Topological quantum matter and crystalline symmetry

NONTECHNICAL SUMMARY

This award supports theoretical research on understanding new kinds of phenomena that can arise due to the presence of crystalline symmetry in quantum systems. Quantum materials typically exist in a crystalline environment due to the periodic arrangement of atoms. Hence, the material under question often also has a crystallographic symmetry that includes rotations, reflections, and translations of the space, meaning that the crystal structure remains the same under such operations.

In the presence of these symmetries, quantum materials can host new types of physical phenomena. For example, if a quantum material has a rotational symmetry, then it is possible for certain kinds of crystal defects to accumulate quantized electric charges that are fractions of the charge of the electron. The project will incorporate methods from mathematical physics and quantum field theory to further our ability to characterize and classify the robust, quantized properties exhibited by quantum systems with crystalline symmetry.

The project will also develop methods to experimentally probe these phenomena in a variety of settings, including two-dimensional quantum materials, photonic systems, and noisy intermediate scale quantum computers.

The project has a strong educational component, as it fully supports a graduate student. The PI will further develop and publicly release lecture notes for his special topics graduate courses on topological quantum matter and on machine learning. In addition, the PI will remain active in various forms of outreach, at the high school level through outreach programs at his institution and through an active social media presence.

Ongoing industry collaborations will further broaden the societal impacts of the project through its applications to quantum computing. TECHNICAL SUMMARY

This award supports theoretical research in developing the theory of topological phases of matter with crystalline symmetry. This theory will apply in the regime of strong interactions, and thus go beyond single-particle band theory, using the tools of topological quantum field theory. The project has several different components and builds directly on significant progress from the last project period.

In the first component, the PI’s group will extend their theories of two-dimensional strongly interacting crystalline topological phases, which applied to the case orientation-preserving wallpaper groups, to include all 17 wallpaper groups. The second component is to perform numerical calculations for models of fractional Chern insulators and quantum spin liquids, specifically using projected quantum wave functions, to understand the emergence of new crystalline symmetry-protected topological invariants and their physical properties.

These include invariants obtained through partial symmetry operations and through fractional charges accumulated at lattice defects. The third component is to develop methods to experimentally probe the PI’s recently discovered crystalline topological invariants in two-dimensional quantum materials, photonic systems, and noisy intermediate scale quantum computers.

Finally, the theoretical advances will be extended to develop a theory of critical phenomena enhanced with crystalline symmetry.

The project has a strong educational component, as it fully supports a graduate student. The PI will further develop and publicly release lecture notes for his special topics graduate courses on topological quantum matter and on machine learning. In addition, the PI will remain active in various forms of outreach, at the high school level through outreach programs at his institution and through an active social media presence.

Ongoing industry collaborations will further broaden the societal impacts of the project through its applications to quantum computing.

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.