Fastest admissible growth of the locality scale in clocal MITE

Determine the fastest asymptotic divergence rate of the locality scale \ell_L (as a function of the system size L) for which the local equivalence between the microcanonical and canonical Gibbs states continues to hold, so that the \ell_L-local microscopic thermal equilibrium (MITE) condition lim_{L\to\infty} max_{\boldsymbol{r}\in\Lambda_L} \|\rho_{L|\ell_L}^{\boldsymbol{r}} - \sigma_{L|\ell_L}^{\boldsymbol{r}}\|_1 = 0 remains valid. Ascertain the maximal growth of \ell_L beyond \omega(L^0) but below o(L), given that \ell_L = \Theta(L) is known to violate local equivalence.

Background

The paper compares their iMATE notion with microscopic thermal equilibrium (MITE). Besides the standard O(L0)-local MITE, they discuss an \ell_L-local formulation where \ell_L can grow with system size, provided it remains subextensive.

The authors note that the local equivalence between microcanonical and canonical Gibbs states fails when \ell_L scales extensively (\ell_L = \Theta(L)), but they leave open the sharp boundary for allowed growth rates within the subextensive regime (\ell_L = o(L)). Clarifying this would precisely delineate the applicability of \ell_L-local MITE.

References

We do not know exactly how fast the divergence of $\ell_{L}$ can be. However, at least when $\ell_{L}=\Theta(L)$, the local equivalence between the microcanonical and the canonical Gibbs states breaks down, which is outside the scope of our interest. Therefore, $\ell_{L}=o(L)$ should be imposed.

— Second law of thermodynamics in closed quantum many-body systems  (2602.06657 - Chiba et al., 6 Feb 2026) in Section 7.1 (iMATE vs. MITE), footnote