Geometrical Structure and Algebraic Structure in Condensed Matter

[1]布洛赫电子和量子群布洛赫电子是在周期势中运动的粒子，其能谱由著名的霍夫施塔特蝴蝶分形给出。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团体研究了这种新的关系。最近，我们成功地对对应的关键方程Bethe Anasterian方程取了一个连续极限，即弱场极限。然后我们讨论了有限尺寸修正，它给出了标准连续统理论的朗道能级结构。它可以进一步说明这个问题与共形场论的关系。[2]量子霍尔平台跃迁中的求和规则量子霍尔平台跃迁是一种典型的量子相变，它发生在磁场强度或随机性强度变化时。这是一个古老的问题，但理论上的理解来解释真实的实验 ...更多信息 s不见了。我们进行了数值计算，特别是侧重于拓扑选择规则。在此基础上，我们建立了解释实验的理论基础，阐明了系综平均和霍尔电导涨落的意义。[3]各向异性超导中的拓扑效应和选择律重点讨论了各向异性超导。我们利用粒子空穴变换建立了这个问题与量子霍尔效应之间的新的对应关系。然后，我们导出了一个拓扑不变量来描述超导态，它给出了一个有趣的限制的选择规则的拓扑转变。[4]Hofstadter问题与d波超导性之间的对偶性我们发现了Hofstadter问题与d波超导性之间的一种新的对偶性，它给出了几个重要的结果。[5]我们用量子蒙特卡罗方法对（1）1、2、3维随机Hubbard模型和（2）复制对称破缺的随机Dirac费米子（手征零模）得到了几个新的结果。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