Theoretical Studies of Quantum Systems with Strong Interactions

University Of Chicago, Chicago IL

Investigators

Abstract

TECHNICAL SUMMARY The Division of Materials Research, the Physics Division, and the Division of Mathematical Sciences contribute funds to this award. This award supports theoretical research and education in the general areas of theoretical condensed matter physics, and statistical physics focusing on geometrical non-equilibrium phenomena in quantum liquids and statistical mechanics. The study emphasizes a role of singularities and instabilities arising in non-equilibrium processes and quantum and statistical aspects of singularities. The first area of research concerns non-linear quantum hydrodynamics of fractional quantum Hall states with a focus on edge states and the relation of fractional quantum Hall states to conformal invariance. The PI will focus on the quantum non-linear hydrodynamics of electronic liquid in the fractional quantum Hall regime and especially on a topological manifestation of the fractional charge of excitations as solitons on fractional quantum Hall edge states emerging as a result of non-linear dynamics. The PI will also investigate realistic experimental settings where edge solitons can be observed. The second area of research focuses on singularities and emergent conformal symmetry in driven processes. PI will study fingering instability in Lapacian Growth building on the idea that singularities of non-equilibrium patterns occurring at small scales give rise to fractal non-equilibrium patterns visible at a large scale. The work will be built on the theory of viscous shocks developed under prior NSF support. The third theme of research focuses on the statistics of geometrical objects in critical phenomena. This study addresses the long-standing problem of the area-length distribution of critical clusters or domains of different phases in critical phenomena. The research addresses important problems of material research, and simultaneously contributes to emergent fields in mathematical physics and mathematics by synthesizing problems and methods of different disciplines. This award also supports training and mentoring graduate and undergraduate students and a postdoctoral fellow. Research results will be integrated into graduate level courses in modern dynamics, and traditional courses in condensed matter physics. NON-TECHNICAL SUMMARY The Division of Materials Research, the Physics Division, and the Division of Mathematical Sciences contribute funds to this award. This award supports research at the interface with mathematics, and mathematical physics. It focuses on the emergent field of non-linear quantum dynamics with an emphasis on the role of geometry in non-equilibrium processes in quantum electronic and atomic liquids. Similar geometrical phenomena also emerge in critical phenomena of statistical mechanics and processes of growth and aggregation of materials. One thrust of the research focuses on developing a theoretical description of a special kind of quantum liquid. Quantum liquids differ from more familiar liquids in that their properties are dominated by the effects of quantum mechanics. The PI will focus on a particular kind of quantum liquid, a quantum Hall liquid that results when electrons confined to two dimensions in an artificial materials structure made of semiconductors and exposed to a high magnetic field. The way the electrons organize themselves leads to an 'edge state' that wraps around the bulk of the liquid. The 'edge state' is also related to a metallic state that arises at the surfaces of a particular class of insulating materials, known as topological insulators. The PI will develop a hydrodynamic theory of this liquid in a way that emphasizes the role of geometry. Another thrust of the research concerns the investigation of geometrical patterns that emerge close to the transformation of one phase into another, for example water to ice. The PI will use sophisticated mathematical methods to determine the number of critical loops containing one phase in the presence of the other with a particular area and length of boundary. The loops are examples of fluctuating random geometrical shapes and patterns that appear not only in phase transitions but also in growth processes of materials and disordered systems. This effort reflects a new approach to critical behavior, for example phase transformations, in two-dimensions. The broader impact of the proposal is a dissemination of methods of material science to the fields of non-equilibrium statistical mechanics, non-linear physics, probability theory and complex analysis, forging links between the disciplines. Interdisciplinary research area is particularly well suited for the training of graduate students and postdoctoral fellows. It requires mathematical sophistication and deep understanding of fundamental aspects of correlated quantum states and statistical mechanics.