# The subtle connection between shape coexistence and quantum phase
transition. The Zr case

J.E. García-Ramos1,2 and K. Heyde3 1Departamento de Ciencias Integradas y
Centro de Estudios Avanzados en Física, Matemática y Computación, Universidad
de Huelva, 21071 Huelva, Spain
2Instituto Carlos I de Física Teórica y Computacional, Universidad de Granada,
Fuentenueva s/n, 18071 Granada, Spain
3Department of Physics and Astronomy, Ghent University Proeftuinstraat, 86
B-9000 Gent, Belgium

###### 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 98Zr
to 100Zr 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 96-110Zr. 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.

Zr isotopes, shape coexistence, quantum phase transition, interacting boson
model.

###### pacs:

21.10.-k, 21.60.-n, 21.60.Fw

##  I Introduction

Nuclei over the full nuclear mass (N,Z) surface are characterized by the
presence of clear cut regularities as the appearance of nuclear shells at
magic numbers. Symmetries, on the other hand, allow to establish hallmarks in
the chart of nuclei defining spherical, rigidly deformed and gamma-unstable
nuclei. Therefore, most of the nuclei can be grouped in one of the previous
classes. The different patterns in the mass table are modulated by the
interplay between the different components of the nuclear force, namely,
monopole interaction, pairing interaction, quadrupole (low multipole in
general) interaction. The monopole part explains the changing energy of the
single-particle levels Tsunoda  _et al._ (2014); pairing tends to stabilize
the atomic nuclei in a spherical shape; while the quadrupole interaction is at
the origin of the appearance of a deformed mean-field producing nuclear
deformation. In short and playing a central role, the detailed balance between
the different components of the nuclear force is the responsible of the final
shape of the nucleus.

Another remarkable aspect of the nuclear chart is the existence of
transitional regions, where the structure of the nuclei evolves from one of
the limiting cases to another as a function of the neutron and/or the proton
number. In certain regions the evolution can be very rapid and the nucleus
changes from being spherical to strongly deformed in just a few mass units,
e.g., in the rare-earth region around $N=90$ Cejnar and Jolie (2009); Cejnar
_et al._ (2010) or in Zr-Sr Ansari  _et al._ (2017); Régis  _et al._ (2017)
region around $N=60$. For such situations the term quantum phase transition
(QPT) was coined Hertz (1976); Sachdev (2011) due to the similarities with the
very well known thermodynamic phase transitions. A QPT implies an abrupt
change in the properties of the ground state that results in the rapid change
of several observables, namely, two-neutron separation energies,
E($4_{1}^{+}$)/E($2_{1}^{+}$), B(E2:$4_{1}^{+}\rightarrow 2_{1}^{+}$), among
others. QPT can be of two types, either of first or of second order. In the
first order case a discontinuity in the first derivative of the energy with
respect to a control parameter is present while in the second order case the
discontinuity hold for the second order derivative. The first-order QPT’s
suppose a more abrupt change than the second order ones.

Figure 1: Energy systematics of even-even Zr isotopes from $A=90$ to $A=110$.
Levels identified as regular are marked in blue and connected with dashed blue
lines, while those identified as intruder are marked in red and connected with
dashed red lines.

To complete the description of the mass table, one has to take into account
that in many cases the change of shape not only occur when moving to neighbour
nuclei, but also the properties in a given nucleus, as a function of the
increasing excitation energy may be changing, giving evidences of deformed
excited states. Hence, it is possible to find states with very different
shapes in the spectrum of a single nucleus. The different shapes correspond to
different n-particle n-hole excitations across a shell closure. These
excitations give rise to an increasing number of active nucleons that can
strongly affect the effective nuclear force. A notable example can be found in
the doubly magic nucleus 40Ca which is spherical for the ground state but
2p-2h and 4p-4h excitations lead to deformed and superdeformed shapes,
respectively Caurier  _et al._ (2007). The existence of states with different
shapes in a narrow range of energies is known as shape coexistence and not
only appears in nuclear physics, but also in molecules and other quantum
systems Heyde and Wood (2011). Shape coexistence is present in many regions of
the nuclear mass table, in general in those around shell or sub-shell closures
of both protons and neutrons. Of particular interest is the region around the
shell closure $Z=82$, i.e., the even-even Pt, Hg, Pb, and Po, where many of
these isotopes have been studied both experimentally and theoretically,
confirming the presence of two and even three families of states with a
different degree of deformation. It is worth to note that in some cases the
interaction among the different families is rather weak as in Hg and Pb while
in the case of Pt and Po is rather strong. Moreover, in certain cases the
intruder configuration, corresponding to 2p-2h excitations become the lowest-
lying configurations, contrary to the 0p-0h configurations now describing
highly-lying states, as is the case for Pt and Po. On the other hand, in the
case of Hg or Pb the ground state always corresponds to a regular 0p-0h
configuration.

QPT and shape coexistence share several aspects, namely, the rapid structural
changes over a chain of isotopes or isotones and the presence of very low-
lying $0^{+}$ states. As a consequence, the excitation energy of the
$2_{1}^{+}$ can drop or the value of the $B(E2:2_{1}^{+}\rightarrow
0_{1}^{+})$ can increase suddenly in few mass units. Besides, in both cases,
the value of the nuclear radius separates from the linear trend or the two-
nucleon transfer reaction intensities experience a rapid increase. However,
some differences are in order, in particular, the existence in the case of the
shape coexistence of a set of states, intruder states, with a parabolic
excitation energy systematics that is centered with respect to the mid shell.
These states present also a different deformation than the ground state band.
This behavior is not observed in the case of a QPT.

The region around $Z=40$, in particular, regarding the Zr isotopes, but also
the Sr ones, is an ideal region to study the interplay between shape
coexistence and the presence of a QPT, that is the reason for focusing on this
mass region in this work. On one hand, this region shows all the hints
pointing to the presence of a QPT but, at the same time, a set of states owns
an energy systematics symmetric with respect to the mid shell and presents a
different deformation than the rest of states, therefore one can say that
shape coexistence is present in the spectrum.

The organization of the present paper is as follows: in Sec. II some hints for
the presence of shape coexistence are provided; in Sec. III the corresponding
evidences for the existence of QPT’s are shown; in Sec. IV a brief summary of
the interacting boson model with configuration mixing (IBM-CM) is presented;
Sec. V is devoted to presenting the results of the IBM-CM calculations;
finally in Secs. VI and VII, discussion, conclusions and outlook are
presented, respectively.

##  II Evidences of shape coexistence in Zr nuclei

Shape coexistence implies that in a narrow range of energy, states with
different deformation coexist. Because deformation is not strictly an
observable, it is needed to rely on observables that are tightly linked to
deformation, such as excitation energy and B(E2) transition rate systematics,
radii, etc. In addition, the existence of highly hindered E2 or E0 transitions
points towards the presence of states with different degree of deformation.

Energy systematics is one of the first hints pointing to the presence of extra
configurations in the spectra of an isotope chain Heyde and Wood (2011). In
particular, in the case of even-even Hg and Pb isotopes a parabolic-like trend
in the energy levels centered at the mid shell, N=104, is clearly observed for
a set of states, together with a rather flat behavior for another group of
states. In both cases the flat systematics corresponds to regular states while
the parabolic one to the intruder ones. There are other isotope chains where
the presence of extra configurations is not so clearly observed. This is
precisely the case of Pt isotopes, where due to the large mixing between
regular and intruder states combined with the crossing of the bandheads of
regular and intruders families makes the presence of intruder states somehow
concealed (see García-Ramos  _et al._ (2011, 2012) for more details). The case
of Po is also similar, though here, the mid shell is barely reached
experimentally García-Ramos and Heyde (2015a, b). In both cases it is hard to
separate a parabolic systematics from a flat one. The case of Zr is very much
the same, as one can readily see in Fig. 1 where the presented energy
systematics of the even-even Zr isotopes span the range from $A=90$ till 110.
According to the experimental information accumulated during last decades (see
references below in this section), it is possible to determine the character
of certain states, hence the regular ones are labeled in blue while the
intruder ones in red, moreover the states with the same internal structure are
connected with lines. In this figure one can easily observe several features,
the first one is the presence of shell and subshell closures for $A=90$ and
$A=96$, respectively. The second is the increase in the density of levels
below E$\approx 1.5$ MeV for $A\geq 100$ owing to the lowering of the intruder
configuration that becomes the ground state for $A=100$ and onwards. However,
the parabolic trend for the intruder energy systematics is not observed.
Anyhow, the existence of low lying $0^{+}$ states is also a good indicator of
the presence of additional particle-hole configurations.

Figure 2: Two-nucleon or $\alpha$ transfer strengths to the $0_{2}^{+}$ state
in the Zr isotopes, relative to the $0_{1}^{+}$ state. (a) Stands for data
from (6Li, 8B) and (b) for data from (14C, 16O) reactions. Data taken from the
corresponding Nuclear Data Sheets and from Heyde and Wood (2011).

The connection of the different shapes with different particle-hole
configurations makes two-neutron, two-proton or $\alpha$ transfer reactions
ideal tools to identity states with rather different shapes. In Fig. 2, the
probability of populating a given state either with two-nucleon or with
$\alpha$ transfer reaction, relative to the probability for populating the
ground state is depicted. This figure shows strong evidences for pairing
collectivity in both the $0^{+}_{1}$ and $0^{+}_{2}$ states, thereby being a
clear evidence of the presence of “coexisting” characteristics in those two
states. These data are pointing out to the underlying pairing collectivity in
these nuclei and they are the result from two-neutron transfer in (6Li, 8B)
(labeled with (a)) and from the (14C, 16O) (labeled with (b)) reactions,
respectively. For more details on the Z$\approx 40$, N$\approx 60$ nuclei see
also section III.A.3 and Fig. 29 and 30 of Ref. Heyde and Wood (2011). All in
all, these data are the hard evidence of the fact that the ground and first
excited $0^{+}$ states exhibit a “coexisting” character, i.e., the transfer
strength goes strongly to more than one “pairing condensate”.

In the last decades a huge experimental effort has been carried out to
complete the picture of deformation in Zr area, mainly through the
determination of excitation energies and lifetimes. For mass $A=94$, besides
the Abriola and Sonzogni (2006) compilation, recent information has been
obtained from lifetime data using neutron scattering Chakraborty  _et al._
(2013); Peters  _et al._ (2013) as well as the $B(E2)$ values extracted from
inelastic electron scattering at low momentum transfer into the
2${}^{+}_{1,2}$ states. This allowed a comparison with the data obtained at
the Kentucky facility from neutron inelastic scattering such as branching
ratios, multipole mixing ratios, and spin assignments indicating very much the
same results, however with an increased precision Scheikh Obeid  _et al._
(2014); Elhami  _et al._ (2008). For mass $A=96$, the major part of data can
be found in the compilation Abriola and Sonzogni (2008), moreover, data on
g-factors have been determined by Coulomb excitation of 96Zr beams in inverse
kinematics. Besides, the lifetime of the 2${}^{+}_{1}$ state has been
redetermined using the Doppler Shift Attenuation (DSA) method Kumbartzki  _et
al._ (2003). Interesting information has been extracted from single and
double-$\beta$ decay Q values among the triplet of nuclei 96Zr, 96Nb and 96Mo
Alanssari  _et al._ (2016); recent high-resolution electron scattering allowed
to extract the $B(E2;0^{+}_{1}\rightarrow 2^{+}_{2})$ value as well as decay
strengths originating from the 2${}^{+}_{2}$ excited states by Kremer et al.
at the S-DALINAC Kremer  _et al._ (2016). The $E2$ collectivity deduced in
this way was moreover discussed by Pietralla et al. Pietralla  _et al._ (2018)
for the heavy Zr nuclei. Moving into the transitional nucleus at $A=98$
(compilation can be found in Singh and Hu (2003)), lifetime data have been
measured using the EXILL-FATIMA array at the cold neutron beam line at the
Institut Laue-Langevin (ILL-Grenoble) by Betterman Bettermann  _et al._ (2010)
and Ansari et al. Ansari  _et al._ (2017), the latter also extracting
lifetimes for the $A=100$ and $A=102$ Zr isotopes. These lifetime data are of
essential use in obtaining a deeper understanding of the changing structure
when moving from $A=96$ towards the isotopes beyond $A=100$. Coming to the
98Zr nucleus, very recent data taken at the ATLAS/CARIBU facility at Argonne
National Laboratory, through Coulomb excitation studies, have allowed to
determine an upper limit of the $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$ value
in 98Zr. Moreover, new lifetime data by Singh et al., using the recoil-
distance Doppler shift method at GANIL, were published Singh  _et al._ (2018).
These data are of major importance to trace the transition from $A=96$ into
the strongly deformed $A=100$ Zr isotope Werner  _et al._ (2018). More details
on that study by Witt et al. have very recently been published Witt  _et al._
(2018). Data on 100Zr are presented in the compilation Singh (2008), the ones
on 102Zr are in DeFrenne (2009), and 104Zr are in Blachot (2007). Data for
106Zr and 108Zr are in Singh (2015a) and Singh (2015b), respectively.
Regarding 106Zr, in Ref. Navin  _et al._ (2014) the energy of the states
$6_{1}^{+}$ and $8_{1}^{+}$ have been measured. This experiment was performed
at GANIL using the large-acceptance spectrometer VAMOS++ trough the reaction
(238U, 9Be) in inverse kinematics with the aim of studying the spectra of
104-105-106Zr. A detailed study by Paul et al. Paul  _et al._ (2017) of the
110Zr nucleus and its lowest-lying states has been carried out very recently.
The above information is the base to separate between regular and intruder
states in Fig. 1.

Figure 3: Evolution of the value of $\rho^{2}(E0:0^{+}_{2}\rightarrow
0^{+}_{1})$, from 94Zr till 100Zr. Black full line for experimental values
while red dashed ones for IBM-CM calculations.

The analysis of the E0 transition rates has been proven to be an ideal tool to
disentangle the presence of different configurations in the spectrum and its
degree of mixing Wood  _et al._ (1999). In particular, a small value of
$\rho^{2}(E0:0^{+}_{i}\rightarrow 0^{+}_{j})$ suggests that states $0^{+}_{i}$
and $0^{+}_{j}$ have largely different degrees of deformation, implying no (or
negligible) mixing between these two $0^{+}$ states. In Fig. 3 the
$\rho^{2}(E0)$ experimental values for the known $0^{+}_{2}\rightarrow
0^{+}_{1}$ transitions together with the theoretical IBM-CM values (see Sec.
IV) are depicted. The values for the range $A=94-98$ are rather small,
suggesting that the involved configurations are rather pure and owing a
different structure. Indeed, the $0_{1}^{+}$ state is assumed to be regular
while the $0_{2}^{+}$ one of intruder nature. However, the $\rho^{2}(E0)$
transition rate for $A=100$ is one of the largest ones observed over the full
nuclear mass table, pointing to the existence of a large mixing between the
intruder and the regular sectors concerning the two first $0^{+}$ states.

##  III Quantum phase transition indicators in Zr nuclei. Is there any hint of
X(5) critical point symmetry?

A QPT develops in systems where the structure of the ground state changes
abruptly for a specific value of the control parameter and the temperature is
equal to zero. In particular, a QPT is related to quantum systems which
Hamiltonians, that express the change in between different symmetries called A
and B, can be written down as,

$\hat{H}=(1-x)\hat{H}(\text{sym}_{A})+x\hat{H}(\text{sym}_{B}),$ (1)

where $\hat{H}(\text{sym}_{A})$ and $\hat{H}(\text{sym}_{B})$ correspond to
Hamiltonians that own a given symmetry, either $A$ of $B$, that in most of
cases can be considered as dynamical symmetries Iachello  _et al._ (1998). In
the familiar Ising model Ising , $\hat{H}(\text{sym}_{A})$ can be identified
with the term leading the system into a ferromagnetic phase, while
$\hat{H}(\text{sym}_{B})$ with that part leading into the paramagnetic phase
Sachdev (2011). The QPT happens for a critical value $x=x_{c}$ where the wave
functions switch from having the symmetry $A$ to having the symmetry $B$. The
existence of a QPT supposes also the abrupt change in the so called order
parameter that presents a null value in the symmetric phase while different
from zero in the broken phase Sachdev (2011). Hence, the order parameter
represents, albeit in a schematic way, the symmetry of a given phase.

Figure 4: Schematic spectra (full black lines and left y-axis) of few $0^{+}$
states for an IBM Hamiltonian undergoing a first order QPT at a value of the
control parameter equal to $9/11$. Value of the order parameter $\beta$
(dashed magenta line and right y-axis). Dashed cyan zone corresponds to the
coexistence region. See Ref. García-Ramos  _et al._ (2001) for details on the
calculation of $\beta$.

Following the classical classification of QPT’s, the transitions can be
separated in first and second order (or continuum) QPT’s Landau and Lifshitz
(1969). In the case of a first order QPT (according to the Erhenfest
classification, the first derivative of the ground state energy with respect
to the control parameter presents a discontinuity), there is a narrow region
around $x_{c}$ where the two symmetries can coexist. However, in the case of a
second order QPT (according to the Erhenfest classification, the second
derivative of the ground state energy with respect to the control parameter
presents a discontinuity), there is no coexistence of symmetries for any
particular value of $x$. Another important feature is that in the case of a
second order QPT, the excitation energy of the first excited state at the
critical point vanishes, while for the first order QPT it takes a very small
value. This is schematically presented for an IBM calculation (with a single
configuration) in Fig. 4 for the case of a first order QPT, where the spectra
corresponding to few $0^{+}$ states and the value of the order parameter for a
schematic IBM Hamiltonian are depicted. The two limiting phases, deformed and
spherical, and the critical point are marked. The aim of presenting this
figure is two fold. First to illustrate how, in general, a QPT induces the
lowering of a set of states and the compression of the spectra and, second, to
see the evolution of the order parameter (calculated in the thermodynamic
limit) García-Ramos  _et al._ (2001), that is connected with the deformation
of the ground state, and that suffers a sudden change passing from zero to
finite values when crossing the critical point.

Figure 5: Indicators of the presence of a QPT. Black full line for
experimental values and red dashed one for IBM-CM calculations. (a) Radii
square referred to $A=90$, (b) two-neutron separation energies, and (c)
E($4_{1}^{+}$)/E($2_{1}^{+}$) ratio.

The position of a QPT in a given model depends on the topology of its
potential energy surface and the case of a first order QPT, that is the one of
interest for this work, implies the existence of two degenerated minima which
own two different shapes Cejnar  _et al._ (2010). In the case of the IBM with
a single configuration, this degeneracy exists for the critical value of the
control parameter, i.e., $9/11$, as shown in Fig. 4. Before the critical
point, the order parameter is different from zero, while after, it is equal to
zero (see magenta line in Fig. 4). The two minima coexist only in a narrow
region around the critical point. The use of the IBM with two configurations,
i.e, the IBM-CM, introduces a net difference with respect to the case of a
single configuration because both configurations coexist for the whole
parameters space of the Hamiltonian. In this case, the first order QPT will
appear where both configurations do cross, but, on the other hand, the lowest
configuration could also experience a QPT. Strictly speaking, a QPT is only
well defined in the thermodynamic limit, i.e., for systems consisting of an
infinite number of particles. Therefore, in nuclear systems, due to its finite
size, all the abrupt changes will be somehow smoothed out Iachello  _et al._
(1998). On the other hand, there is an additional difficulty when dealing with
QPT’s in nuclei that is the absence of a real control parameter because the
Hamiltonian of a given nucleus cannot be changed externally. This drawback is
partially solved considering that the number of neutrons (protons), or A, in a
chain of isotopes (isotones) acts as an effective control parameter. Note,
that doing so, the order parameter is not a continuous but a discrete variable
Cejnar  _et al._ (2010). We mention that the variation of A may involve the
change of several Hamiltonian parameters and not only one, as illustrated in
the schematic case of Eq. (1). In fact, in this work 12 parameters should be
determine in every Zr isotope. The way to do so has been extensively discussed
in great detail in Ref. García-Ramos and Heyde (2019) and in an abridged way
in Section IV. In Fig. 5 different observables that can be considered as
indicators of the presence of a QPT are presented. In Fig. 5(a) the
experimental and theoretical values (see Section IV and García-Ramos and Heyde
(2019)) of the nuclear radii are presented with $A=90$ taken as the reference
nucleus. This observable is related with the evolution of the order parameter
and it shows a sudden increase at $A=100$. In Fig. 5(b) the two-neutron
separation energy is depicted and both the experimental and the IBM-CM
theoretical values are shown. This observable is tightly connected with the
derivative of the ground state energy and exhibits a sudden change in its
trend at $A=100$. Finally, in Fig. 5(c) the ratio
E($4_{1}^{+}$)/E($2_{1}^{+}$) (experimental and IBM-CM theoretical values) is
presented. This ratio is connected with the evolution of the order parameter.
Indeed the observed trend is similar to the behavior of the order parameter in
Fig. 4. The value of this observable experiences a sudden increase at $A=100$.

According to the previous figures, it is a logical step to assume that Zr
undergoes a QPT for $A\approx 100$ being possible to define an order parameter
(Fig. 5(c). Moreover, there exists a kind of discontinuity in the first
derivative of the energy, i.e., as observed in the S2n (Fig. 5(b).

In the areas where a QPT develops it could be of application the concept of
critical point symmetry. This concept was introduced twenty years ago by
Iachello, first for the transition between spherical and gamma-unstable shapes
Iachello (2000), named as E(5) critical point symmetry and later for a
transition from sphericity to axial deformation, called X(5) Iachello (2001).
In our case the relevant critical point symmetry is X(5) because the
competition is between a spherical and an axially deformed shape. The
interesting conclusion follows from the X(5) critical point because it
provides the value of parameter free excitation energy and transition rate
ratios that can be directly compared with the experimental information. In the
case of Zr, the most promising nuclei to be compared with X(5) are 100Zr and
102Zr. In Table 1, some X(5) key ratios are compared with 100Zr and 102Zr. The
experimental data for 152Sm are also given for comparison purposes. Because of
the scarcity of data, in some cases we have resorted to theoretical IBM-CM
calculations (see Sec. IV) and in this case they appear marked with an
asterisk. Note that in the case of 100Zr all the considered states belong to
the intruder band, hence in some cases the index for a given L is a unit
higher than for the X(5) one, e.g., the 0${}_{3}^{+}$ and 2${}_{3}^{+}$
states. In the case of 102Zr, the 0${}_{2}^{+}$ state is the one used to be
compared with X(5) although theoretically has a regular character (76%),
however the 0${}_{3}^{+}$ state is not known experimentally, but it is
expected to presents a major intruder character (76%). The $0_{3}^{+}$ state
presents a theoretical value of E($0_{3}^{+}$)/E($2_{1}^{+}$) equal to $6.87$.
The 2${}_{2}^{+}$ state is mainly of intruder character and, therefore this is
the one used in the comparison with X(5).

Table 1: Comparison of key observables in the X(5) limit with 152Sm, 100Zr and 102Zr. ∗ stands for theoretical IBM-CM values in absence of experimental data. X(5) |  X(5) |  152Sm |  100Zr |  102Zr
---|---|---|---|---
E($4_{1}^{+}$)/E($2_{1}^{+}$) | 2.91 | 3.01 | 2.66 | 3.16
E($6_{1}^{+}$)/E($2_{1}^{+}$) | 5.45 | 5.80 | 5.00 | 6.39
E($0_{2,3}^{+}$)/E($2_{1}^{+}$) | 5.67 | 5.62 | 3.91 | 5.92
$\frac{B(E2:4_{1}^{+}\rightarrow 2_{1}^{+})}{B(E2:2_{1}^{+}\rightarrow 0_{1}^{+})}$ | 1.58 | 1.50 | 1.39 | 1.59
$\frac{B(E2:2_{2,3}^{+}\rightarrow 2_{1}^{+})}{B(E2:2_{1}^{+}\rightarrow 0_{1}^{+})}$ | 0.09 | 0.04 | 0.14∗ | 0.18∗
$\frac{B(E2:2_{2,3}^{+}\rightarrow 4_{1}^{+})}{B(E2:2_{1}^{+}\rightarrow 0_{1}^{+})}$ | 0.36 | 0.13 | 0.006∗ |  $6~{}10^{-7}$∗
$\frac{B(E2:0_{2,3}^{+}\rightarrow 2_{1}^{+})}{B(E2:2_{1}^{+}\rightarrow 0_{1}^{+})}$ | 0.63 | 0.23 | 0.37∗ | 0.11∗
$\frac{B(E2:2_{2_{3}}^{+}\rightarrow 0_{1}^{+})}{B(E2:2_{1}^{+}\rightarrow 0_{1}^{+})}$ | 0.02 | 0.007 | 0.017∗ | 0.027∗

Regarding 100Zr, the agreement with the X(5) limit is only acceptable (similar
to the case of 152Sm) for E($4_{1}^{+}$)/E($2_{1}^{+}$),
E($6_{1}^{+}$)/E($2_{1}^{+}$), $\frac{B(E2:4_{1}^{+}\rightarrow
2_{1}^{+})}{B(E2:2_{1}^{+}\rightarrow 0_{1}^{+})}$,
$\frac{B(E2:2_{2,3}^{+}\rightarrow 2_{1}^{+})}{B(E2:2_{1}^{+}\rightarrow
0_{1}^{+})}$, $\frac{B(E2:0_{2,3}^{+}\rightarrow
2_{1}^{+})}{B(E2:2_{1}^{+}\rightarrow 0_{1}^{+})}$ and
$\frac{B(E2:2_{2_{3}}^{+}\rightarrow 0_{1}^{+})}{B(E2:2_{1}^{+}\rightarrow
0_{1}^{+})}$, however it fails for the most important ratio
E($0_{2,3}^{+}$)/E($2_{1}^{+}$) which is too low. In the case of 102Zr the
comparison is similar with the difference that in this case the agreement for
E($0_{2,3}^{+}$)/E($2_{1}^{+}$) is reasonable but not so for
E($6_{1}^{+}$)/E($2_{1}^{+}$). In short, the excited $0^{+}$ states in 100Zr
are too low in excitation energy while the yrast band in 102Zr is too rigid.
Hence, it is obvious that there is no crystal-clear correspondence with the
X(5) critical symmetry, however the agreement for certain ratios is similar
when comparing with the corresponding ratios for the case of 152Sm and
therefore, it is fair to claim that 100-102Zr presents certain resemblance
with the X(5) critical symmetry.

##  IV The Interacting Boson Model with configuration mixing formalism

The selected formalism to perform theoretical calculations in the region of
interest, i.e., 94-110Zr, is an enlarged version of the IBM Iachello and Arima
(1987). The original version of the IBM was proposed in the mid 1970’s by
Iachello and Arima (1987) for dealing with medium and heavy mass even-even
nuclei and it supposes a strong reduction of the available Hilbert space for
the nucleons because only the degrees of freedom corresponding to pairs of
nucleons coupled either to angular momentum $L=0$ (S pairs) or to angular
momentum $L=2$ (D pairs) are considered. Moreover, these pairs are bosonized
and, hence, the building blocks of the model are s and d bosons. The number of
effective bosons, N, corresponds to half the number of nucleon pairs,
regardless of its proton or neutron nature (this is strictly true for the
simplest version of the model, IBM-1, but in other versions it is possible to
define proton and neutron bosons). The IBM-1 model was enlarged for treating
nuclei where more than one particle-hole configuration was used Duval and
Barrett (1981, 1982). In this model, the original boson space $[N]$ is
enlarged to $[N]\oplus[N+2]$ where in the case of Zr isotopes $N$ is the boson
number, corresponding to the number of active protons outside the $Z=40$, zero
in the case of Zr, plus the number of active neutrons outside the shell
closure $N=50$ divided my two. The $[N+2]$ space corresponds to considering
two extra bosons that come from the promotion of a pair of protons across the
shell closure $Z=40$, generating an extra boson made of proton holes and
another made of proton particles.

The Hamiltonian in this formalism can be written as,

$\hat{H}=\hat{P}^{{\dagger}}_{N}\hat{H}^{N}_{\rm
ecqf}\hat{P}_{N}+\hat{P}^{{\dagger}}_{N+2}\left(\hat{H}^{N+2}_{\rm
ecqf}+\Delta^{N+2}\right)\hat{P}_{N+2}\ +\hat{V}_{\rm mix}^{N,N+2}~{},$ (2)

where $\hat{P}_{N}$ and $\hat{P}_{N+2}$ are projection operators onto the
$[N]$ and the $[N+2]$ boson spaces, respectively, $\hat{V}_{\rm mix}^{N,N+2}$
is describing the mixing between the $[N]$ and the $[N+2]$ boson subspaces,
and

$\hat{H}^{i}_{\rm
ecqf}=\varepsilon_{i}\hat{n}_{d}+\kappa^{\prime}_{i}\hat{L}\cdot\hat{L}+\kappa_{i}\hat{Q}(\chi_{i})\cdot\hat{Q}(\chi_{i}),$
(3)

is a simplified form of the general IBM Hamiltonian called extended
consistent-Q Hamiltonian (ECQF) Warner and Casten (1983); Lipas  _et al._
(1985) with $i=N,N+2$, $\hat{n}_{d}$ being the $d$ boson number operator,

$\hat{L}_{\mu}=[d^{\dagger}\times\tilde{d}]^{(1)}_{\mu},$ (4)

the angular momentum operator, and

$\hat{Q}_{\mu}(\chi_{i})=[s^{\dagger}\times\tilde{d}+d^{\dagger}\times
s]^{(2)}_{\mu}+\chi_{i}[d^{\dagger}\times\tilde{d}]^{(2)}_{\mu}~{},$ (5)

the quadrupole operator. The parameter $\Delta^{N+2}$ accounts for the energy
needed to promote a pair of protons across the shell closure $Z=40$, corrected
for the pairing and the monopole interaction energy Heyde  _et al._ (1985,
1987). The operator $\hat{V}_{\rm mix}^{N,N+2}$ describes the mixing between
the $N$ and the $N+2$ configurations and it is defined as

$\hat{V}_{\rm mix}^{N,N+2}=\omega_{0}^{N,N+2}(s^{\dagger}\times
s^{\dagger}+s\times s)+\omega_{2}^{N,N+2}(d^{\dagger}\times
d^{\dagger}+\tilde{d}\times\tilde{d})^{(0)}.$ (6)

This approach has been used successfully for describing the spectroscopic
properties of Pt García-Ramos and Heyde (2009); García-Ramos  _et al._ (2011,
2012), Hg García-Ramos and Heyde (2014, 2015c), Po García-Ramos and Heyde
(2015a, b), and Zr isotopes García-Ramos and Heyde (2018, 2019). The good
reproduction of the available experimental data justify the use of only 0p-0h
and 2p-2h excitations in the previous cited works. In the Zr isotopic series,
a few data in the heavier Zr isotopes may hint to the need for including also
4p-4h excitations, not used in the present paper.

The parameters of the Hamiltonian and $\hat{T}(E2)$ operator need to be fixed
either from microscopical considerations, e.g., through a mapping from a shell
model Hamiltonian, or from a phenomenological procedure, i.e., performing a
least-squares fit to energy levels and $B(E2)$ transition probabilities. In
this work, the second procedure has been used to obtain the Hamiltonian’s and
the $\hat{T}(E2)$ parameters, as explained in detail in Section IV.B of Ref.
García-Ramos and Heyde (2019). In Table III of Ref. García-Ramos and Heyde
(2019) the fitted parameters of the Hamiltonian and the $\hat{T}(E2)$
operators are given and explained.

##  V Results

###  V.1 Mean-field energy surfaces and the evolution of deformation

One of the standard ways to study QPT’s or shape coexistence is through the
density functional theory (DFT) that allows to obtain a mean-field energy
surface and provides a clear picture about the deformation of the system. In
particular, in the IBM it is possible to calculate mean-field energy surfaces
defining the so called intrinsic state formalism Ginocchio and Kirson (1980);
Dieperink and Scholten (1980); Dieperink  _et al._ (1980); Gilmore (”1974”)
that is not more than a condensate of bosons (or coherent state), namely, a
condensate of deformed bosons that depend on two deformation parameters
$\beta_{B}$ and $\gamma_{B}$, where the sub-index B stands for the parameters
corresponding to bosons. These deformation values should not be confused with
the $\beta$ and $\gamma$ deformation parameters as describing a quadrupole
deformed nuclear surface. The intrinsic state formalism is defined as,

$|N;\beta_{B},\gamma_{B}\rangle={1\over\sqrt{N!}}\left({1\over\sqrt{1+\beta_{B}^{2}}}\left(s^{\dagger}+\beta_{B}\cos\gamma_{B}\,d^{\dagger}_{0}+{1\over\sqrt{2}}\beta_{B}\sin\gamma_{B}\,(d^{\dagger}_{2}+d^{\dagger}_{-2})\right)\right)^{N}|0\rangle$
(7)

and, therefore, the mean-field energy corresponds to the matrix element
$E(N,\beta_{B},\gamma_{B})=\langle
N;\beta_{B},\gamma_{B}|\hat{H}|N;\beta_{B},\gamma_{B}\rangle$. The value of
$\beta_{B}$ and $\gamma_{B}$ which minimize the expectation value of the
energy provide the shape of the nucleus with a given number $N$ of bosons.

The latter formalism needs to be extended when dealing with IBM-CM, defining a
matrix within coherent state bands,

$H_{CM}=\left(\begin{array}[]{cc}E(N,\beta_{B},\gamma_{B})&\Omega(\beta_{B})\\\
\Omega(\beta_{B})&E(N+2,\beta_{B},\gamma_{B})\end{array}\right).$ (8)

which lowest eigenvalue will correspond to the mean-field energy surface of
the model (see Frank  _et al._ (2002, 2004, 2006); Morales  _et al._ (2008)
for a detail description of the formalism). One can easily compute this energy
surfaces once the parameters of the Hamiltonian are given. In our case we use
the ones obtained in García-Ramos and Heyde (2019).

Figure 6: (a) Comparison of the scaled value of $\beta$ from the IBM-CM mean-
field energy (“Min.”) and from Eqs. (10) and (11) (“From q2”). The scaled
value of $\beta$ corresponding to the unperturbed “regular”, [N], and the
unperturbed “intruder”, [N+2], configurations are also depicted. (b) Energy of
the unperturbed regular and intruder IBM bandheads, compared with the ground-
state energy (full IBM-CM diagonalization).

The comparison of the IBM-CM energy surface with the one from mean-field DFT
using a realistic interaction needs some care because the IBM deformation
parameters should be rescaled. Therefore, one needs to establish a connection
between $(\beta_{B},\gamma_{B})$ and $(\beta,\gamma)$. To do so we closely
follow the technique presented in García-Ramos  _et al._ (2014), using the
ansatz,

$\beta=1.18\frac{2}{A}\beta_{B}(N+2~{}(1-\omega))~{}\delta+\xi.$ (9)

where $\delta$ and $\xi$ are fitting parameters to be obtained from the
comparison of the IBM deformation parameters with the ones of the collective
model. On the other hand, it is assumed that $\gamma_{B}=\gamma$.

Figure 7: Matrix coherent-state potential energy surface along the axial axis
for even-even Zr isotopes, from 94Zr up to 110Zr. The deepest energy minimum
is set to zero.

The main novelty of this work with respect to García-Ramos  _et al._ (2014) is
that in the present study, the value of $\beta$ is extracted from the
quadrupole shape invariants using the wave function obtained from the IBM-CM
calculation with the parameters given in García-Ramos and Heyde (2019) while
in García-Ramos  _et al._ (2014) it was obtained from a mean-field energy
surface using a Gogny-D1S interaction. The equations involved to extract
$\beta$ from the quadrupole shape invariants are Kumar (1972); Cline (1986),

$q_{2}=\sqrt{5}\langle 0^{+}|[\hat{Q}\times\hat{Q}]^{(0)}|0^{+}\rangle,$ (10)
$\beta=\frac{4\,\pi\,\sqrt{q_{2}}}{3\,Z\,e\,r_{0}^{2}\,A^{2/3}},$ (11)

where $r_{0}=1.2A^{1/3}$ fm and $e$ the charge of the proton.

Figure 8: Matrix coherent-state potential energy surface in the $\beta-\gamma$
plane for even-even Zr isotopes, from 94Zr up to 110Zr. The energy spacing
between adjacent contour lines is $100$ keV and the deepest energy minimum is
set to zero, corresponding to the red color.

As next step we perform a least-squares fit using the value of $\beta$
extracted from the quadrupole shape invariants and the value of $\beta_{B}$
from IBM corresponding to the minimum of the mean-field energy surface. This
procedure leads to the following values of the fitting coefficients including
its respective errors $\delta=1.98(15)$, $\xi=0.110(17)$.

In Fig. 6(a) the scaled $\beta$ values extracted from Eq. (9) are compared
with the ones obtained from the quadrupole shape invariant (10,11). We also
show the ones corresponding to the unperturbed [N] and [N+2] configurations.
Note that the IBM-CM $\beta$ value for $A=100$ corresponds to a spherical
minimum, although the deformed one was used for performing the least-squares
fit previously described, because it is known that for $A=100$ Zr is already
strongly deformed. The spherical minima (for $A<100$) have values of
$\beta\neq 0$ because the obtained intercept in the least-squares fit is also
different from $0$. According to the presented results, the regular
configuration always present a very small value of $\beta$ and therefore we
identify them with spherical states. The intruder configuration shows a rather
different trend. On one hand, it is strongly perturbed because of the vicinity
of the $N=50$ and $N=56$ shell and subshell closures, respectively. On the
other hand, it presents the rapid change from a spherical shape towards a
deformed one, around $A=100$. The agreement between the mean-field results and
the ones extracted for the quadrupole shape invariants is remarkable. The
switch in $\beta$ from the regular to the intruder configuration is perfectly
understandable according to Fig. 6(b) where the unperturbed energies of the
regular and intruder configurations cross at $A=100$. Note that the value of
$\beta$ obtained from the quadrupole shape invariant already corresponds to a
deformed shape for $A=100$ while not the one from mean field. This is due to
the existence of two almost degenerate minima at the IBM-CM mean-field level
for $A=100$, one spherical and the other deformed, being the spherical minimum
a little deeper than the deformed one (see Figs. 7(d) and 8(d)). However, the
calculation in the laboratory frame leads to a deformed shape for $A=100$.

In Fig. 7 the axial energy surface for the whole chain of isotopes is
depicted, while in Fig. 8 the energy surfaces in the $(\beta,\gamma)$ plane is
presented. Note that the rescaled $\beta$ value has been used in all the
figures. These two sets of figures confirm the existence of a single minimum
of spherical nature for 94-98Zr; for 100Zr two minima coexist with almost the
same energy, though the spherical is the deepest one; in 98-102Zr a rapid
transition from a spherical to well-deformed shapes is observed; from 102Zr
onwards two minima coexist, although in this case the deformed is the deepest
one and the spherical minimum is very shallow. This fact is more clearly
observed in Fig. 7 as compared with Fig. 8. Note that the deformed minimum for
the heavier isotopes is always prolate.

It can be of interest to compare the results presented in the present study
with the mean-field energy surfaces obtained with realistic interactions. In
the literature, we encountered several recent calculations in which the Zr
isotopes are studied. In the Bruyères-le-Châtel compilation Hilairea and Girod
(2008); CEA , Hartree-Fock-Bogoliubov calculations based on the Gogny-D1S
effective interaction are presented. The obtained values of $\beta$ in the
deformed cases are quite similar to the ones obtained in the present study,
although the minima of 98-100Zr are oblate (almost degenerate with the prolate
one), and, in general, prolate and oblate minima are connected through a
triaxial valley. The same holds for the other isotopes. In general, in the
well deformed cases $\beta_{\text{oblate}}\approx 0.2$ while
$\beta_{\text{prolate}}\approx 0.4$. The IBM-CM calculations for these nuclei
present $\beta\approx 0.4$. In Ref. Nomura  _et al._ (2016), HFB energy
surfaces using a Gogny-D1M interaction are used and mapped into an IBM-CM with
three different particle-hole configurations. Here, also prolate and oblate
almost degenerate minima coexist, but also a spherical minima, although at
very high energy, exists. The main drawback of this approach is that it
predicts a deformed shape for 96Zr and 98Zr. In Ref. Togashi  _et al._ (2016),
Monte Carlo shell-model calculations are carried out for the Zr isotopes.
Within this approach it is possible to generate total energy surfaces and to
extract information on the deformation of the states. Here, the deformed
nuclei corresponds to prolate shapes and the potential energy is flat in the
$\gamma$ direction, with presence of a local minimum in the oblate side, but
less deformed than the prolate one. The value of $\beta$ for 100-101Zr is
approximately $0.35$ which is slightly smaller than the value obtained with a
Gogny interaction. In Ref. Rodríguez-Guzmán  _et al._ (2010), the interaction
Gogny-D1S has also been used and an oblate-prolate transition is observed,
passing from $A=98$ to $A=100$, resulting in a prolate shape with
$\beta\approx 0.44$ for 100Zr. In Abusara and Ahmad (2017) a relativistic EDF
is considered, obtaining a general prolate-oblate coexistence for the deformed
part of the Zr chain and, in the case of 100Zr, also a spherical-prolate
coexistence. Moreover, in the heaviest isotope the deepest minimum corresponds
to the oblate one.

In summary, a comparison between the realistic mean-field calculations and the
IBM-CM energy surfaces points to certain similarities, as for example, the
spherical shape of the lightest isotopes and the presence of a local spherical
minimum for 100Zr, although at the same time an oblate minimum is present in
the realistic mean-field calculations together with a shallow valley
connecting with the prolate one, which is almost degenerate with the first
one. However, in the IBM only the spherical and the prolate minima are
present, which seems to be a limitation of the used approach.Most of the
realistic mean-field calculations present an oblate shape (global minimum) for
the heaviest isotopes while the IBM energy surface calculations always
generate a prolate one.

Figure 9: U(5) decomposition ($n_{d}$ content, $NDC$, see text for details) of
the yrast band and the states $0_{2}^{+}$, $2_{2}^{+}$, $4_{2}^{+}$, and
$6_{2}^{+}$ of 94Zr (panels (a) and (b)), 96Zr (panels (c) and (d)), 98Zr
(panels (e) and (f)). Columns labeled with “Regular” correspond to the
decomposition in the regular sector, while labeled with “Intruder” to the
decomposition in the intruder sector.

###  V.2 Wave function structure

To know about the structure of the different eigenstates of Zr isotopes, it is
very enlightening to express the wave function using the U(5) basis in both
the regular and the intruder spaces. This basis allows to distinguish between
states with a spherical character, i.e., states described with wave functions
having an almost single U(5) component as well as states with a broad
distribution of U(5) components. To do so, we express explicitly the
eigenfunctions in the U(5) basis, making use of the corresponding U(5) quantum
numbers, namely, $n_{d}$, $\tau$, and $n_{\Delta}$, for the N and N+2
subsystems, respectively,

$\displaystyle\Psi(k,JM)$ $\displaystyle=$
$\displaystyle\sum_{n_{d},\tau,n_{\Delta}}a^{k}_{n_{d},\tau,n_{\Delta}}(J;N)\psi((sd)^{N}_{n_{d},\tau,n_{\Delta}};JM)$
(12) $\displaystyle+$ $\displaystyle\
\sum_{n_{d},\tau,n_{\Delta}}b^{k}_{n_{d},\tau,n_{\Delta}}(J;N+2)\psi((sd)^{N+2}_{n_{d},\tau,n_{\Delta}};JM)~{}.$

Here, $n_{d}$ corresponds to the number of $d$ bosons, $\tau$ to the boson
seniority, $n_{\Delta}$ to the number of $d$ boson triplets coupled to zero,
and $k$ is a rank number to label the state. The $n_{d}$ content of the wave
function, that we will call $NDC$, either in the regular or the intruder
sector are defined as,

$\displaystyle NDC(n_{d},k,N)$ $\displaystyle=$
$\displaystyle\sum_{\tau,n_{\Delta}}\left(a^{k}_{n_{d},\tau,n_{\Delta}}(J;N)\right)^{2}$
(13) $\displaystyle NDC(n_{d},k,N+2)$ $\displaystyle=$
$\displaystyle\sum_{\tau,n_{\Delta}}\left(b^{k}_{n_{d},\tau,n_{\Delta}}(J;N+2)\right)^{2}$
(14)

fulfilling the normalization constraint,

$\sum_{n_{d}}NDC(n_{d},k,N)+\sum_{n_{d}}NDC(n_{d},k,N+2)=1.$ (15)

Here, the sum extends over all allowed $n_{d}$ values in the Hilbert spaces.
$NDC$ provides precise information on the structure of the considered states.
On one hand, if only one, or very few, of the $n_{d}$ components have a
sizeable value, it implies that the state has a vibrational or spherical
character, either in the regular or in the intruder sector. On the other hand,
if the $NDC$ values are broadly distributed over several $n_{d}$ components,
the state can be defined to exhibit a rotational or deformed character. This
is a possible way to express the underlying structure of the wave functions
derived from the IBM-CM, but the $NDC$ values do not correspond in any direct
way with observables as measured in the various experiments.

Figure 10: Same caption than Fig. 9, but for 100Zr (panels (a) and (b)), 102Zr
(panels (c) and (d)), 104Zr(panels (e) and (f)).

In Figs. 9 and 10 we depict the value of $NDC$ for the first and the second
member of each angular momentum $0^{+}$, $2^{+}$, $4^{+}$, $6^{+}$. We present
the results for the isotopes where the most rapid changes in structure are
expected. In panels (a) and (b) of Fig. 9, 94Zr is presented. It is easily
appreciated how the states $0_{1}^{+}$, $2_{1}^{+}$, $4_{1}^{+}$ own a regular
and spherical character, while $6_{1}^{+}$ is also of spherical nature but
situated mainly in the intruder sector. On the other hand, the states
$0_{2}^{+}$, $2_{2}^{+}$, $4_{2}^{+}$ are indicative of a deformed behavior
(the distribution is spread over several components) in the intruder sector,
although containing a small contribution in the regular side. Finally, the
structure of the state $6_{2}^{+}$ is intruder and spherical. The nuclei 96Zr
and 98Zr (panels (c), (d), (e), and (f)) exhibit a similar structure, namely,
$0_{1}^{+}$, $4_{1}^{+}$, and $6_{1}^{+}$ present a regular and spherical
behavior; $2_{1}^{+}$ and $2_{2}^{+}$ also present a spherical behavior but
with components in the regular and in the intruder sectors; finally,
$0_{2}^{+}$, $4_{2}^{+}$, and $6_{2}^{+}$ owns an intruder, but spherical
character. In Fig. 10 we enter into the region where deformation quickly
appears and consequently, the yrast band of the three nuclei (panels (a), (c),
and (e)) exhibit an almost $100\%$ intruder component for all analyzed angular
momenta, with $NDC$ values spread over many components, again acting as a
strong hint for the presence of a deformed shape. For the non-yrast states,
the situation is slightly different in every isotope. In the case of 100Zr
(panel (b)), the states $0_{2}^{+}$ and $2_{2}^{+}$ present a spherical and
regular character, while $4_{2}^{+}$ and $6_{2}^{+}$ an intruder and deformed
one. In the case of 102Zr (panel (d)), only the $0_{2}^{+}$ state presents a
regular and spherical character, while the rest of the band exhibits an
intruder and rather deformed character, with the $NDC$ distributed over many
$n_{d}$ components. Finally, in 104Zr (panel (f)), all the states show an
intruder character with a rather spread $NDC$ distribution, but also with a
very large component at the lowest possible value of $n_{d}$. The
corresponding figures for 104-110Zr (not shown) present the same trend as in
102Zr.

In summary, one observes how the structure of the regular states evolves from
being spherical for 94,96,98,100,102Zr to being slightly deformed in 104Zr and
onwards (not shown in figures). Concerning the structure of the intruder
states, they start with a deformed structure in 94Zr, but then the internal
structure changes into a spherical one for 96-98Zr, in 100Zr it presents a
mixed structure, finally, becoming deformed in 102-104-106Zr and onwards.

###  V.3 Density of states

The density of low lying states is a property that is expected to be strongly
perturbed by the presence of intruder states or by the existence of a QPT.
Hence, it is worth to be analyzed. As a matter of fact, Bucurescu and Zamfir
(2018) propose the density of states as an indicator to distinguish between
shape coexistence and the presence of a QPT.

Figure 11: Number of states below an excitation energy E=$4$ MeV as a function
of A, computed using IBM-CM calculations García-Ramos and Heyde (2018). Black
full line corresponds to the number of $0^{+}$ states and red dashed one to
the total number of states, up to and including L$=8$.

In this section we present a quantity that is related with the density of
states as it is the number of states below a certain energy, namely $4$ MeV.
To such an end we rely on the IBM-CM results from García-Ramos and Heyde
(2018). Owing to that the presence of low lying $0^{+}$ states is a hint for
the existence of shape coexistence we compute the number of $0^{+}$ states
together with the total number of states up to and including $L=8$ below $4$
MeV in Fig. 11.

In Fig. 11 the theoretical number of $0^{+}$ and the total number of states
are presented. Both present a sudden increase when reaching the critical
point, $A=100$. The number of $0^{+}$ states remains rather constant after the
critical point although with a smooth decrease, but it is barely affected by
the presence of the mid shell. Concerning the total number of states below 4
MeV, it also suffers a rapid increase at the QPT point, but in this case there
are also abrupt changes after this point.

Bucurescu and Zamfir (2018) suggest that the behavior of the density of states
can provide some hints to distinguish between the presence of shape
coexistence from the presence of a QPT. The key point is that a QPT implies
the existence of a maximum, with a peak, in the density of states, while the
existence of shape coexistence supposes a rapid increase in the density of
states but rather with a plateau than with a peak.

According to our results it seems that the number of $0^{+}$ states points
towards the presence of shape coexistence. However, the peak in the total
number of states rather suggests the existence of a QPT. Hence, no clear
conclusions can be obtained from this analysis.

##  VI Discussion

Figure 12:  (a) Absolute energy systematics of $J^{\pi}=0^{+}$ states for an
IBM Hamiltonian with two configurations, one with constant parameters and
another one that experiences a QPT (see text); (b) absolute energy systematics
of $J^{\pi}=0^{+}$ states for an IBM Hamiltonian with a single configuration
that experiences a QPT (see text); (c) the same as in panel (a) but for the
relative energy; (d) the same as in panel (b) but for the relative energy; (e)
regular component of the first three low-lying states corresponding to panel
(a); in panel (f), the $\langle n_{d}\rangle$ value for the first three low-
lying states corresponding to panel (b). The green curve stands for the
derivative of the ground state energy. All energies are given in arbitrary
units.

So far we have accumulated evidences pointing either to the presence of shape
coexistence or to the existence of a QPT and most probably the reader got the
impression that both phenomena are, indeed, difficult to be separated because
both are tightly connected. To shed some light on this issue it is useful to
perform a set of schematic calculations comparing both situations but at the
same time being close to what happens experimentally for the Zr isotopes. This
is illustrated in Fig. 12 where the spectrum of an IBM-CM mixing calculation
is compared with an IBM single configuration one. In panels (a), (c), and (e)
we used a configuration mixing Hamiltonian with $\varepsilon_{N}=1$ (in
arbitrary units) and the remaining of the regular Hamiltonian parameters are
put equal to zero, while $\varepsilon_{N+2}=x$,
$\kappa_{N+2}=\frac{x-1}{N+2}$, $\chi=-\sqrt{7}/2$, and the remaining of the
intruder parameters are put equal to zero. The number of bosons is set to
$N=18\;(N+2=20)$ and the mixing term to
$\omega_{0}^{N,N+2}=\omega_{2}^{N,N+2}=0.02$. This small value will ensure a
tiny mixing between the regular and the intruder sectors in the wave function.
The shift parameter is fixed to $\Delta^{N+2}=0.75$ to guaranty a crossing of
the regular and the intruder $0^{+}$ bandheads at $x=9/11\approx 0.8181$ for
$\omega_{0}^{N,N+2}=\omega_{2}^{N,N+2}=0.0$. Moreover, in panels (b), (d), and
(f) we present the calculation for a single IBM configuration with
$\varepsilon=x$, $\kappa=\frac{x-1}{N}$, $\chi=-\sqrt{7}/2$, with the
remaining parameters equal to zero and using $N=20$. Note that $x$ acts as a
control parameter in both Hamiltonians.

The schematic IBM-CM Hamiltonian was built in order to mimic the Zr
systematics, with a regular configuration which Hamiltonian remains
essentially constant and an intruder configuration that undergoes a QPT. The
mass number $A=100$ qualitatively should correspond to the control parameter
$x=9/11$, simply because, as pointed out in Section III, 100Zr corresponds to
a critical point. Moreover, increasing (decreasing) values of $x$
qualitatively correspond to decreasing (increasing) values of A because the
nuclei become more spherical (deformed) as $x$ increases. On the other hand,
the IBM single configuration calculation only aims at reproducing what happens
in the Zr ground states. Once more, there is a QPT at $x=9/11$. To help the
understanding of Fig. 12, we have added the labels 98Zr, 100Zr, and 102Zr as
an extra reference to guide the eye inspecting the content of this figure.

What strikes in the comparison is that for $x<9/11$ and $E<0$ both spectra are
almost identical. The reason is that for this area the leading Hamiltonians in
left and right panels are approximately the same. However, above $E=0$ clear
differences are present because the almost harmonic spectrum of the regular
Hamiltonian, i.e., E=$0,2,3,4,\ldots$, is present in panels (a) and (c), but
not in (b) and (d). These vibrational states are clearly observed because of
the small mixing that has been used. Besides, in panels (a) and (c) one can
note how the spherical states repel the deformed ones and avoided crossings
are present, happening in particular for the ground state. Regarding panels
(b) and (d), the global spectrum undergoes an evolution from deformed to
spherical shapes because of the dependence of the Hamiltonian on $x$. To
better appreciate the analogies and differences between both calculations, the
lowest three states are marked. Concerning these three states, for $x<9/11$,
the left panel and the right panel show essentially the same trend because for
this zone the spherical states (regular) are not yet influencing the lower
part of the spectrum. However, for $x=9/11$ the slope of the energy curves
changes more rapidly in the IBM-CM case as compared with the single
configuration one. The reason is due to the crossing and repulsion between
regular and intruder families of states. The most evident consequence is that
the change in the first derivative (green long dashed curves) of the ground
state energy is much faster in the left than in the right panels. Second
consequence refers to the phonon structure of the states for $x>9/11$. In
panel (a) and (c) the first and the second states have $\langle
n_{d}\rangle\approx 0$, while the third has $\langle n_{d}\rangle\approx 2$
(not shown in the figure). This follows from the fact that the first and the
third states have intruder nature while the second one is mainly of regular
character, as shown in panel (e). In panel (b) and (d), however, the ground
state has $\langle n_{d}\rangle\approx 0$, the second $\langle
n_{d}\rangle\approx 2$, while the third $\langle n_{d}\rangle\approx 3$, as
illustrated as a function of x in panel (f). As a consequence, the presence in
the spectrum of two low lying $0^{+}$ states with $\langle n_{d}\rangle\approx
0$ is a direct consequence of the existence of two families of states. The
existence of two families of states is responsible of the more rapid lowering
of the first excited state as compared with the case of a single configuration
[see panels (c) and (d)].

The quantity presented in panels (e) and (f), the regular component and
$\langle n_{d}\rangle$, respectively, are both ideal candidates to serve as an
order parameter. The observed changes are much more rapid in the case of two
configurations than in the case of a single one, as already observed for the
ground state energy. Therefore, an extra consequence of the presence of two
configurations is that the induced QPT happens abrupt as compared with the
case when considering only a single configuration. This is also observed in
the green long dashed line of Fig. 12(a) and Fig. 12(b).

In left panels, one observes that the QPT that the intruder configuration
undergoes favours the more rapid lowering of the intruder states. It is
evident that the crossing of two configurations with a constant Hamiltonian
induces a certain discontinuity in the derivative of the ground state energy
with respect to a control parameter, but this discontinuity will increase if
on top of the crossing, the intruder ground state evolves from being spherical
to becoming deformed. This fact can be explained inspecting Fig. 6(b) in which
one observes a large energy gain in binding energy being due to the onset of
deformation.

Figure 13: Energy systematics for Zr isotopes from the IBM-CM calculation,
separating in states which are mainly regular (full lines with full symbol)
and mainly of intruder nature (dashed lines with open symbols), compared with
an IBM calculation using a single configuration García-Ramos  _et al._ (2005)
(dotted line and starts).

What has been presented so far in the schematic calculations turns out to be
equivalent to the comparison of the IBM-CM results used in this work and
extracted from Ref. García-Ramos and Heyde (2019) with the results of an IBM
calculation for Zr using a single configuration, e.g, the one used in Ref.
García-Ramos  _et al._ (2005) (only calculated up to $A=104$). In the latter
paper, the IBM parameters were obtained through a least-squares fit to the
available information (available in 2005), with special emphasis in the two-
neutron separation energy, that was considered as the main indicator hinting
towards the existence of a QPT. In Ref. García-Ramos  _et al._ (2005) it was
proven that the even-even Zr isotopes undergo a QPT with 100Zr being the
critical nucleus. To study how both approaches are compared, in Fig. 13 we
depict part of the spectra of both Hamiltonians. The regular and the intruder
states in the IBM-CM calculation are separated taking into account the value
of its regular component, i.e., if larger than $0.5$ are marked as regular
while if smaller than $0.5$ are marked as intruder. A first relevant fact is
that the single configuration calculation gives rise to too high excitation
energies for $L>2$ in the lightest isotopes. Moreover, it follows reasonably
well that the excitation energy of the $2_{1}^{+}$ state all the way and the
one of the $4_{1}^{+}$ and $6_{1}^{+}$ states from $A=100$ and onwards. A
second important result corresponds to the good reproduction of the
$0_{2}^{+}$ excitation energy for $A=94$, $96$, $98$ and $104$, but not for
$A=100$ and $102$. Therefore, some states present in the spectrum, namely, the
$0_{2}^{+}$ state for 100Zr, cannot be correctly reproduced using a single
configuration. Consequently, in Ref. García-Ramos  _et al._ (2005) it was
explained that the experimental $0_{2}^{+}$ state of 100Zr, which is of
approximate spherical character, was excluded from the least-squares fit. This
is, therefore, the main drawback of the single-configuration calculation,
i.e., some states are beyond the model space and cannot be correctly
reproduced, particularly the very low-lying excited $0^{+}$ states. In the
case of the IBM-CM calculation, very low-lying $0^{+}$ states are correctly
explained because the crossing of two different configurations, intruder and
regular.

Therefore, one can conclude that the differences between the energy
systematics of both approaches can be understood considering the level
repulsion and quantum integrability as explained in Arias  _et al._ (2003).
For a single configuration, the existence of a sizeable interaction matrix
elements between the first two $0^{+}$ states prevents its approaching, and
instead, a large repulsion results. However, in the case of regular and
intruder configurations, the existence of a quantum number with different
value for both, namely N and N+2, allows the crossing of both configurations
in the case of no mixing term, or the presence of a small repulsion for
sufficiently small values of the mixing term, leading to a much more abrupt
change in the slope of the ground state energy than in the case of a QPT with
a single configuration.

##  VII Conclusions and outlook

In the present study, a connection between shape coexistence and QPT has been
systematically studied from different points of view and exemplified for the
case of Zr nuclei. First, we have reviewed the many experimental evidences for
shape coexistence and for QPT. Next, the possible fulfillment of the X(5)
critical point symmetry has been considered concluding that the evidences are
rather mixed and that 100Zr and 102Zr present certain characteristics of the
X(5) symmetry but not all of them. The IBM-CM mean-field energy surfaces have
been presented, both in the axial direction and in the $\beta-\gamma$ plane.
The scale that connects the IBM deformation parameters and the one from the
collective model has been obtained for every isotope. With the goal of
analyzing in depth the structure of the wave function, a decomposition in the
U(5) basis was carried out, observing how the structure of the intruder states
changes from being spherical to being well deformed. Also, the evolution of
the density of states as a function of mass number has been explored, without
obtaining a clear conclusion. Finally, a schematic calculation has been
conducted to obtain some hints to distinguish between shape coexistence and
QPT.

In short, in the even-even Zr isotopes, ample experimental evidence exists for
a QPT showing up , with 100Zr as the “critical nucleus”. This conclusion is
based on the experimental information [S2n, $E(4_{1}^{+})/E(2_{1}^{+})$, or
B(E2:$4_{1}^{+}\rightarrow 2_{1}^{+}$) evolution] and supported by several
theoretical models, including the IBM used in the present work. It is also
self-evident, in view of the available experimental information, that states
with a very different structure coexist in a narrow region of energy and,
consequently, shape coexistence shows up in the Zr isotopic chain of nuclei
and is responsible for the evolution of the nuclear structure in this region
of the nuclear mass table. It is fair to say that the existence of a QPT is
triggered by shape coexistence and is the result of the crossing in the
$0^{+}$ ground state of two configurations corresponding with different
structural properties, i.e., spherical and deformed character. The reduce
mixing between regular and intruder configurations and the evolution of the
intruder configuration from a spherical shape to a deformed one when passing
from 98Zr to 100Zr is at the origin of the unique properties of the shape
evolution observed in this mass area. Similar conclusions have been formulated
in a recent conference contribution by Poves (2018) and in Caurier  _et al._
(2007) with a discussion for the doubly-magic 40Ca nucleus. A possible
conclusion is that if the shell gap at $Z,N=20$ single-particle levels would
have been a little smaller, this doubly magic nucleus would have turned up as
deformed in its ground state band Heyde and Wood (2011).

The study of the possible relationship between shape coexistence and QPT in Zr
isotopes has been also carried out in Refs. Gavrielov  _et al._ (2019a, b),
obtaining similar conclusions, although in these works the authors claimed
that there is an addition evolution into gamma-unstable shapes in 106Zr an
onwards.

The presented analysis can encourage further studies in this mass region, for
example in the Sr isotopes, but also at the border of this deformation area,
i.e., for Kr and Mo isotopes.

##  VIII Acknowledgment

We are very grateful to A. Leviatan and J.L. Wood for enlightening
discussions. This work was supported (KH) by the InterUniversity Attraction
Poles Program of the Belgian State-Federal Office for Scientific, Technical
and Cultural Affairs (IAP Grant P7/12) and it has also been partially
supported (JEGR) by the Ministerio de Ciencia, Innovación y Universidades
(Spain) under project number PID2019-104002GB-C21, by the Consejería de
Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía
(Spain) under Group FQM-370 and by European Regional Development Fund (ERDF),
ref. SOMM17/6105/UGR. Resources supporting this work were provided by the
CEAFMC and Universidad de Huelva High Performance Computer (HPC@UHU) funded by
ERDF/MINECO project UNHU-15CE-2848.

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