Geometrical Structure and Algebraic Structure in Condensed Matter

点击翻译按钮获取中文摘要

[1] Bloch Electrons and the Quantum GroupBloch electrons are particles moving in a periodic potential whose energy spectrum is given by the famous fractal known as a Hofstadter's Butterfly. It also has surprising relation to a mathematical new concept, the quantum group. We studied this new relation with several other groups in the world. Recently, we have succeeded to take a continuum limit, that is, week field limit for the Bethe Ansatz equation which is a key equation for the correspondence. Then we discussed the finite size correction, which gives the Landau level structure for the standard continuum theory. It may give further relation between the problem and the conformal field theory.[2] Sum rule in the Quantum Hall plateau transitionThe Quantum Hall Plateau transition is a typical quantum phase transition which occurs when the field strength of the magnetic field or the randomness strength changes. It's an old problem but the theoretical understanding to explain real experiment … More s were missing. We have carried out numerical calculations especially focusing on the topological selection rules. Then we have established theoretical understanding to explain the experiments in which we clarified the meaning of the ensemble average and fluctuation of the Hall conductance as well.[3] Topological effect in the anisotropic superconductivity and selection ruleAnisotropic superconductivity is focused. We have established a new correspondence between this problem and the Quantum Hall effect by using a particle Hole transformation. Then we have derived a topological invariant to characterize the superconducting states which gives an interesting restriction for the selection rule for the topological transition.[4] Duality between the Hofstadter problem and the d-wave superconductivityWe have discovered a new duality, between the Hofstadter problem and the d-wave superconductivity which gives several important consequences.[5] Other related resultsWe have obtained several new results for (1) random Hubbard models in 1, 2, and 3 dimensions by the Quantum Monte Carlo method and (2) Random Dirac Fermions with replica symmetry breaking (Chiral zero modes). Less