Akemann Gernot, Mielke Adam, Päßler Patricia
Faculty of Physics, Bielefeld University, Postfach 100131, 33501 Bielefeld, Germany.
Technical University of Denmark, Asmussens Allé, Building 303B, 2800 Kgs. Lyngby, Denmark.
Phys Rev E. 2022 Jul;106(1-1):014146. doi: 10.1103/PhysRevE.106.014146.
A random matrix representation is proposed for the two-dimensional (2D) Coulomb gas at inverse temperature β. For 2×2 matrices with Gaussian distribution we analytically compute the nearest-neighbor spacing distribution of complex eigenvalues in radial distance. Because it does not provide such a good approximation as the Wigner surmise in 1D, we introduce an effective β_{eff}(β) in our analytic formula that describes the spacing obtained numerically from the 2D Coulomb gas well for small values of β. It reproduces the 2D Poisson distribution at β=0 exactly, that is valid for a large particle number. The surmise is used to fit data in two examples, from open quantum spin chains and ecology. The spacing distributions of complex symmetric and complex quaternion self-dual ensembles of non-Hermitian random matrices, that are only known numerically, are very well fitted by noninteger values β=1.4 and β=2.6 from a 2D Coulomb gas, respectively. These two ensembles have been suggested as the only two symmetry classes, where the 2D bulk statistics is different from the Ginibre ensemble.
针对二维（2D）库仑气体在逆温度β下提出了一种随机矩阵表示。对于具有高斯分布的2×2矩阵，我们解析地计算了复本征值在径向距离上的最近邻间距分布。由于它不像一维中的维格纳推测那样能提供很好的近似，我们在解析公式中引入了一个有效βeff(β)，它能很好地描述在β较小时从二维库仑气体中数值得到的间距。它在β = 0时精确地再现了二维泊松分布，这对于大量粒子是有效的。该推测用于拟合来自开放量子自旋链和生态学的两个例子中的数据。非厄米随机矩阵的复对称和复四元数自对偶系综的间距分布仅通过数值已知，它们分别被二维库仑气体中β = 1.4和β = 2.6的非整数值很好地拟合。这两个系综被认为是仅有的两个对称类，其中二维体统计与吉尼贝系综不同。
