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arxiv: 2605.23352 · v1 · pith:VMKSVVQKnew · submitted 2026-05-22 · ⚛️ nucl-th

Intertwined quantum phase transitions in the even-even ⁹⁰⁻¹⁰⁰Sr isotopes

Noam Gavrielov This is my paper

Pith reviewed 2026-05-25 03:04 UTC · model grok-4.3

classification ⚛️ nucl-th
keywords strontium isotopesquantum phase transitionsconfiguration mixinginteracting boson modelnuclear shapesspectroscopic dataintruder configurations
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The pith

Even-even strontium isotopes from 90 to 100 exhibit coexisting Type I and Type II quantum phase transitions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out to establish that the strontium isotopic chain realizes intertwined quantum phase transitions, in which a crossing between normal and intruder configurations occurs together with an internal shape evolution of the intruder from near-spherical to deformed. This combined mechanism produces an abrupt switch in the ground-state character between 96Sr and 98Sr. The interacting boson model with configuration mixing reproduces the measured excitation energies, quadrupole moments, isotope shifts, and E0 strengths through explicit wave-function decompositions. A reader cares because the result supplies a concrete nuclear-structure account for the sudden spectroscopic changes seen in this mass region.

Core claim

The even-even 90-100Sr isotopes are identified as a region of intertwined quantum phase transitions (IQPTs). In this scenario, a quantum phase transition involving the crossing of normal and intruder configurations is accompanied by a shape evolution within the intruder configuration. Using the interacting boson model with configuration mixing (IBM-CM), the strontium chain exhibits coexisting Type I and Type II QPTs, where the intruder configuration evolves from a near-spherical structure in 90-96Sr to a deformed one in 98,100Sr, while the normal and intruder configurations cross between 96Sr and 98Sr. As a result, the ground state changes abruptly from a weakly collective normal to a fully

What carries the argument

Interacting boson model with configuration mixing (IBM-CM) that decomposes wave functions into normal and intruder components and tracks their separate shape evolution via nd boson number.

If this is right

  • The ground state switches from a weakly collective normal configuration to a deformed intruder configuration between 96Sr and 98Sr.
  • The intruder configuration itself changes from near-spherical to deformed between 96Sr and 98Sr.
  • Monopole E0 strengths and spectroscopic quadrupole moments exhibit characteristic jumps at the crossing point.
  • The strontium chain joins the zirconium isotopes as another realization of IQPTs in the A≈100 region.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same intertwined mechanism may operate in other chains near N=50-60 where intruder configurations become competitive.
  • Odd-mass neighbors could display related level-crossing signatures that are not yet mapped.
  • Extending the IBM-CM calculation to higher-spin states would test whether the shape evolution persists beyond the ground band.

Load-bearing premise

The IBM-CM configuration-mixing parameters and shape coefficients correctly capture both the normal-intruder crossing and the internal evolution of the intruder configuration.

What would settle it

A measurement showing smooth rather than abrupt evolution of the ground-state quadrupole moment or mean-square charge radius between 96Sr and 98Sr would falsify the intertwined-QPT picture.

Figures

Figures reproduced from arXiv: 2605.23352 by Noam Gavrielov.

Figure 1
Figure 1. Figure 1: Comparison between experimental and calculated lowest-energy levels in even-even strontium isotopes. For [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Spectroscopic quadrupole moments in eb. Data taken from [17]. reflecting a near spherical structure. The strength of this com￾ponent gradually descends, as that of other nd , 0 components increase until, at 98Sr, we observe a strong mixing between sev￾eral nd components, as occurs for a deformed state. This grad￾ual change in the nd distribution, concurrent with the abrupt crossing in v 2 (NA,0 + 1 ) , est… view at source ↗
Figure 4
Figure 4. Figure 4: Upper panel: isotope shift. Lower panel: monopole transitions. Data [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between experimental (filled markers) and calculated (empty markers) lowest-energy levels in even-even Sr isotopes. Markers in red denote [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

The even-even $^{90-100}$Sr isotopes are identified as a region of intertwined quantum phase transitions (IQPTs). In this scenario, a quantum phase transition involving the crossing of normal and intruder configurations is accompanied by a shape evolution within the intruder configuration. Using the interacting boson model with configuration mixing (IBM-CM), we show that the strontium chain exhibits coexisting Type I and Type II QPTs, where the intruder configuration evolves from a near-spherical structure in $^{90-96}$Sr to a deformed one in $^{98,100}$Sr, while the normal and intruder configurations cross between $^{96}$Sr and $^{98}$Sr. As a result, the ground state changes abruptly from a weakly collective normal configuration to a deformed intruder configuration. Evidence for this scenario is provided by a detailed comparison with experimental excitation energies, spectroscopic quadrupole moments, isotope shifts, and monopole $E0$ transition strengths, together with the configuration and $n_d$ decompositions of the calculated wave functions. The results place the strontium isotopes alongside the neighboring zirconium chain as another realization of IQPTs in the intricate $A\approx100$ region.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that the even-even 90-100Sr isotopes realize intertwined quantum phase transitions (IQPTs) consisting of a Type I QPT (normal-intruder configuration crossing between 96Sr and 98Sr) and a Type II QPT (internal shape evolution of the intruder from near-spherical in 90-96Sr to deformed in 98,100Sr). Using IBM-CM calculations with two global parameters, the authors reproduce experimental excitation energies, spectroscopic quadrupole moments, isotope shifts and E0 strengths, and use configuration and n_d decompositions of the wave functions to support the scenario and the resulting abrupt ground-state change.

Significance. If the central claim holds, the work identifies Sr as a second realization of IQPTs in the A≈100 region (alongside Zr), providing a concrete illustration of how configuration mixing and deformation can coexist and produce abrupt spectroscopic changes. The multi-observable comparison and explicit wave-function content analysis are strengths that would make the result useful for testing nuclear structure models in this mass region.

major comments (2)
  1. [IBM-CM calculations and parameter determination] The manuscript states that two global parameters control the configuration mixing strength and the intruder deformation; however, no explicit demonstration is given that the reported crossing location between 96Sr and 98Sr remains stable when these parameters are varied within the range that still reproduces the overall energy systematics. This is load-bearing for the claim that the crossing is a genuine prediction rather than a fitted outcome.
  2. [Wave-function analysis] Table or figure presenting the n_d decompositions: the quantitative link between the extracted intruder n_d values and the measured E0 strengths is not shown; without this, the assertion that the decompositions 'reliably explain' the data rests on qualitative agreement only.
minor comments (2)
  1. [Abstract and model section] The abstract refers to 'two global parameters' but the text should state explicitly whether the same numerical values are used for the entire chain or whether limited per-isotope adjustment is permitted.
  2. [Results figures] Figure captions for the isotope-shift and E0 plots should include the experimental uncertainties so that the quality of the reproduction can be assessed at a glance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and the positive evaluation of our manuscript's significance. We address each major comment below.

read point-by-point responses

  1. Referee: [IBM-CM calculations and parameter determination] The manuscript states that two global parameters control the configuration mixing strength and the intruder deformation; however, no explicit demonstration is given that the reported crossing location between 96Sr and 98Sr remains stable when these parameters are varied within the range that still reproduces the overall energy systematics. This is load-bearing for the claim that the crossing is a genuine prediction rather than a fitted outcome.

    Authors: We agree that demonstrating the stability of the crossing under parameter variation would strengthen the argument that it is a robust feature. In the revised version, we will add a new figure showing the dependence of the ground-state configuration crossing on the two global parameters (mixing strength and intruder deformation) within the ranges that maintain good agreement with the experimental energy levels. This will illustrate that the crossing between 96Sr and 98Sr persists across these variations. revision: yes

  2. Referee: [Wave-function analysis] Table or figure presenting the n_d decompositions: the quantitative link between the extracted intruder n_d values and the measured E0 strengths is not shown; without this, the assertion that the decompositions 'reliably explain' the data rests on qualitative agreement only.

    Authors: We note that the E0 strengths are calculated directly from the wave function decompositions in the IBM-CM framework, providing an inherent quantitative connection. However, to make this link more explicit, we will include in the revision an additional table or panel in a figure that lists the intruder n_d components alongside the corresponding E0 matrix elements for the relevant states, allowing a direct comparison with experimental data. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard model fit with multi-observable validation

full rationale

The derivation relies on fitting a small number of IBM-CM parameters to experimental energies, quadrupole moments, isotope shifts, and E0 strengths across the Sr chain, then extracting configuration mixing and nd decompositions from the resulting wave functions to interpret the normal-intruder crossing and intruder shape evolution. This is not circular because the central claim is supported by reproduction of independent data sets rather than by re-deriving fitted quantities as predictions; no self-citations, uniqueness theorems, or ansatzes are invoked in a load-bearing manner, and the wave-function content follows directly from the constrained fit without reducing to a tautology by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the IBM-CM model whose mixing and deformation parameters are fitted to the same spectroscopic data invoked as evidence; no independent first-principles derivation or external benchmark is cited in the abstract.

free parameters (2)
  • IBM-CM configuration mixing strength
    Controls the crossing between normal and intruder configurations and is adjusted to place the transition between 96Sr and 98Sr.
  • Intruder deformation parameter
    Determines the evolution from near-spherical to deformed intruder structure and is fitted to match quadrupole moments and isotope shifts.
axioms (1)
  • domain assumption The interacting boson model with configuration mixing provides a faithful description of both configuration crossing and shape evolution in the A approximately 100 region.
    All conclusions about intertwined QPTs are obtained inside this model; its validity is presupposed rather than derived.

pith-pipeline@v0.9.0 · 5730 in / 1342 out tokens · 27130 ms · 2026-05-25T03:04:40.570854+00:00 · methodology

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Reference graph

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