SYK Model — Maximal Quantum Chaos
Sachdev-Ye-Kitaev: random all-to-all interactions, non-Fermi liquid, holographic black holes
Spectral Density ρ(E)
Spectral Form Factor K(t)
OTOC Growth C(t) = -⟨[W(t),V]²⟩
Green's Function & Self-Energy
SYK model (Sachdev-Ye 1993, Kitaev 2015): H = (1/4!)Σᵢⱼₖₗ Jᵢⱼₖₗ ψᵢψⱼψₖψₗ
where Jᵢⱼₖₗ are random Gaussian couplings. It is maximally chaotic:
the Lyapunov exponent saturates the MSS bound λ_L = 2π/β (Maldacena-Shenker-Stanwood 2015).
The spectral density → semicircle at high E (random matrix), but shows non-Fermi liquid behavior:
Σ(ω) ~ ω^{1/2} (IR fixed point). The Spectral Form Factor K(t) shows a dip-ramp-plateau
structure — the linear ramp encodes level repulsion (quantum chaos). SYK is dual to a near-extremal
black hole (Jackiw-Teitelboim gravity), making it a solvable model of quantum gravity.