Subtle connection between shape coexistence and quantum phase transition: The Zr case

Background: Zr region is characterized by very rapid changes in the ground-state structure of the nuclei. In particular, the onset of deformation when passing from Zr-98 to Zr-100 is one of the fastest ever observed in the nuclear chart. It has been probed both experimentally and theoretically that certain low-lying excited states of Zr isotopes own different shapes than the ground state. Purpose: We intend to disentangle the interplay between the sudden changes in the ground-state shape, i.e., the existence of a quantum phase transition, and the presence in the spectra of coexisting states with very different deformation, i.e., the presence of shape coexistence. Method: We rely on a previous calculation using the interacting boson model with configuration mixing (IBMCM), which reproduces in detail the spectroscopic properties of Zr96-110. This IBM-CM calculation allows to compute mean-field energy surfaces, wave functions, and any other observable related with the presence of shape coexistence or with a quantum phase transition. Results: We obtain energy surfaces and the equilibrium value of the deformation parameter beta, the U(5) decomposition of the wave functions, and the density of states. Conclusions: We confirm that Zr is a clear example of quantum phase transition that originates from the crossing of two configurations with a very different degree of deformation. Moreover, we observe how the intruder configuration exhibits its own evolution, which resembles a quantum phase transition too.