Relation between SU(2) charge sharpening and strong-to-weak symmetry breaking

Determine the relationship between charge sharpening in one-dimensional monitored SU(2)-symmetric dynamics and strong-to-weak spontaneous symmetry breaking in two-dimensional systems undergoing SU(2)-preserving dephasing.

Background

The paper notes that charge sharpening in monitored dynamics and strong-to-weak spontaneous symmetry breaking appear closely related for U(1) symmetry because they share a replica effective field theory. For SU(2), noncommuting fusion projectors prevent a direct reduction of the monitored dynamics to dephasing in a fixed local basis, and the relation between the two phenomena is therefore unresolved.

References

For $SU(2)$ the two appear to be distinct: fusion projectors on overlapping bonds share no common eigenbasis, so the fusion record does not define a joint classical probability distribution, and the monitored dynamics cannot be reduced to dephasing in any fixed local basis. We leave a study of their relation to future work.

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

Unlike the $U(1)$ case, SWSSB of nonabelian continuous symmetries is still quite poorly understood . An enticing direction along the lines of this work would be to study the SWSSB properties of dephased ground states with e.g. $SU(2)$ symmetry, and Kraus operators given by $SU(2)$ singlet operators like $\vec{S}_i \cdot \vec{S}_j$ or $S_i2$.

— Strong-to-Weak Spontaneous Symmetry Breaking of Dephased Fermions  (2609.37965 - Sarma, 29 Sep 2026) in Section Conclusions