Comment on "Resonance-induced growth of number entropy in strongly disordered systems"

We comment on the recent paper by Ghosh and Žnidarič (Phys. Rev. B 105, 144203 (2022)) which studies the growth of the number entropy $S_N$ in the Heisenberg model with random magnetic fields after a quantum quench. The authors present arguments for an intermediate power-law growth in time $t$ and a sub-ergodic saturation value, claiming consistency of their results with many-body localization (MBL) for strong disorder. We show that these interpretations are inconsistent with other recent studies and discuss specific issues with the analysis of the numerical data. We point out, in particular, that (i) the saturation values $\widetilde{S}_N(L,W)$ for fixed length $L$ are only bounded from above by 'the ergodic value' and are already far below this value for $W\ll 1$. Furthermore, the saturation values can show non-monotonic scaling with $L$. (ii) Power-law fits $S_N(t)\sim 1/t^α$ -- with $α=1$ expected based on the resonance model described in the paper -- yield a system-size dependent exponent $α$ while fits $S_N\sim \frac{1}{W^3}\ln\ln t$ do hold independent of system size and over several orders of magnitude in time. (iii) We also argue that for the cases where the effective resonance model works best and predicts a saturation of the number entropy, the same applies to the von-Neumann entropy, i.e.~the dynamics at the considered scales is of single particle type and unrelated to MBL.

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