Kumar Sachin, Ahmed Zafar
Theoretical Physics Section, Bhabha Atomic Research Centre, Mumbai 400 085, India.
Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India.
Phys Rev E. 2017 Aug;96(2-1):022157. doi: 10.1103/PhysRevE.96.022157. Epub 2017 Aug 30.
Real nonsymmetric matrices may have either real or complex conjugate eigenvalues. These matrices can be seen to be pseudosymmetric as ηMη^{-1}=M^{t}, where the metric η could be secular (a constant matrix) or depending upon the matrix elements of M. Here we construct ensembles of a large number N of pseudosymmetric n×n (n large) matrices using N[n(n+1)/2≤N≤n^{2}] independent and identically distributed random numbers as their elements. Based on our numerical calculations, we conjecture that for these ensembles the nearest level spacing distributions [NLSDs, p(s)] are sub-Wigner as p_{abc}(s)=ase^{-bs^{c}}(0<c<2) and the distributions of their eigenvalues fit well to D(ε)=A[tanh{(ε+B)/C}-tanh{(ε-B)/C}] (exceptions also discussed). These sub-Wigner NLSDs are encountered in Anderson metal-insulator transition and topological transitions in a Josephson junction. Interestingly, p(s) for c=1 is called semi-Poisson, and we show that it lies close to the form p(s)=0.59sK_{0}(0.45s^{2}) derived for the case of 2×2 pseudosymmetric matrix where the eigenvalues are most aptly conditionally real, E_{1,2}=a±sqrt[b^{2}-c^{2}], which represent characteristic coalescing of eigenvalues in parity-time (PT) -symmetric systems.
实非对称矩阵可能有实特征值或复共轭特征值。这些矩阵可被视为伪对称矩阵，因为ηMη⁻¹ = Mᵗ，其中度规η可以是长期的（常数矩阵），也可以取决于M的矩阵元素。在这里，我们使用N[n(n + 1)/2≤N≤n²]个独立同分布的随机数作为元素，构造了大量N个n×n（n很大）伪对称矩阵的系综。基于我们的数值计算，我们推测对于这些系综，最近能级间距分布[NLSDs, p(s)]是亚维格纳分布，即pₐbₑ(s) = ase⁻ᵇˢᶜ(0 < c < 2)，并且它们的特征值分布很好地符合D(ε) = A[tanh{(ε + B)/C} - tanh{(ε - B)/C}]（也讨论了例外情况）。这些亚维格纳NLSDs出现在安德森金属 - 绝缘体转变和约瑟夫森结中的拓扑转变中。有趣的是，c = 1时的p(s)被称为半泊松分布，并且我们表明它接近于为2×2伪对称矩阵情况导出的形式p(s) = 0.59sK₀(0.45s²)，其中特征值最恰当地是条件实的，E₁,₂ = a ± sqrt[b² - c²]，这代表了宇称 - 时间（PT）对称系统中特征值的特征合并。
