Spectral Form Factor K(t)

K(t) = |⟨Tr e^{iHt}⟩|² / Z² — dip-ramp-plateau structure distinguishes integrable from chaotic spectra

80
40
GUE (chaotic)
Poisson (integrable)
GOE
Dip: t_d ~ N^{-1/2}
Ramp: K(t) ~ t/2π (GUE)
Plateau: K → 1 at t_H (Heisenberg time)

Poisson: K(t)=1 (flat, no correlations)
GUE: rigid spectrum, level repulsion → ramp structure
Heisenberg time t_H =
Dip position: