论文标题
$ 2 \ times 2 $通用随机矩阵合奏团的混乱与集成性交叉的动态签名
Dynamical Signatures of Chaos to Integrability Crossover in $2\times 2$ Generalized Random Matrix Ensembles
论文作者
论文摘要
我们介绍了一个普通化$ 2 \ times 2 $真实的对称随机矩阵的两参数合奏，称为$β$ -Rosenzweig-porter合奏（\ brpe），由$β$参数化，这是一种类似的库仑气体模型的虚拟临界温度，以及$γ$，控制着混乱的相对强度。 \ brpe \涵盖了戴森所有三重对称类别的RPE：正交，单一和符合性的$β= 1,2,4 $。首先，我们通过计算最近的邻居间距（NNS）的密度和第二矩来研究能量相关性，并稳健地量化各种水平排斥程度之间的交叉。其次，动力学特性是根据忠诚度的时间演变确定的，从而可以识别特征性的th thouless和平衡时间尺度。平均保真度中相关孔的相对深度是从混乱到集成性的交叉的动态特征，使我们能够在$γ$ - $β$平面中构造\ brpe \的相图。我们的结果与数值计算的保真度定性协议，以$ n \ gg2 $矩阵集合。此外，我们观察到，对于$ n $，NNS的第二矩和相关孔的相对深度显示出二阶相变的相对深度，$γ= 2 $。
We introduce a two-parameter ensemble of generalized $2\times 2$ real symmetric random matrices called the $β$-Rosenzweig-Porter ensemble (\brpe), parameterized by $β$, a fictitious inverse temperature of the analogous Coulomb gas model, and $γ$, controlling the relative strength of disorder. \brpe\ encompasses RPE from all of the Dyson's threefold symmetry classes: orthogonal, unitary and symplectic for $β=1,2,4$. Firstly, we study the energy correlations by calculating the density and 2nd moment of the Nearest Neighbor Spacing (NNS) and robustly quantify the crossover among various degrees of level repulsions. Secondly, the dynamical properties are determined from an exact calculation of the temporal evolution of the fidelity enabling an identification of the characteristic Thouless and the equilibration timescales. The relative depth of the correlation hole in the average fidelity serves as a dynamical signature of the crossover from chaos to integrability and enables us to construct the phase diagram of \brpe\ in the $γ$-$β$ plane. Our results are in qualitative agreement with numerically computed fidelity for $N\gg2$ matrix ensembles. Furthermore, we observe that for large $N$ the 2nd moment of NNS and the relative depth of the correlation hole exhibit a second order phase transition at $γ=2$.
