论文标题
与$ k $ body互动的嵌入式高斯单位合奏的两点相关功能的双变量矩
Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with $k$-body interactions
论文作者
论文摘要
具有$ k $ - 体相互作用的嵌入式随机矩阵合奏已建立，适合许多量子系统。对于这些整体，尽管这些集团在50年前引入了两个点相关函数，但尚未得出。随机矩阵集合的特征值中的两点相关函数是两个特征值的特征值密度的集合平均值，说$ e $和$ e^\ prime $。数字方差和dyson-mehta $δ_3$统计量之类的波动度量是由两点函数定义的，因此合奏中的水平运动的方差也是如此。最近，人们认识到，对于具有$ k $ body交互的嵌入式合奏，单点函数（集合平均特征值）遵循所谓的$ q $ - 正常分布。这样，可以通过从$ q $ - 正常的形式开始并使用相关的$ q $ - hermite polyenmials $he__ζ（x | q）$来扩展特征值密度。协方差$ \上线{s_ζs_{ζ^\ prime}} $（横层表示膨胀系数的集合平均值）$s_ζ$，$ζ\ ge 1 $在这里确定两个点功能，因为它们是两分矩的线性组合$σ_{pq} $σ_{pq} $ twot Twip-twop twip-twopoint of Twip twip-pq} $的两点功能。除了描述所有这些内容外，在本文中，派生的是双变量矩的公式$σ_{pq} $，带有$ p+q \ le 8 $，是两点相关功能的，用于与$ k $ - body互动的嵌入式高斯单位合奏，适用于$ m $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $。用于获得公式的是$ su（n）$ wigner-racah代数。这些带有有限$ n $校正的公式用于在渐近限制中为协方差$ \ overline {s_ζs_{ζ^\ prime}} $得出公式。
Embedded random matrix ensembles with $k$-body interactions are well established to be appropriate for many quantum systems. For these ensemble the two point correlation function is not yet derived though these ensembles are introduced 50 years back. Two-point correlation function in eigenvalues of a random matrix ensemble is the ensemble average of the product of the density of eigenvalues at two eigenvalues say $E$ and $E^\prime$. Fluctuation measures such as the number variance and Dyson-Mehta $Δ_3$ statistic are defined by the two-point function and so also the variance of the level motion in the ensemble. Recently, it is recognized that for the embedded ensembles with $k$-body interactions the one-point function (ensemble averaged density of eigenvalues) follows the so called $q$-normal distribution. With this, the eigenvalue density can be expanded by starting with the $q$-normal form and using the associated $q$-Hermite polynomials $He_ζ(x|q)$. Covariances $\overline{S_ζS_{ζ^\prime}}$ (overline representing ensemble average) of the expansion coefficients $S_ζ$ with $ζ\ge 1$ here determine the two-point function as they are a linear combination of the bivariate moments $Σ_{PQ}$ of the two-point function. Besides describing all these, in this paper derived are formulas for the bivariate moments $Σ_{PQ}$ with $P+Q \le 8$, of the two-point correlation function, for the embedded Gaussian unitary ensembles with $k$-body interactions [EGUE($k$)] as appropriate for systems with $m$ fermions in $N$ single particle states. Used for obtaining the formulas is the $SU(N)$ Wigner-Racah algebra. These formulas with finite $N$ corrections are used to derive formulas for the covariances $\overline{S_ζS_{ζ^\prime}}$ in the asymptotic limit.
