连通谱形状因子的流体力学理论
Hydrodynamic Theory of the Connected Spectral Form Factor
摘要：
量子混沌的一种表现形式是随机矩阵状的细粒度能谱。在逆电平间隔时间之前，随机矩阵理论预测频谱形状因子的连接部分中的方差增加的“斜坡”。然而，在现实的量子混沌系统中，频谱形状因子的有限时间动力学要丰富得多，纯随机矩阵斜坡只出现在足够晚的时间。在这篇文章中，我们提出了一个流体动力学理论的连接频谱形状因子之前的反水平间隔时间。我们从讨论精确对称性和光谱的拉伸和折叠开始。然后，我们推导出一个一般公式的频谱形状因子的系统几乎保守的部门在返回概率和频谱形状因子在每个部门。接下来，我们认为，波动流体动力学理论可以适应从通常的Schwinger-Keldysh轮廓的频谱形状因子所需的周期性时间设置，我们明确表明，恢复的能量扩散的情况下的一般公式。我们还开始在这个修改后的流体动力学框架的相互作用的影响的研究，并显示如何无干扰的时间，定义为频谱形状因子接近纯随机矩阵的结果所需的时间，是由慢流体动力学模式控制。
英文摘要：
One manifestation of quantum chaos is a random-matrix-like fine-grained energy spectrum. Prior to the inverse level spacing time, random matrix theory predicts a `ramp' of increasing variance in the connected part of the spectral form factor. However, in realistic quantum chaotic systems, the finite time dynamics of the spectral form factor is much richer, with the pure random matrix ramp appearing only at sufficiently late time. In this article, we present a hydrodynamic theory of the connected spectral form factor prior to the inverse level spacing time. We start from a discussion of exact symmetries and spectral stretching and folding. We then derive a general formula for the spectral form factor of a system with almost-conserved sectors in terms of return probabilities and spectral form factors within each sector. Next we argue that the theory of fluctuating hydrodynamics can be adapted from the usual Schwinger-Keldysh contour to the periodic time setting needed for the spectral form factor, and we show explicitly that the general formula is recovered in the case of energy diffusion. We also initiate a study of interaction effects in this modified hydrodynamic framework and show how the Thouless time, defined as the time required for the spectral form factor to approach the pure random matrix result, is controlled by the slow hydrodynamics modes.
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