Understanding the footprints of chaos in quantum-many-body systems has been under debate for a long time. In this work, we study the echo dynamics of the Sherrington-Kirkpatrick (SK) model with transverse field under effective time reversal. We investigate numerically its quantum and semiclassical dynamics. We explore how chaotic many-body quantum physics can lead to exponential divergence of the echo of observables and we show that it is a result of three requirements: i) the collective nature of the observable, ii) a properly chosen initial state and iii) the existence of a well-defined chaotic semi-classical (large-$N$) limit. Under these conditions, the echo grows exponentially up to the Ehrenfest time, which scales logarithmically with the number of spins $N$. In this regime, the echo is well described by the semiclassical (truncated Wigner) approximation. We also discuss a short-range version of the SK model, where the Ehrenfest time does not depend on $N$ and the quantum echo shows only polynomial growth. Our findings provide new insights on scrambling and echo dynamics and how to observe it experimentally.

中文翻译：

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Sherrington-Kirkpatrick模型中的量子回波动力学

理解量子多体系统中混沌的足迹已经有很长时间了。在这项工作中，我们研究了在有效时间反转下具有横向场的Sherrington-Kirkpatrick（SK）模型的回波动力学。我们在数值上研究其量子和半经典动力学。我们探讨如何混乱的多体量子物理学会导致可观的回波的指数分歧，我们表明它是三个要求的结果：1）可观察的集体性质，II）中，合理的初始状态和iii）存在明确定义的混沌半经典（大$ N $）限制。在这些条件下，回声以指数形式增长直至Ehrenfest时间，该时间与自旋数$ N $成对数关系。在这种制度下 半经典（截断的维格纳）近似很好地描述了回声。我们还讨论了SK模型的短距离版本，其中Ehrenfest时间不取决于$ N $，量子回波仅显示多项式增长。我们的发现为加扰和回声动力学以及如何通过实验观察提供了新的见解。