Intertwined quantum phase transitions in the even-even $^{90-100}$Sr isotopes

Using the interacting boson model with configuration mixing, this study identifies the even-even $^{90-100}$Sr isotopes as a region of intertwined quantum phase transitions where a crossing between normal and intruder configurations coincides with a shape evolution within the intruder configuration, a scenario strongly supported by comprehensive comparisons with experimental data.

The Gist

Imagine the atomic nucleus not as a solid ball, but as a bustling dance floor filled with pairs of dancers (protons and neutrons). In the world of physics, these dancers can arrange themselves in different "styles" or shapes: sometimes they move in a loose, spherical circle (like a calm waltz), and other times they stretch out into a long, spinning oval (like an energetic tango).

This paper investigates a specific group of atoms called Strontium isotopes (specifically those with mass numbers 90 to 100). The researchers found that as you add more neutrons to these atoms, the dance floor undergoes a dramatic, double-layered transformation. They call this an "Intertwined Quantum Phase Transition" (IQPT).

Here is the story of what happens, broken down into simple concepts:

1. The Two Dance Styles (Configurations)

In these atoms, the dancers can be in one of two main "costumes" or configurations:

• The Normal Costume: This is the standard, calm style. For lighter Strontium atoms, the nucleus stays mostly spherical and weakly active.
• The Intruder Costume: This is a special, excited style where the dancers have jumped to a higher energy level. In the heavier Strontium atoms, this style becomes very energetic and deformed (stretched out).

2. The Double Transformation (The "Intertwined" Part)

Usually, a change in an atom happens in one way: either the shape changes slowly, or the dancers swap costumes. But in Strontium, both happen at the same time, creating a "double switch."

• The Shape Shift (Type I): As the atoms get heavier, the "Intruder" dancers slowly change their style. They start as a loose, spherical group (in lighter atoms) and gradually stretch out into a tight, spinning oval (in heavier atoms). It's like a group of people slowly turning from a circle into a line.
• The Costume Swap (Type II): At the same time, there is a "tug-of-war" between the two costumes. For a while, the "Normal" costume is the favorite (it's the ground state). But suddenly, around a specific number of neutrons, the "Intruder" costume becomes the favorite. The ground state of the atom abruptly switches from being "Normal" to being "Intruder."

3. The Critical Moment (The Tipping Point)

The paper identifies a specific "tipping point" between the atoms Strontium-96 and Strontium-98.

• Before the switch (Strontium 90–96): The atom is mostly "Normal" (spherical). The "Intruder" dancers are there, but they are just watching from the sidelines, mostly spherical themselves.
• The Switch (Strontium 96 to 98): The "Intruder" dancers suddenly stretch out (become deformed) and they win the tug-of-war, taking over the main stage. The atom's ground state flips from a weak, spherical shape to a strong, stretched-out shape.
• After the switch (Strontium 98–100): The atom is now fully "Intruder" and fully deformed.

4. How They Knew This Happened

The researchers didn't just guess; they used a mathematical model (the Interacting Boson Model with Configuration Mixing) to simulate the dance floor and compared their predictions to real-world experiments. They looked at four key "clues":

1. Energy Levels: How much energy it takes to make the dancers move. The data showed a sudden drop in energy, signaling the costume swap.
2. Shape Measurements: They measured the "quadrupole moments" (essentially, how round or oval the atom is). The data showed a sudden jump from round to oval.
3. Size Changes: They measured how the size of the atom changes as neutrons are added. The size jumped unexpectedly at the critical point, confirming the shape change.
4. Electrical Signals: They looked at how the atom emits energy (monopole transitions). A huge spike in this signal occurred exactly at the moment the two configurations crossed paths.

The Big Picture

The authors conclude that Strontium is a perfect example of this "Intertwined" phenomenon. It joins a small club of elements (like its neighbor Zirconium) where the nucleus doesn't just slowly change shape or just slowly swap costumes—it does both simultaneously in a dramatic, abrupt leap.

Think of it like a car that, while driving down a highway, suddenly switches from a sedan to a sports car and accelerates from 30 mph to 100 mph in the blink of an eye. That is the "Intertwined Quantum Phase Transition" happening inside the Strontium atom.

Technical Summary

Technical Summary: Intertwined Quantum Phase Transitions in the Even-Even 90–100Sr Isotopes

Problem Statement
The region of nuclei with mass number $A \approx 100$ is characterized by abrupt structural changes around neutron number $N=60$. Traditionally, the ground state for $N < 60$ is interpreted as a spherical vibrator, while for $N \ge 60$, it exhibits dense spectra associated with deformed rotors. However, these structural evolutions are complicated by the crossing of different proton configurations (normal vs. intruder), leading to shape coexistence. While Quantum Phase Transitions (QPTs) in this region have been studied, specifically in zirconium isotopes, the specific nature of these transitions in the strontium chain ($Z=38$) requires a unified theoretical description that distinguishes between shape evolution within a single configuration (Type I QPT) and the crossing of multiple configurations (Type II QPT). The paper aims to identify and characterize "Intertwined Quantum Phase Transitions" (IQPTs) in the even-even $^{90-100}$Sr isotopes.

Methodology
The study employs the Interacting Boson Model with Configuration Mixing (IBM-CM) to describe the low-lying quadrupole states of the even-even strontium isotopes.

• Model Space: The model utilizes two configurations: a "normal" configuration ($A$) corresponding to 0p-2h proton excitations relative to the $Z=40$ subshell, and an "intruder" configuration ($B$) corresponding to 2p-4h proton excitations across the same subshell. The boson numbers are $N_B = N_\pi + N_\nu$ for the normal state and $N_B + 2$ for the intruder state.
• Hamiltonian: The total Hamiltonian is a block matrix containing separate Hamiltonians for the normal ($\hat{H}^{(A)}$) and intruder ($\hat{H}^{(B)}$) spaces, coupled by a mixing interaction $\hat{W}$. The individual Hamiltonians include terms for the $d$-boson energy ($\epsilon_d$), quadrupole-quadrupole interaction ($\kappa \hat{Q} \cdot \hat{Q}$), and angular momentum ($\kappa' \hat{L} \cdot \hat{L}$). An energy offset $\Delta_p$ accounts for the monopole contribution of the proton-neutron interaction.
• Parameters: Parameters were fitted to experimental data, including excitation energies, spectroscopic quadrupole moments, isotope shifts, and monopole $E0$ transition strengths. A key feature of the parameterization is the simplicity of the trends: most parameters remain constant or follow a well-defined trend across the chain, with the mixing parameter $\omega$ held constant at 0.017 MeV.
• Analysis: The wave functions are diagonalized to determine occupation probabilities of the normal and intruder configurations ($v^2$) and the $d$-boson number ($n_d$), which serves as an indicator of deformation.

Key Contributions and Results
The analysis confirms that the strontium chain exhibits IQPTs, characterized by the simultaneous occurrence of Type I and Type II QPTs:

1. Type II QPT (Configuration Crossing): Between neutron numbers $N=58$ ($^{96}$Sr) and $N=60$ ($^{98}$Sr), the ground state configuration undergoes an abrupt inversion. The ground state changes from a weakly collective normal configuration (dominant in $^{90-96}$Sr) to a deformed intruder configuration (dominant in $^{98-100}$Sr). This crossing is marked by a sudden drop in the $2^+_1$ energy and a sharp change in the isotope shift and monopole $E0$ transition strength.
2. Type I QPT (Shape Evolution): Within the intruder configuration itself, a shape evolution occurs. In $^{90-96}$Sr, the intruder configuration maintains a near-spherical structure (dominant $n_d=0$ component). In $^{98,100}$Sr, the intruder configuration evolves into a deformed structure, evidenced by a spread in $n_d$ occupation and the emergence of a rotational band structure.
3. Validation against Experiment: The model successfully reproduces experimental data across the chain:
   • Energy Levels: The calculated spectra match experimental levels, capturing the abrupt drop in the $2^+_1$ energy at $N=60$ and the evolution of the intruder band.
   • Quadrupole Moments: The calculated spectroscopic quadrupole moments ($Q_s$) show a transition from near-zero values (spherical) to large negative values (prolate deformed) at $N=60$, consistent with the onset of deformation in the intruder ground state.
   • Isotope Shifts and Monopole Strengths: The isotope shift $\Delta \langle r^2 \rangle$ shows a peak at the critical point ($^{98}$Sr), and the monopole $E0$ strength for the $0^+_2 \to 0^+_1$ transition peaks at $^{98}$Sr, signaling the configuration crossing.
4. Critical Point Analysis: Detailed examination of $^{96}$Sr and $^{98}$Sr reveals that $^{96}$Sr represents the pre-critical point where the intruder band is weakly deformed and the normal configuration is ground, while $^{98}$Sr represents the post-critical point where the intruder configuration is the ground state with strong deformation. The mixing between configurations remains small throughout, allowing the distinct identification of the two QPT types.

Significance
The paper establishes the even-even strontium isotopes as a second realization of Intertwined Quantum Phase Transitions in the $A \approx 100$ region, placing them alongside the neighboring zirconium chain. The study demonstrates that the structural changes in this region are not merely a single phase transition but a complex interplay where a shape evolution within an excited configuration (Type I) coincides with a crossing of that configuration with the ground state configuration (Type II). By utilizing a simple parameter set within the IBM-CM framework, the work provides a unified picture of the nuclear structure in this region, reinforcing the necessity of incorporating multiple configurations to understand the rapid onset of deformation near $N=60$. The results suggest that similar IQPT scenarios may exist in neighboring odd-mass isotopes, motivating further theoretical and experimental investigation into the interplay between collective and single-particle degrees of freedom in yttrium and strontium isotopes.