Equilibrium and Non-Equilibrium Quantum Statistical Mechanics

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Bruno Nachtergaele will work on problems in equilibrium and non-equilibrium quantum statistical mechanics. The main focus is on quantum spin models, but also some other quantum many-body systems will be studied. There are four groups of problems in the project. 1) Rigorous study of Chern-Simons phases in quantum spin models, in particular in the Freedman model. 2) Bosonization and quantum central limit theorems, with applications to the analysis of the asymptotics for large spin of the low-lying spectrum of the XXZ Heisenberg chain. 3) Microscopic models for magnetic interface dynamics. This includes the study of bound states of domain walls and droplets at impurities, diffusion of domain walls, domains in dimensions higher dimensions and electron scattering in a magnetic wire with domain walls. 4) Non-equilibrium stationary states, entropy production, and the microscopic description of thermodynamic currents. For the last two topics, Nachtergaele will study concrete dynamical questions of simple models that have low-energy states with non-trivial geometric structures such as interfaces. One of the long-term goals is to make progress in the theory of non-equilibrium statistical mechanics.In order to quantitatively understand many properties of electronic components and measuring devices at the nanometer scale, analytic methods are needed that allow us to study models that take into account both quantum mechanical effects and interface phenomena. Nachtergaele will work on model problems to develop such methods. Particular emphasis will be put on interfaces between magnetic domains and their dynamics. This is a fundamental problem in understanding the dynamics of magnetic memory devices, such as computer hard disks. The same mathematical techniques also apply to the new field of spintronics. The approach is to study a number of specific models with the expectation that a sufficiently detailed solution of these models will point the way to answers for similar questions in more general systems. The project contains a number of attractive research problems for students from a wide variety of backgrounds, as well as for mathematics students interested in opportunities to become familiar with areas outside of mathematics where mathematics is applied at a high level. Modern materials science, which very much relies on computational modeling and design of materials, is an example.