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

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.

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.

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.

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.

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.

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,

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.

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.

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 S_2n (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).

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.

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,

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

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,

the angular momentum operator, and

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

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.

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.

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.

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 [S_2n, $E(4_{1}^{+})/E(2_{1}^{+})$, 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.

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.