Concrete K-theory operations for topological physics

Detecting and measuring the stability of waves, be they sound waves, wave patterns of electrons or light, is a critical task in modern physics. Specifically, stable waves, bound to a position, are a central topic in the study of quantum materials. Low-power transistors and small lasers are some of the devices that can be designed from quantum materials. The critical mathematical tool for detecting these stable waves is called K-theory, a topic that cuts across many of the main subject areas of mathematics. Using tools from operator algebras, a subfield of mathematical analysis, a form of K-theory has recently been discovered to be useful in the development of mathematical probes of computer models of materials. Loring’s group will be developing the subject of K-theory for operator algebras, with particular emphasis on new mathematical techniques that can be implemented in software used by physicists. This project will also create opportunities for students at the University of New Mexico and Florida A&M University to do research and participate in internships at Sandia National Laboratories. The mathematics to be developed in this project will focus on multivariable spectrum for noncommuting operators and associated invariants in real and complex K-theory. These invariants can be applied to detect local topology in a variety of quantum materials. These invariants depend on a local gap in the Hamiltonian, a concept that can be made precise using the Clifford spectrum of the various position operators and the Hamiltonian. Local gaps can exist in topological metals, and in composite systems where a metal lead abuts a topological insulator. The fundamental challenge is to find invariants for matrix models of locally gapped free-fermion systems. Using unsuspended E-theory as a model for real and complex K-homology, Loring will develop simple formulas for these invariants and determine if these invariants are complete. 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.