# Spectral form factor The spectral form factor (SFF) is an important quantity that can be used to diagnose [[0008 Quantum chaos|quantum chaos]]. In terms of the analytically continued thermal partition function,$Z(\beta+i t)=\sum_{n} e^{-(\beta+i t) E_{n}},$it can be defined as$g(\beta, t)=\left|\frac{Z(\beta+it)}{Z(\beta)}\right|^2\\ =\frac{1}{Z(\beta)^2}\sum_{m, n} e^{-\beta\left(E_m+E_n\right)} e^{i\left(E_m-E_n\right) t}.$ It characterises two-point correlations $\langle\rho(E)\rho(E')\rangle$ of the eigenvalue spectrum. The ramp captures the long-range level repulsion of eigenvalues whereas the plateau arises from the discreteness of the spectrum (short-range). <!-- - variant: $Y_{E, \Delta}(t)=\sum_{n} e^{-\frac{\left(E_{n}-E\right)^{2}}{2 \Delta^{2}}} e^{-i E_{n} t}$ --> ## Refs - [[2022#Blommaert, Kruthoff, Yao]] integrable structure - relation to thermal effects at late time - [[2023#Winer, Swingle]] - relation to [[0052 Pseudo-entropy|pseudo-entropy]] - [[2024#He, Lau, Zhao]] - in scattering: - [[2024#Bianchi, Firrotta, Sonnenschein, Weissman]] - sonic systems - [[WinerSwingle2022]][](https://arxiv.org/pdf/2211.09134.pdf) - from replica trick - [[2024#Cao, Faulkner]] ## Dominant contributions - $Z(\beta+it)$: for large $t$, due to phase cancellations from high energy states, the temperature effectively lowers (why) and at some $t\sim O(\beta)$ the quantity will be dominated by thermal AdS rather than BH - $Y_{E,\Delta}(t)$: thermal AdS will not contribute for high enough energy, so can study BH at a longer time scale ## Annealed v.s. quenched - annealed easier and more correct - annealed: $\frac{\left\langle|Z(\beta, t)|^{2}\right\rangle}{\left\langle Z(\beta)^{2}\right\rangle}$ - quenched: $\left\langle\frac{|Z(\beta,t)|^{2}}{Z(\beta)^{2}}\right\rangle$ ## Related - [[0308 Half-wormhole]] - [[0154 Ensemble averaging]]