混沌系统的Fisher零点与谱形式因子的涨落
Fisher zeroes and the fluctuations of the spectral form factor of chaotic systems
AI总结：
针对量子混沌系统谱形式因子的涨落问题，本文通过Fisher零点揭示了尖峰的产生机制，并在引入能级排斥的修改型随机能级模型及SYK模型中验证了该机制。
摘要：
量子混沌系统的谱形式因子具有熟悉的“斜坡+平台”形式。人们已经设计出了在半经典或热力学极限下确定其形式的技术，这两种情况都基于对能量范围或系统系综的平均。对于单个实例，涨落很大，在极限下不会消失，并且取决于系综元素本身，因此似乎对整个程序提出了质疑。将谱因子视为复逆温度β_R+iβ_I（β_I≡τ表示时间）下配分函数的模，它具有Fisher零点区域，这是复温度平面上Yang-Lee零点的类似物。谱因子中的大尖峰实际上是由β_I参数化的线对这些零点的近距离擦过所导致的。最大的尖峰确实是广延的且对细节极其敏感，但我们表明它们既呈指数级罕见又呈指数级狭窄。受此启发，并在Derrida关于随机能量模型工作的启发下，我们在此研究了一个修改后的随机能级模型，在其中我们引入了能级排斥。我们还验证了在SYK模型中产生尖峰的机制是相同的。
英文摘要：
The spectral form factor of quantum chaotic systems has the familiar `ramp $+$ plateau' form. Techniques to determine its form in the semiclassical or the thermodynamic limit have been devised, in both cases based on the average over an energy range or an ensemble of systems. For a single instance, fluctuations are large, do not go away in the limit, and depend on the element of the ensemble itself, thus seeming to question the whole procedure. Considered as the modulus of a partition function in complex inverse temperature $β_R+iβ_I$ ($β_I \equiv τ$ the time), the spectral factor has regions of Fisher zeroes, the analogue of Yang-Lee zeroes for the complex temperature plane. The large spikes in the spectral factor are in fact a consequence of near-misses of the line parametrized by $β_I$ to these zeroes. The largest spikes are indeed extensive and extremely sensitive to details, but we show that they are both exponentially rare and exponentially thin. Motivated by this, and inspired by the work of Derrida on the Random Energy Model, we study here a modified model of random energy levels in which we introduce level repulsion. We also check that the mechanism giving rise to spikes is the same in the SYK model.
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