CM In the literature: open

Does the relaxation rate scale as $g^{2}$ in the thermodynamic limit

In plain words

Simple theory says a weak perturbation of strength g makes a solvable system relax at a rate proportional to g squared. The question is whether this holds exactly for very large systems or whether the rate is parametrically smaller.

Precise statement

For $H = H_{\mathrm{XXZ}} + g V$ with $V$ a generic local integrability-breaking perturbation not of the weak-integrability-breaking type (for that type the rate scales as $g^4$, Surace and Motrunich 2023), at infinite temperature, compute the exponent a in $\Gamma(g) \sim g^a$ for the decay rate of the slowest quasi-conserved local current, with $L \to \infty$ taken before $g \to 0$. The golden-rule expectation is $a = 2$. Answer: the value of a, with evidence that finite-size effects are controlled (they dominate when the golden-rule rate is below the inverse traversal time of the system, or for $g$ below the size-dependent chaos threshold).

What would settle it

An infinite-system computation of $\Gamma(g)$ (memory-matrix or recursion-method continued fraction) agreeing with real-time numerics on systems longer than the decay length at the smallest $g$ studied.

Status in the literature

Unverified note

$g^{4}$ scaling is established for weak-integrability-breaking perturbations (Surace and Motrunich 2023; Vanovac, Surace, Motrunich 2024); the generic case is open.