Coincidence of spin-sharpening and entanglement transitions

Determine whether the spin-sharpening transition and the entanglement transition in the one-dimensional monitored circuit with conserved SU(2) spin genuinely coincide.

Background

Numerical studies of monitored spin-1/2 chains with SU(2) symmetry found that the spin-sharpening threshold is numerically indistinguishable from the entanglement-transition threshold. The paper develops a replica statistical-mechanics theory that proposes the two transitions are a single transition, but the numerical evidence alone had not established their exact coincidence.

References

Strikingly, this spin-sharpening threshold is numerically indistinguishable from the entanglement-transition threshold, $p_\sharp\approx p_c$, though whether the two genuinely coincide was left open.

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