论文标题
Fisher零和混沌系统光谱形式的波动
Fisher zeroes and the fluctuations of the spectral form factor of chaotic systems
论文作者
论文摘要
量子混沌系统的光谱形式具有熟悉的“坡道$+$ platau”表格。在两种情况下，基于能量范围或系统集合的平均值，都设计了在半经典或热力学极限中确定其形式的技术。对于一个实例，波动很大，不要在极限上消失，而取决于整体本身的元素，因此似乎质疑整个过程。被认为是分区功能在复杂的反温度$β_R+iβ_i$（$β_i\equivτ$的时间）中的模量，光谱因子具有Fisher Zeroes的区域，该区域是复杂温度平面的Yang-Lee Zeroe的类似物。光谱因子中的大尖峰实际上是由$β_i$参数为这些零的近似线的结果。最大的尖峰确实是广泛的，并且对细节非常敏感，但我们表明它们既呈指数稀少又呈指数薄。受到这一点的动机，并受到德里达在随机能量模型中的工作的启发，我们在这里研究了一个随机能量水平的修改模型，在其中我们引入了水平排斥。我们还检查SYK模型中引起尖峰的机制是否相同。
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.
