Theoretical Studies of Quantum Systems with Strong Interaction: Geometry and Topology of Quantum States and Flows

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NONTECHNICAL SUMMARYThis award supports fundamental research and education in the properties of electrons inside what are called "topological" solids. Whereas in metals, electrons interact with each other only weakly, so that their motion resembles that of molecules in a gas, in these newly appreciated materials, the electrons interact strongly, forming a "quantum liquid." Just as the science of hydrodynamics can describe the turbulent motion of water and other familiar liquids, this project develops a hydrodyanmics of quantum liquids.In the last decade, materials physicists have studied the topological nature of quantum fluids, referring to unusual global properties, including resistanceless current flows around edges, that are robust against sample imperfections and impurities. Most theoretical studies in this subject are focused on properties of the ground (lowest-energy) state and on linear-transport theory, according to which, for example, the electrical current in a wire is proportional to the applied voltage. However, it has became apparent that the fundamental features of quantum fluids lie in hydrodynamics, beyond linear-response theory, and involve not just topology but also geometry. Geometric, hydrodynamic, properties reflect a response of a quantum system to local bending of the sample. The project aims to push forward two novel research directions in quantum materials physics: (i) a geometric theory of quantum topological fluids, specifically of the fractional quantum Hall effect, and (ii) a hydrodynamic description of motions of such fluids.Hydrodynamics is, perhaps, the most developed branch of physics, where fundamental laws interwind with vast applications to phenomena at all scales. At the same time, hydrodynamics hosts notoriously difficult unsolved problems. One is turbulence; another is turning classical hydrodynamics into a quantum theory. Both problems are commonly considered intractable. At the same time, nature confronts us with experimentally accessible and beautiful quantum fluids like superfluid helium, which flows without resistance, and the fractional quantum Hall effect. Earlier attempts to quantize hydrodynamics, going back to Landau and Feynman, led to the semiclassical theory. Today, the emergence of electronic and atomic quantum fluids calls for a full scope of quantization. This research brings advanced methods of theoretical physics combined with methods of modern geometry to material science with a focus on experimentally-measurable signatures of geometric phenomena in flows. Graduate students will receive broad training in theoretical techniques, advanced mathematical methods, and communication skills.TECHNICAL SUMMARYThe award supports research into two novel themes in condensed matter theory: the geometric and hydrodynamic approaches to the theory of quantum fluids. The motivating case study is the fractional quantum Hall effect (FQHE). The theory targets precise quantization of transport coefficients and the fundamental role of geometry in quantized fluids. The work also aims to develop experimental settings for observing effects of geometry in semiconductors, cold gases, classical chiral flows, and chiral metafluids. Another theme of the work is the search for the local conformal symmetries in chiral quantum liquids and at the same time in classical turbulent flows. A related theme investigates physical applications of quantum anomalies, primarily the gravitational anomaly, in chiral quantum flows. Finally, the work aims to develop a hydrodynamic approach to non-linear flows in quantum systems with topological characterizations and search for the relation with turbulence.The work exploits advanced methods of modern geometry, conformal field theory, anomalies of quantum field theory, and hydrodynamics adapted to study quantum materials. The foci of the project are quantum systems with a topological characterization. The transport coefficients in such systems are quantized with unmatched precision. A reason for that is that these coefficients are topological invariants of holomorphic bundles. Recent developments show that the quantization is related to the geometry of quantum states, focusing on local properties. In turn, topological properties are merely the global reflection of geometric properties. The geometric properties describe a transformation of quantum states under a variation of the underlying metric. As a result, they govern the motion of the quantum fluid and in the end determine its hydrodynamics. The work will advance understanding of non-linear aspects of non-equilibrium states in topological quantum systems. There are many indications that the nature of flows in topological quantum fluids is deeply connected with that of turbulent flows in classical fluids. The work provides insights into the geometry of turbulent flows and builds a platform for the engineering of topological metamaterials. Another target of the project is the quantization of hydrodynamics, a long-standing fundamental problem of quantum theory, often considered intractable. Recent understanding of the role of the gravitational anomaly in the FQHE suggests a clear path to overcoming difficulties of quantization. The project seeks to develop a comprehensive scheme of quantization of hydrodynamics of two- and three-dimensional chiral incompressible flows. The work adapts the methods of modern geometry to materials science and non-equilibrium statistical mechanics, forging links between different disciplines. The proposed work has an interdisciplinary character: it addresses fundamental problems of materials science and at the same time contributes to the fields of hydrodynamics and modern geometry. Education and mentoring are integral parts of the proposal. The work provides a quality platform to attract and to train theoretically minded students and to prepare them for careers in academia and science-related industry. Students will receive broad training in a full scope of research: analytical reasoning, mastering advanced mathematical methods by applying them to physical systems, projecting theoretical research to realistic materials, and communication with experimentalists and researchers in adjacent fields.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.