What does "instant thermalization" in large-$q$ SYK models mean?

Abstract

Motivated by the Planckian thermalization rate $\Gamma \sim T$ observed in the two-body interacting SYK model, we study thermalization in the $q/2$-body case. Previous studies of such systems have established the notion of instantaneous thermalization to leading order in $1/q$. It was conjectured that the thermalization may still be Planckian but with a divergent rate $\Gamma \sim Tq$, explaining the ``instantaneous'' part. For an analytically and numerically tractable system, we calculate the effective temperatures after a quench at time $t=0$ and indeed find a Planckian rate, albeit with the unexpected decaying behavior $\Gamma \sim T{q}^{-1}$. This is contrasted with the behavior of the causal Green's function $\mathcal{G}(t_1,t_2)$, which instantly acquires a thermal form in the two-time plane block $t_1,t_2>0$. The resulting picture is that instant thermalization is a meaningful concept only for this block, while the off-diagonal blocks $t_1\cdot t_2<0$ are inherently non-thermal at any $q$. Since the effective temperature is obtained from correlations spanning all blocks, it inherits a finite rate set by the off-diagonal. We illustrate this directly by studying the non-thermal correlations' time dependence.