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We investigate the properties of a frustrated spin-5/2 chain withnext-nearest neighbor two and three-site interactions, with two questions inmind: the nature of the transition into the dimerized phase induced by thethree-site interaction, and the possible presence of a critical floating phaseat intermediate values of the next-nearest neighbor interaction. We providestrong evidence that the continuous transition into the dimerized phase, whichhas been found to be generically in the Wess-Zumino-Witten SU(2)$_{2S}$universality class up to spin $S=2$, is SU(2)$_5$ only at two isolated pointsof the phase diagram, and that it is SU(2)$_3$ in between, in agreement withthe presence of two relevant operators allowed by symmetry for SU(2)$_5$, andwith the conservation of the parity of the level index along therenormalization flow between SU(2)$_k$ theories with different values of $k$.We also find that the dimerization induced by the next-nearest neighborinteraction is a three step process, with first a small partially dimerizedphase followed by a broad critical floating phase with incommensuratecorrelations before the fully dimerized phase is reached. Implications for theiron oxide ${\mathrm{Bi}}_{3}{\mathrm{FeMo}}_{2}{\mathrm{O}}_{12}$ are brieflydiscussed.

From SU(2)$_5$ to SU(2)$_3$ Wess-Zumino-Witten transitions in a frustrated spin-5/2 chain