in this pursuit. The generic goals are: to classify the gapped bulk Hamiltonians and explore their physical responses; establish K-theoretic bulk-boundary correspondence principles and detect those systems with topological boundary spectra; and to use numerical algorithms that implement ideas from Non-Commutative Geometry to exemplify and generate real-world concepts and materials.

It has been long known that the bulk dynamics of a patterned material occur inside a well-defined algebra of physical observables, which derives from the hull of the pattern. This is a topological space associated with and constructed from the given pattern. It augments physical dimensions and can greatly enrich the bulk-boundary correspondence. For example, specially patterned 1-dimensional systems can manifest the physics of the Integer Quantum Hall Effect in 2-dimensions and display topological quantum pumping. Similarly, specially patterned 2-dimensional systems can manifest the physics of the Integer Quantum Hall Effect in 4-dimensions and display the quantized piezo-magneto-electric effect. These systems are much easier to investigate, both theoretically and numerically, in the presence of correlations because of their reduced dimensionality, hence the classification and thorough characterization of correlated aperiodic systems becomes feasible.

When the K-theories of the algebras of physical observables can be computed explicitly, the classification of topological phases becomes exhaustive, with the generators of the K-groups providing all fundamental topological models. Non-Commutative Geometry was found to provide the necessary tools to define topological invariants and to connect them to physical responses, as well as to explore their fate in regimes where spectral gaps are replaced by mobility gaps. This project will integrate these general ideas into a search-and-discovery program in materials science that goes well beyond the class of disordered crystals and covering many other thermodynamic phases of matter, algorithmically synthesized meta-materials and complex systems found in nature. This program is aimed to provide accurate and efficient numerical algorithms, which will be systematically used to channel theoretical advances to experimental laboratories.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Agency
National Science Foundation (NSF)
Institute
Division of Materials Research (DMR)
Application #
1823800
Program Officer
Daryl Hess
Project Start
Project End
Budget Start
2019-09-01
Budget End
2022-08-31
Support Year
Fiscal Year
2018
Total Cost
$252,000
Indirect Cost
Name
Albert Einstein College of Medicine
Department
Type
DUNS #
City
New York
State
NY
Country
United States
Zip Code
10033