Quantum Statistical Mechanics of Lattice Systems

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ABSTRACTNachtergaele will work on a variety of problems in quantum statistical mechanics. In a first group of problems he will study quantum spin systemsas models for magnetic materials. The main focus here is on equilibriumand non-equilibrium properties of domain walls and other structuressuch as droplets. With Messager, he proposes a new quantum solid-on-solid model for the 111 interface in the Heisenberg XXZ ferromagnet. For this model the stability of the interface at strictly positive temperatures will be studied. The universality of fluctuations of interfaces will be investigated in a class of one-dimensional models. Another goal is to prove the existence of droplets in quantum spin models at zero temperature and to make first steps towards the study of their dynamics. In continuum quantum statistical mechanicsNachtergaele will work on the derivation of the Euler equations from theSchroedinger dynamics of interacting fermions, and on a characterization of the various types of Bose condensation. The latter is a project incollaboration with Zagrebnov and Bru. The first is work already begun with HT Yau. Quantum statistical mechanics is an area of growing interest because many of the phenomena that are being discovered and studied today in condensed matterphysics and material science are intrinsically quantum mechanical: Bosecondensation, superfluidity, superconductivity, quantum optics, quantumcomputation, and many of the most interesting aspects of magnetism at themicroscopic scale. In addition to the implications in material science, thiswork will provide new insights in fundamental problems of quantum statisticalmechanics and lead to analytical methods for their solution. The mathematicalmodels of quantum statistical mechanics, though related to problems in partialdifferential equations, representation theory, spectral theory of operators, and probability theory, pose new mathematical challenges of their own. Many fascinating questions have so far resisted existing analytical and numerical methods. The results of this project will contribute to ourunderstanding of the fundamental mathematical structures of quantum mechanicalmodels and, in the long run, may lead to innovations in numerical workon quantum mechanical models, which is increasingly needed in pure and applied science.