论文标题
't Hooft限制的光谱形式 - 中间人与普遍性
The spectral form factor in the `t Hooft limit -- Intermediacy versus universality
论文作者
论文摘要
光谱外形（SFF）是表征具有离散光谱系统特征值统计量的便捷工具，因此是量子混沌性的代理。这项工作介绍了Chern-Simons矩阵模型（CSMM）的SFF的分析计算，该计算首先是为了描述Mobility Edge的无序电子的中间水平统计。 CSMM的特征是参数$ 0 \ leq Q \ leq 1 $，其中圆形单一合奏（CUE）以$ q \ to 0 $恢复。后来发现CSMM是$ u（n）$ chern-simons理论的矩阵模型描述，这是$ s^3 $的，这是对拓扑字符串理论的双重描述。频谱外形与$（2n，2）$ - 圆环链接的彩色homfly不变性成正比，其两个组件分别带有基本和反量的表示。我们检查是否服用$ n \ to \ infty $，同时保留$ q <1 $将连接的SFF降低到单位坡度的精确线性斜坡，这证实了CSMM的特定情况的Arxiv：2012.11703的主要结果。然后，我们考虑't Hooft限制，其中$ n \ to \ infty $和$ q \ to 1^ - $，这样$ y = q^n $仍然是有限的。当我们采用$ q \至1^ - $时，这构成了提示限制的相反极端。在h Hooft限制中，连接的SFF变成了一个非凡的多项式序列，据作者意识到，到目前为止，这些序列尚未出现在文献中。差距在频谱中打开，并且通过恒定重新进行重新展开后，连接的SFF近似于所有$ y $ $ y $ y $ y \ y \ 1 $的线性斜坡，其中连接的SFF为零。因此，我们发现，尽管引入了CSMM来描述中间统计信息，并且'thooft限制是提示的相反限制，但我们仍然以$ y $ $ y $ y $ y \ y \ y \约1 $恢复Wigner-dyson通用性。
The Spectral Form Factor (SFF) is a convenient tool for the characterization of eigenvalue statistics of systems with discrete spectra, and thus serves as a proxy for quantum chaoticity. This work presents an analytical calculation of the SFF of the Chern-Simons Matrix Model (CSMM), which was first introduced to describe the intermediate level statistics of disordered electrons at the mobility edge. The CSMM is characterized by a parameter $ 0 \leq q\leq 1$, where the Circular Unitary Ensemble (CUE) is recovered for $q\to 0$. The CSMM was later found as a matrix model description of $U(N)$ Chern-Simons theory on $S^3$, which is dual to a topological string theory characterized by string coupling $g_s=-\log q$. The spectral form factor is proportional to a colored HOMFLY invariant of a $(2n,2)$-torus link with its two components carrying the fundamental and antifundamental representations, respectively. We check that taking $N \to \infty$ whilst keeping $q<1$ reduces the connected SFF to an exact linear ramp of unit slope, confirming the main result from arXiv:2012.11703 for the specific case of the CSMM. We then consider the `t Hooft limit, where $N \to \infty$ and $q \to 1^-$ such that $y = q^N $ remains finite. As we take $q\to 1^-$, this constitutes the opposite extreme of the CUE limit. In the `t Hooft limit, the connected SFF turns into a remarkable sequence of polynomials which, as far as the authors are aware, have not appeared in the literature thus far. A gap opens in the spectrum and, after unfolding by a constant rescaling, the connected SFF approximates a linear ramp of unit slope for all $y$ except $y \approx 1$, where the connected SFF goes to zero. We thus find that, although the CSMM was introduced to describe intermediate statistics and the `t Hooft limit is the opposite limit of the CUE, we still recover Wigner-Dyson universality for all $y$ except $y\approx 1$.
