Topological and C*-Algebraic Quantum Matter

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A series of revolutionary developments across physics reveal that the phenomena of quantum mechanics not only play a crucial role for very small systems, but that large systems can also behave in a quantum way. Physicists and mathematicians are only beginning to understand a vast array of quantum phases of matter: classes of systems that may look different microscopically but share common macroscopic properties. This project aims to solidify the connection between topological phases of matter as thought of by condensed matter physicists and their conjectured mathematical classification. The intended approach to this problem is expected to have immediate applications to condensed matter physics and quantum information science via new examples of topological phases and the development of new theoretical methods. The award will also contribute to US workforce development through the training of graduate students. The principal scientific goal of this project is to use algebraic quantum mechanics to develop a framework to study the relationship between condensed matter physics and topology. In condensed matter physics quantum systems equipped with a time evolution or Hamiltonian are often presented using lattice models: these are simplified models of a material at low temperature. When the energy gap of a quantum system is non-zero, some properties are robust to deformations of the system, leading to the notion of equivalence classes of systems known as gapped phases of matter. Some gapped phases, called invertible, are believed to be well approximated by topological quantum field theories, although this connection is not understood rigorously. However, Kitaev has proposed a more fundamental connection between topology and condensed matter physics, suggesting that invertible gapped phases are classified by a loop-spectrum in the sense of homotopy theory. The project team will make this precise by introducing a cohomology theory of invertible gapped phases. Physically, this corresponds to studying quantum systems with parameters that are allowed to vary, with some predetermined restrictions and without changing the fundamental properties of the system. This award supports investigations aimed toward constructing this theory and answering related fundamental questions in the mathematical physics of gapped phases of matter.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.