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Andreas Osterloh
Bethe Ansatz Solvability, Correlated Hopping, and the Connection to Generalized Exchange Statistics
Supervisor: Prof. Dr. Ulrich Eckern [Theoretical physics II]
Date of oral examination: 08/25/2000
107 pages, english , Shaker Verlag, Aachen 2001; ISBN 3-8265-8351-5
Integrability is one of the most striking characteristics of a model, not primary since it is seeded extraordinary sparsely in the field of models. In fact, it represents a very strong demand on the dynamics of a model, which many of the models with physical relevance do not fulfill. Yet, there are a lot of integrable models, such as the harmonic oscillator, some (classical) vertex models, (1+1)-dimensional field theories, conformal invariant models in two or (1+1) dimensions, the Kondo model and the one-dimensional XXZ and Hubbard model, which all have physical importance. Though mainly living in one spatial dimension, integrable systems even became closer to physical reality, since quasi one-dimensional systems can be prepared and explored experimentally. Important examples are Quantum Hall samples at very high magnetic fields, polymers, extremely anisotropic media and carbon nano-tubes. Finally, they have been used to construct link polynomials which are used to characterize and distinguish knots and links in Knot theory.

Another important concept in quantum physics is the exchange statistics of the constituent particles. In three and higher dimensions there are only two classes of exchange statistics, namely bosonic and fermionic statistics. Though electrons are clearly fermions, it is often an advantage in the presence of interactions to not deal with the electrons themselves, but instead with excitational complexes partly including the interaction. In less than three dimensions, these quasi particles can follow any statistics and have therefore been termed as anyons in dimension two.

In this thesis the effect of modifying the particle statistics on the integrability of a model is discussed. Focusing on the XXZ- and the Hubbard model, two approaches are prosecuted. The first constitutes in an elaboration of the coordinate Bethe ansatz technique for particles with generalized statistics. In the second approach a mapping between generalized statistics and so-called correlated hopping is presented. The hopping term is the kinetic part of the Hamiltonian for lattice models, which is called correlated if it depends on the positions of the particles. The effect of including these correlations on the integrability of a model is then studied systematically, yielding a set of necessary and sufficient conditions, the correlations have to fulfill in order to conserve integrability. These conditions are directly mapped onto necessary and sufficient criteria for integrable statistics. Together with these criteria, the modified Bethe equations are obtained and discussed briefly.