We study quantum phase transitions (QPTs) involving octupole degrees of freedom within the $spdf$ interacting boson model ($spdf$-IBM) by examining the potential-energy surface (PES) in the thermodynamic limit. Our analysis of a transitional Hamiltonian allows us to map the phase diagram for paths between spherical, quadrupole-deformed, and octupole-deformed shapes, identifying their critical points and order. The results confirm the established nature of the ${\text{U}}_{pdf}\left(5\right)\to {\text{O}}_{spdf}\left(6\right)$ and ${\text{U}}_{pdf}\left(5\right)\to {\text{SU}}_{pdf}\left(3\right)$ transitions. Furthermore, we uncover novel phenomena, including a sequential transition with an intermediate phase for the ${\text{U}}_{pdf}\left(5\right)\to {\text{SU}}_{spdf}\left(3\right)$ path, as well as transitions occurring entirely within the octupole sector. Finally, we connect these ground-state QPTs to precursors in the excitation spectra of finite-$n$ systems.

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• Nuclear shapes and moments
• Nuclear structure & decays
• Models based on symmetries