Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices
Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices
复制标题
DOI：
10.1103/physrevlett.112.254101
复制
发表时间：
2014-06-26
影响因子：
8.6
通讯作者：
Vivo, Pierpaolo
中科院分区：
文献类型：
--
作者：
Marino, Ricardo;Majumdar, Satya N.;Vivo, Pierpaolo
我们考虑N × N高斯随机矩阵，其特征值的平均密度具有[-root 2，root 2]上的Wigner周期形式。对于这样的矩阵，使用库仑气体技术，我们计算概率P-N，P-L（N-L）的大N行为，N-L特征值位于盒[-L，L]。该概率缩放为P-N，P-L（N-L = kappa N-L）近似于exp（-beta N-2 psi（L）（kappa（L），其中beta是系综的Dyson指数，psi（L）（kappa（L））是我们精确计算的与beta无关的速率函数。当L变化时，我们确定了三种机制：（i）N-1根2（尾）。我们发现了数量方差V-N（L）作为L的函数的一个戏剧性的非单调行为：在与ln（NL）成比例的对数增长之后（当L类似于O（1/N）时），V-N（L）随着L接近周期的边缘而突然减小，然后随着L > root 2而衰减为拉伸指数。V-N（L）在边缘处的这种“衰减”由波浪线（beta）上的缩放函数（V）来描述，该缩放函数（V）在主体（i）和尾部（iii）之间平滑地插值。对于beta = 2，我们根据Airy核显式地计算（V）在波浪号（2）上。这些分析结果，通过数值模拟验证，直接提供β = 2的粒子数波动在零温度下的一维无自旋费米子在谐波陷阱的完整统计。
We consider N x N Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over [-root 2, root 2]. For such matrices, using a Coulomb gas technique, we compute the large N behavior of the probability P-N,P-L(N-L) that N-L eigenvalues lie within the box [-L, L]. This probability scales as P-N,P-L(N-L = kappa N-L) approximate to exp (-beta N-2 psi(L)(kappa(L))), where beta is the Dyson index of the ensemble and psi(L)(kappa(L)) is a beta-independent rate function that we compute exactly. We identify three regimes as L is varied: (i) N-1 root 2 (tail). We find a dramatic nonmonotonic behavior of the number variance V-N(L) as a function of L: after a logarithmic growth proportional to ln(NL) in the bulk (when L similar to O(1/N), V-N(L) decreases abruptly as L approaches the edge of the semicircle before it decays as a stretched exponential for L > root 2. This "dropoff" of V-N(L) at the edge is described by a scaling function (V) over tilde (beta) that smoothly interpolates between the bulk (i) and the tail (iii). For beta = 2 we compute (V) over tilde (2) explicitly in terms of the Airy kernel. These analytical results, verified by numerical simulations, directly provide for beta = 2 the full statistics of particle-number fluctuations at zero temperature of 1D spinless fermions in a harmonic trap.