Research on Random Matrix Theory:Mathematical aspects and its application to quantum physics

无序系统的数学方面为了建立一个有效的理论来描述具有任意无序和相互作用强度的多体量子系统的能级统计，我们应用了以下技术来解决这个问题。无序系统相互作用的数值模拟方法通过开发一个新的框架的副本技巧的基础上可积系统，包括Pfiran晶格层次，我们已经部分再现了数字生成的过渡水平统计的ID无序系统的全球对称类AIII和CI。2。AdS/CFT激励方法。作为对1.的补充，我们将弦理论中唯一的主要概念规范-引力对应应用于从无序系统导出谱非线性σ模型。通过选择无序系统所处边界的特定高维弯曲时空，我们试图导出一个描述后者的标度不变有效理论， ...更多信息 我们通过Dirac算符的谱统计，数值分析了QCD的手征跃迁和禁闭跃迁，即各向异性格点上的猝灭SU（2）规范理论。通过蒙特-卡罗模拟，在不同温度和边界条件下，我们精确测量了QCD相变前后Kogut-Susskind Dirac算符本征值在谱体中的能级间距分布和二能级关联函数，以及相应本征函数的分形性质。测得的能级间距分布与作者提出的描述局域化跃迁临界能级统计量的正交变形RMT的分析结果相吻合，并与后者有很好的匹配。所测得的电导参数与三维安德森模型的电导参数非常接近，这为Diakonov-Petkov猜想（通过夸克波函数的局域化恢复手征对称性）提供了有力的支持。少

Mathematical aspects of disordered systemsAiming to construct an effective theory describing the energy level statistics of many-body quantum systems with arbitrary strength of both disorder and interactions, we have applied the following techniques to the problem.1. Replica method for interacting disordered system. By developing a novel framework for the replica trick based on integrable systems including Pfaffian lattice hierarchy, we have partly reproduced numerically- generated transitional level statistics of ID disordered systems of global symmetry classes AIII and CI.2. AdS/CFT-motivated method. As a complementary to 1., we are applying the gauge-gravity correspondence, the single major concept in string theory, to the derivation of spectral nonlinear sigma model from disordered systems. By choosing ad-hoc higher dimensional curved spacetimes whose boundary the disordered system resides on, we have tried to derive a scale-invariant effective theory describing the latter atop the … More second-order phase transition.Application of RMT to quantum physicsWe have numerically analyzed the chiral and confinement transition of QCD through the spectral statistics of the Dirac operator, namely for the quenched SU(2) gauge theory on nonisotropic lattices. Monte-Carlo-simulating at varied temperatures and boundary conditions, we have precisely measured the level spacing distribution and the two-level correlation function of Kogut-Susskind Dirac operator eigenvalues in the spectral bulk, and the fractality of the associated eigenfunctions, at and around the QCD phase transition. The measured level spacing distribution is fitted against the analytical result of the orthgonal deformed RMT that has been shown by the author to describe the critical level statisitics at the localization transition, and exhibited an excellent match with the latter. That the conductance parameter determined turns out to be very close to that of the 3D Anderson model provides a strong support for the Diakonov-Petkov conjecture (restoration of chiral symmetry by the localization of the quark wave functions). Less