Theoretical Studies of Quantum Systems with Strong Interactions

University Of Chicago, Chicago IL

Investigators

Abstract

TECHNICAL SUMMARY: This award is funded by the Division of Materials Research, the Division of Mathematical Sciences and the Physics Division. This project falls under the umbrella of the NSF-wide Mathematical Sciences Priority Area. This award supports theoretical research focused on singularities arising in non-equilibrium processes. Advances made under the PIs previous award will be used to extend the field into the domain of conformal kinetic and non-linear effects in degenerate quantum systems. The PI aims to define and develop a theory of conformally invariant kinetic processes. Many important systems far from equilibrium show conformal invariance similar to conformal invariance of critical phenomena. However non-equilibrium processes are different. Conformal invariance inevitably leads to singular patterns occurring at small scales. In turn singularities give rise to fractal non-equilibrium patterns visible at a large scale. Fingering instability in Laplacian Growth, fractal clusters in critical systems, hydrodynamic instability in degenerate and coherent quantum systems are subjects of the study. Themes of the study include the origin, statistics, and regularization of singularities, and fractal geometry of stochastic patterns of kinetic processes. The PI will address long-standing problems in established fields and emerging trends, these include: - Statistics of singularities in non-equilibrium classical and quantum processes; - Stochastic geometry of critical systems; - Stochastic growth and aggregation; and - Nonlinear-transport and singularities in correlated quantum systems. Special emphasis will be given to Stochastic Loewner Evolution, an emerging field that provides new tools and poses new questions for criticality in two dimensions and to fractal structures emerging as results of stochastic growth phenomena. The results of the proposed research will enhance knowledge and understanding of complex condensed matter systems. NON-TECHNICAL SUMMARY: This award is funded by the Division of Materials Research, the Division of Mathematical Sciences and the Physics Division. This project falls under the umbrella of the NSF-wide Mathematical Sciences Priority Area. This award supports theoretical condensed matter physics research at an interface with mathematics that is focused on advancing our understanding of complex condensed matter systems. The research focuses on non-equilibrium processes. An important aspect of the PIs work involves growth processes that display snowflake-like fingers that penetrate from one phase into another, as happens in the growth of alloys and semiconductor structures. The PI seeks a fundamental understanding of how these fingering patterns emerge in the growth process. The PI also plans to capitalize on recent advances in mathematics and theoretical physics to study other processes in which random geometric structures play an important role and to study how matter restricted to two-dimensions reorganizes itself through a phase transition. The PI will integrate education and research through training and mentoring graduate and undergraduate research students, and making novel contributions to the Research Experiences for Undergraduates, and Mathematics Educators programs. The results of the proposed research will enhance knowledge and understanding of complex condensed matter systems.