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

Abstract

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 98 Zr to 100 Zr is one of the fastest ever observed in the nuclear chart. It has been probed both experimental 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 (IBM-CM) which reproduces in detail the spectroscopic properties of 96−110 Zr. 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 β, 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.

Zr Isotopes as a region of intertwined quantum phase transitions

“…Here we compare our results with representative large scale shell model calculations: the Monte-Carlo shell model (MCSM) [22] and the complex excited VAMPIR model (EVAM) [21] and with other IBM-CM calculations: mean-field based (IBM-MF) [18] and an independent calculation [23,24] similar to ours, but with a different fitting protocol, denoted henceforth by IBM-CM-2. We focus the comparison on the 98,100 Zr isotopes, which lie near the critical-point of both Type I and Type II QPTs.…”

“…The shape of the symbol [•, , ♦], indicates the closest dynamical symmetry [U(5), SU(3), SO (6)] to the level considered, based on Eq. (24). Note that the calculated values start at neutron number 52, while the experimental values include the closed shell at 50.…”

“…These dramatic structural changes have attracted considerable theoretical and experimental interest in the Zr chain. Different theoretical approaches have studied them, including mean-field based methods, both nonrelativistic [17,18] and relativistic [19], large-scale shellmodel calculations [20,21], the Monte-Carlo shell-model (MCSM) [22] and algebraic models [8,9,23,24]. Recently, several experimental investigations have also come to light [25][26][27][28][29][30][31][32][33], opening the door for understanding the properties of both yrast and non-yrast states.…”

“…[33] for more details). The IBM-CM in boldface and IBM-CM-2 are the current IBM-CM calculation and that of [23,24], respectively.…”

Effects of the Tensor Force on the Ground Properties of Zr Isotopes

“…Table 1 and Figure 2). For example, it was pointed out that the region of Zr isotopes with A = 90-110 show a quantum phase transition phenomena and shape coexistence of prolate and triaxial shapes based on the large-scale Monte Carlo shell model calculations [25] and the interacting boson model with the configuration mixing method [26].…”