Analytical explanation of unsaturated spin sharpening at the measurement-only point

Construct an analytical description of the measurement-only limit of the SU(2)-symmetric monitored circuit that captures the numerically observed absence of saturation of spin sharpening after times of order $L^2$ in the sharp phase.

Background

The large-loop-fugacity treatment neglects singlet measurement outcomes whose weights are subleading in the expansion but generate replica correlations at the measurement-only point. Consequently, the theory does not provide a meaningful description of that point. Numerical results show that spin sharpening in the sharp phase does not appear to saturate after the diffusive time scale t∼L2t\sim L^2, and explaining this behavior analytically is explicitly left unresolved.

References

Indeed, the spin sharpening observed numerically in Ref. shows no sign of saturation after $t\sim L2$ in the sharp phase. Capturing this behavior analytically lies beyond the current theory, and we leave it for future study.

— Statistical Mechanics of Non-Abelian Learnability Transitions  (2608.19325 - Ma et al., 19 Aug 2026) in Discussion