Quantum Statistical Mechanics of Lattice Systems

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9706599 Nachtergaele The main focus of the project is on quantum spin models that arise in condensed matter physics. Nachtergaele will uork on three classes of problems in quantum spin (and related) systems. The first is the characterization of the complete set of ground states in infinite volume. Here, ground states are defined in the sense of stability under arbitrary local perturbations. This problem is best tractable for models with a unique ground state, such as the AKLT model. A harder but also more important problem is to characterize the complete set of ground states when this set includes non-periodic states, such as interface states, which is the case, e.g., for the XXZ Heisenberg model. A second problem is to study the fluctuation properties and low-lying excitations of interface states in quantum statistical mechanics models both at zero and non-zero temperature. The proposal is to prove stability of the 111 interface at sufficiently low temperatures in the Falicov-Kimball model, and, in a second phase, to study 111 interfaces in the XXZ Heisenberg model, where new phenomena related to continuous spectrum should be expected. The third proposed line of research is to analyze and use a new implementation of the Density Matrix Renormalization Group method to study various phenomena in quasi-one-dimensional systems. Quantum statistical mechanics is an area of growing interest because many of the phenomena that are being discovered and studied today are intrinsically quantum mechanical: Bose condensation, superfluidity, superconductivity, quantum optics, quantum computation, and many of the most interesting aspects of magnetism. Quantum spin models are models for the magnetic properties of a large variety of materials, many of which have only been recently synthesized. The importance of magnetic properties is not just that they allow us to understand the behavior of magnets. Often there is an interaction between magnetic (spin) structure and elect ronic and geometric structure of the material. The novel spin-Peierls materials that in the last couple of years received so much attention are a good example of this interaction. It has been known for some time that antiferromagnetic correlations are an important aspect of high-temperature superconductors. Another example where magnetic and transport properties influence each other are the Giant Magneto Resistance materials. The approach in this project is mathematical, and is complementary to the experimental and theoretical research on the same materials by chemists and physicists. This proposal addresses fundamental questions about the mathematical structure of quantum lattice models. Nachtergaele will also work on the implementation of new mathematical insights into more efficient computational methods. This research will provide hands-on learning opportunities for graduate and undergraduate students who not only want to become well-versed in mathematics but also have an interest in concrete applications. In the process these students will become familiar with the language of physicists and chemists working on the same or related problems. This will prepare them better for future careers in an interdisciplinary environment.