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

随机矩阵理论研究：数学方面及其在量子物理中的应用

• 批准号: 16540342
• 负责人: MOCHIZUKI Shinsuke
• 金额: $ 2.48万
• 依托单位: Shimane University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540342/
• 关键词: Random Matrix Theory; Energy Level Statistics; Gauge Field Theory; Quantum Chaos; Quantum Field Theory; Anderson Localization; Lattice Gaufre Theory; 局在; レプリカ法; 超対称法; カイラルラグランジアン; ブラウン運動; 粒子間相互作用

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

无序系统的数学方面构建了一种有效的理论，该理论描述了具有无序和相互作用的任意强度的多体量子系统的能量水平统计，我们已将以下技术应用于问题。1。相互作用系统相互作用的复制方法。通过为基于PFAFFIAN晶格层次结构在内的集成系统开发一个新颖的框架，我们部分复制了全球对称类AIII和CI.2的ID无序系统的单独生成的过渡水平统计。 AD/CFT动机方法。作为1的完整性，我们将量规 - 重力对应（字符串理论中的单个主要概念）应用于频谱非线性Sigma模型从无序系统中的推导。 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关于非异型晶格的理论。在不同的温度和边界条件下，我们精确地测量了频谱体积中Kogut-Susskind Dirac操作员特征值的水平间距分布和两级相关函数，以及相关特征功能的分形，在QCD相位和周围的QCD相位转移。测得的水平间距分布适用于作者已显示的矫形器RMT的分析结果，该结果已显示出描述定位过渡时的临界水平统计信息，并与后来的临界值匹配。确定的电导参数非常接近3D Anderson模型的电导参数为Diakonov-Petkov猜想（通过夸克波函数的定位恢复手性对称性）提供了强有力的支持。较少的

项目成果

Replica Treatment of Calogero-Sutherland Model

Calogero-Sutherland 模型的复制处理

• 发表时间: 2004
• 作者: Shinsuke;Nishigaki;S.M. Nishigaki;T. Nagao;S. Nishigaki;T. Nagao;S.M. Nishigaki;S.M. Nishigaki;S.M. Nishigaki;西垣 真祐;西垣 真祐;S.M. Nishigaki;S.M. Nishigaki
• 通讯作者: S.M. Nishigaki

ChPT for QCD-like Theories : from universality to phase diagram

类 QCD 理论的 ChPT：从普适性到相图

• 发表时间: 2006
• 作者: Shinsuke;Nishigaki;S.M. Nishigaki;T. Nagao;S. Nishigaki;T. Nagao;S.M. Nishigaki;S.M. Nishigaki
• 通讯作者: S.M. Nishigaki

対称性と乱雑さ

对称与无序

• 发表时间: 2006
• 期刊: パリティ 21・5
• 作者: Taro;Nagao;T. Nagao;西垣 真祐;西垣 真祐
• 通讯作者: 西垣 真祐

Aspects of Random Matrices

随机矩阵的方面

• 发表时间: 2007
• 期刊: Mathematical Sciences 524
• 作者: Taro;Nagao
• 通讯作者: Nagao

Random Matrix Theory for Dirac Spectra

狄拉克谱的随机矩阵理论

• 发表时间: 2004
• 作者: Shinsuke;Nishigaki;S.M. Nishigaki;T. Nagao;S. Nishigaki;T. Nagao;S.M. Nishigaki;S.M. Nishigaki;S.M. Nishigaki
• 通讯作者: S.M. Nishigaki