Geometrical Structure and Algebraic Structure in Condensed Matter

[1]Bloch电子和量子群Bloch电子是在周期势中运动的粒子，其能谱由著名的Hofstadter‘s蝴蝶给出。它还与一个数学新概念--量子群--有着惊人的联系。我们与世界上其他几个组织研究了这种新的关系。最近，我们成功地得到了Bethe Anatz方程的一个连续极限，即弱场极限，它是对应的一个关键方程。然后我们讨论了有限尺寸修正，它给出了标准连续理论的朗道能级结构。[2]量子霍尔平台相变中的求和规则量子霍尔平台相变是一种典型的量子相变，当磁场强度或随机强度发生变化时，量子相变就会发生。这是一个古老的问题，但解释真实实验…的理论理解还有更多的S失踪。我们进行了数值计算，特别是对拓扑选择规则进行了研究。然后，我们建立了理论上的理解来解释实验，在这些实验中，我们也阐明了系综平均和霍尔电导涨落的含义。[3]各向异性超导中的拓扑效应和选择规则。通过粒子空穴变换，我们在这个问题和量子霍尔效应之间建立了新的对应。[4]Hofstadter问题和d波超导性之间的对偶性我们发现了Hofstadter问题和d波超导电性之间的一个新的对偶性，它给出了几个重要的结果。[5]其他相关结果我们用量子蒙特卡罗方法得到了(1)一、二、三维随机Hubbard模型和(2)具有复制对称性破缺(手征零模)的随机Dirac费米子的几个新结果。较少

[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