Source
Yang and Abanin (2025), LIT-1958, §IV (eqs. 13–23, Fig. 5), pp. 6–8; read in NOTE-1534.
What was actually shown
The argument. Write L†[Q] = −γ_Q Q + ε₁J₁ (eq. 13), with ε₁ ∼ γ for an exact IOM and ε₁ ∼ ‖δH‖ for an approximate one. An eigenoperator V = Q + V⊥ satisfies ‖V⊥‖ ≲ |ε₁|∫₀^∞ e^{γ_V t}‖J₁(t)‖dt (eq. 20). Without noise ‖J₁(t)‖ is constant and the integral diverges for any ε₁. With noise, if J₁ grows ballistically, ‖J₁(t)‖ ∼ e^{−γv_B t²/2} for v_B t < L and e^{−γLt} after (eq. 21), so the integral converges and ‖V⊥‖ ≲ ε₁/E_g^< with E_g^< ∼ min(γL, √(v_Bγ)) (eq. 22), which does not shrink with L and is ≫ γ at small γ and large L. Hence “we reasonably expect the perturbation theory to hold as long as ε ≲ γ ≪ 1” (p. 8).
The check. Fig. 5: for MFIM with hz = 0.05 (a perturbed TFIM), L = 9, and A₁ as the approximate IOM, γN(t) converges to an O(1) value and (E_g^<)⁻¹ behaves as γ⁻¹ at small γ. Appendix D (Fig. 9, TFIM, γ = 0.1) shows bulk decay rates growing with L while the slow ones do not, at small L.
What could have come out otherwise: the integral in Fig. 5 could have failed to converge, or (E_g^<)⁻¹ could have scaled otherwise in γ.
What this does not say
- That the gap exists. The authors’ first comment (p. 8): modulated energy densities H_k have O(k) commutators with H, giving “a continuum of slow modes” in the Mori limit, so the gap “must be defined in sectors that are orthogonal to all hydrodynamic modes … and might not exist at all”. The argument then needs the parallel subspace widened to all relevant IOMs, which the authors call “challenging” in practice.
- That it covers integrable models. It assumes ballistic growth of the complement as in “fully chaotic systems”; the relaxation to “some form of operator growth such that J₁(t) has a larger size than Q at late times” is asserted, not shown.
- That large L has been reached. The only sizes computed are L ≤ 9.
- The time-scale ratio as a result. That the slow and fast rates separate by a factor of order √(v_B/γ) follows from eq. 22 under the same assumptions, and is the record’s reading for CLAIM-478, not a number the paper reports.