论文标题
广义Rosenzweig-Porter模型的复制方法
Replica approach to the generalized Rosenzweig-Porter model
论文作者
论文摘要
具有真实（GOE）偏高条目的广义Rosenzweig-Porter模型可以说是最简单的随机矩阵集合，显示具有分形特征态的相位，我们在此使用replica方法在此表征。我们首先在平均光谱密度的限制下得出分析表达式，其中矩阵的大小$ n $很大但有限。然后，我们以有限的间隔关注特征值的数量，并计算其累积的生成函数以及水平的可压缩性，即前两个累积物的比率：这些是描述局部水平统计信息的有用工具。特别是，当将系统探测到无用能量的尺度上时，我们会明确地计算该水平的压缩性通过通用缩放函数来描述。有趣的是，发现相同的缩放函数来描述该制度中复合物（GUE）Rosenzweig-Porter模型的水平压缩性。我们通过数值测试确认结果。
The generalized Rosenzweig-Porter model with real (GOE) off-diagonal entries arguably constitutes the simplest random matrix ensemble displaying a phase with fractal eigenstates, which we characterize here by using replica methods. We first derive analytical expressions for the average spectral density in the limit in which the size $N$ of the matrix is large but finite. We then focus on the number of eigenvalues in a finite interval and compute its cumulant generating function as well as the level compressibility, i.e., the ratio of the first two cumulants: these are useful tools to describe the local level statistics. In particular, the level compressibility is shown to be described by a universal scaling function, which we compute explicitly, when the system is probed over scales of the order of the Thouless energy. Interestingly, the same scaling function is found to describe the level compressibility of the complex (GUE) Rosenzweig-Porter model in this regime. We confirm our results with numerical tests.
