Breakdown of Simple Scaling in Entanglement Dynamics: Power-Law Relaxation of the Effective Exponent in Quantum Spin Chains

We investigate the scaling relation between the physical entanglement time τent and the modular time τmod in the dynamics of the integrable Heisenberg XXX spin chain. Contrary to the hypothesis of a stable universal exponent γ ≈ 2, we find no evidence for a time-independent power-law scaling. Instead, the effective exponent γ(t) exhibits a systematic drift toward smaller values at longer evolution times, following a robust power-law relaxation γ(t) −γ∞ ∼ t−α with α ≈ 0.65, supported by an excellent data collapse with relative spread below 6%. Remarkably, the amplitude of this relaxation is independent of system size N within the accessible window N ∈ {14,16,18,20}. We interpret this behavior as an effective renormalization group flow of the scaling exponent in time, suggesting that γ is not a fixed scaling dimension but a running quantity controlled by a marginally irrelevant operator. Our results demonstrate that previously reported values γ ≈ 2 originate from preasymptotic regimes rather than a true scaling law, and reveal a dynamical mechanism by which simple scaling breaks down in quantum entanglement evolution.