Intertwined quantum phase transitions in the zirconium and niobium isotopes

Nuclei in the $A\approx100$ region exhibit intricate shape-evolution and configuration crossing signatures. Exploring both even-even and their adjacent odd-mass nuclei gives further insight on the emergence of deformation and shape-phase transitions. We employ the algebraic frameworks of the interacting boson model with configuration mixing and the new interacting boson-fermion model with configuration mixing in order to investigate the even-even zirconium with neutron number 52-70 ($^{40}$Zr) and odd-mass niobium ($_{41}$Nb) isotopes with 52-62. We compare between the evolution in energy levels, configuration and symmetry content of the wave functions, two neutron separation energies and $E2$ transition rates. The comparisons between the two chains of isotopes denote the occurrence of intertwined quantum phase transitions (IQPTs) in both chains. Such a situation occurs when two configurations, normal and intruder, cross through the critical point of a Type II quantum phase transition (QPT), and the intruder configuration undergoes on its own a Type I shape-evolution QPT from a spherical shape (weak coupling scenario) to axially deformed rotor (strong coupling scenario) in the Zr (Nb) isotopes.