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arxiv: 2508.19248 · v3 · submitted 2025-08-26 · ❄️ cond-mat.str-el · cond-mat.mtrl-sci

Disorder-induced proximate quantum spin ice phase in Pr2Sn2O7

Pith reviewed 2026-05-18 20:36 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mtrl-sci
keywords quantum spin icepyrochloredisorderspin freezingPr2Sn2O7non-Kramers ionsneutron scatteringground state
0
0 comments X p. Extension

The pith

In Pr2Sn2O7, disorder induces spin freezing at 0.15 K while quantum spin-ice correlations persist down to low temperatures.

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

The paper examines the non-Kramers pyrochlore Pr2Sn2O7 through bulk measurements and neutron scattering. Below about 1 K the material displays anisotropic spin-ice correlations and two separate dynamical timescales, signatures associated with quantum spin-ice physics. Upon cooling further the system undergoes a complete spin-freezing transition at 0.15 K, at which point the nuclear Schottky anomaly recovers fully, a gapped magnetic excitation appears, and weak (100) magnetic correlations develop. Placing the compound near the spin-frozen edge of a disorder-perturbed phase diagram through comparison with other Pr-based pyrochlores leads to the conclusion that structural disorder can drive the evolution from quantum spin-ice behavior into a frozen state while the system remains close to a quantum spin liquid.

Core claim

The central claim is that in the non-Kramers pyrochlore Pr2Sn2O7, structural disorder produces a proximate quantum spin-ice phase in which hallmark quantum spin-ice correlations and dynamics coexist with a disorder-induced spin-freezing transition at 0.15 K and the recovery of a full nuclear Schottky anomaly.

What carries the argument

The disorder-perturbed phase diagram for non-Kramers pyrochlores, which positions Pr2Sn2O7 near the spin-frozen boundary while still allowing quantum spin-ice correlations to survive.

If this is right

  • Quantum spin-ice behavior can evolve into frozen ground states through the action of structural disorder.
  • Proximity to a quantum spin liquid can coexist with disorder-induced spin freezing in non-Kramers pyrochlores.
  • The ground-state selection in these materials is sensitive to small amounts of disorder that shift the system across phase boundaries.
  • Two distinct dynamical timescales and anisotropic correlations remain observable even after the system freezes.

Where Pith is reading between the lines

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

  • Similar disorder effects could be tested by deliberately introducing controlled defects into other non-Kramers pyrochlores to map the same boundary.
  • The framework suggests that sample-to-sample variations in freezing temperatures reported across the Pr pyrochlore family may largely reflect differences in disorder levels rather than differences in the underlying exchange parameters.
  • If the phase diagram is general, then reducing disorder in Pr2Sn2O7 should push the system deeper into the quantum spin-ice regime without changing the high-temperature correlations.

Load-bearing premise

The spin freezing at 0.15 K is caused primarily by disorder rather than by an intrinsic feature of the ideal lattice.

What would settle it

A measurement on a substantially cleaner single crystal of Pr2Sn2O7 that shows the absence of spin freezing or a much lower freezing temperature would falsify the disorder-driven interpretation.

Figures

Figures reproduced from arXiv: 2508.19248 by Adam A. Aczel, Andrey A. Podlesnyak, Benjamin A. Frandsen, Brenden R. Ortiz, Joseph A. M. Paddison, Jue Liu, Matthias D. Frontzek, Miles Knudtson, Si Athena Chen, Stephen D. Wilson, Yi Luo.

Figure 1
Figure 1. Figure 1: FIG. 1. A qualitative phase diagram of the antiferromagnetic Ising [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Pair distribution function (PDF) analysis of Pr [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Magnetic diffuse scattering of Pr [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Bulk characterization of flux-grown single-crystal Pr [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Neutron spectroscopy of single-crystal Pr [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

Magnetic pyrochlores with non-Kramers rare-earth ions provide a platform for exploring emergent gauge physics and quantum spin-ice behavior, yet the influence of structural disorder on their ground states remains insufficiently understood. Here we combine bulk characterization and single-crystal neutron-scattering measurements to investigate the non-Kramers pyrochlore Pr2Sn2O7. At temperatures below ~1 K, the system exhibits key hallmarks of quantum spin-ice physics, including anisotropic spin-ice correlations and two distinct dynamical timescales. Upon further cooling, however, we observe a complete spin-freezing transition at T_f ~ 0.15 K, accompanied by recovery of the full nuclear Schottky anomaly, the emergence of a gapped magnetic excitation, and the development of incipient (100) magnetic correlations. Comparison with related Pr-based pyrochlores places Pr2Sn2O7 near the spin-frozen boundary of a disorder-perturbed phase diagram. These results establish a disorder-driven framework for how quantum spin-ice behavior evolves into frozen ground states, demonstrating that proximity to a quantum spin liquid can coexist with disorder-induced spin freezing in non-Kramers pyrochlores.

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

1 major / 2 minor

Summary. The manuscript reports bulk characterization and single-crystal neutron-scattering measurements on the non-Kramers pyrochlore Pr2Sn2O7. Below ~1 K the system displays anisotropic spin-ice correlations and two dynamical timescales, consistent with quantum spin-ice physics. Further cooling reveals a spin-freezing transition at Tf ~ 0.15 K, accompanied by recovery of the full nuclear Schottky anomaly, a gapped magnetic excitation, and incipient (100) magnetic correlations. The authors interpret these findings as a disorder-induced proximate quantum spin ice phase and place Pr2Sn2O7 near the spin-frozen boundary of a disorder-perturbed phase diagram for Pr-based pyrochlores.

Significance. If the central interpretation is confirmed, the work supplies a concrete experimental example of how structural disorder can drive the evolution from quantum spin-ice correlations to a frozen state while preserving proximity to a quantum spin liquid. The combination of bulk thermodynamics and neutron data on a single non-Kramers pyrochlore adds a useful data point to the growing literature on disorder effects in these materials.

major comments (1)
  1. [Abstract and final paragraph] Abstract and final paragraph: The assertion that the Tf ≈ 0.15 K spin freezing is disorder-induced and that the system lies near the spin-frozen boundary relies on qualitative comparison with other Pr pyrochlores and schematic phase-diagram placement. No quantitative disorder metric—such as refined cation/oxygen occupancies from diffraction or an effective random-field strength extracted from specific-heat modeling—is reported for the measured crystal. Without this link the causal attribution to disorder remains an inference rather than a direct consequence of the data.
minor comments (2)
  1. [Methods/Experimental details] The manuscript should provide a dedicated section or appendix detailing the neutron data reduction, background subtraction, and error propagation procedures that underlie the reported anisotropic correlations, two timescales, and gapped mode.
  2. [Figures and captions] Figure captions and text should explicitly state the temperature ranges, wave-vector ranges, and fitting models used to extract the two dynamical timescales and the gapped excitation energy.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the positive assessment of its significance. We address the major comment below and propose revisions to strengthen the presentation of our interpretation.

read point-by-point responses
  1. Referee: [Abstract and final paragraph] Abstract and final paragraph: The assertion that the Tf ≈ 0.15 K spin freezing is disorder-induced and that the system lies near the spin-frozen boundary relies on qualitative comparison with other Pr pyrochlores and schematic phase-diagram placement. No quantitative disorder metric—such as refined cation/oxygen occupancies from diffraction or an effective random-field strength extracted from specific-heat modeling—is reported for the measured crystal. Without this link the causal attribution to disorder remains an inference rather than a direct consequence of the data.

    Authors: We appreciate the referee's point regarding the need for a more direct link between the observed spin freezing and disorder. While our manuscript does not include a refined structural analysis for disorder parameters in the specific crystal studied (as the neutron scattering experiment was optimized for magnetic correlations rather than nuclear structure), the interpretation is grounded in the systematic comparison with other Pr pyrochlores where such metrics have been established in the literature. In the revised version, we will modify the abstract and the final paragraph to explicitly state that the placement near the spin-frozen boundary is inferred from this comparative analysis and the characteristic signatures of disorder effects (such as the low freezing temperature and gapped excitations). Additionally, we will consider adding a brief modeling of the specific-heat data to estimate an effective random-field strength if it can be done without introducing new assumptions. This will make the causal attribution clearer while remaining faithful to the data presented. revision: yes

Circularity Check

0 steps flagged

No circularity: purely experimental observations and external comparisons

full rationale

The manuscript is an experimental report based on bulk characterization and single-crystal neutron scattering. All central observations (anisotropic spin-ice correlations below ~1 K, complete spin freezing at T_f ~ 0.15 K, gapped excitations, incipient (100) order, and recovery of the nuclear Schottky anomaly) are presented as direct data products. The interpretive claim of a 'disorder-induced proximate quantum spin ice phase' is reached by qualitative placement relative to other Pr-based pyrochlores and a schematic disorder-perturbed phase diagram; no equations, fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citation chains appear in the provided text. The derivation chain is therefore self-contained against external benchmarks and contains no reductions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Experimental condensed-matter study that relies on established interpretation of neutron scattering and thermodynamic data; no new free parameters, axioms beyond standard physics, or invented entities are introduced.

axioms (1)
  • domain assumption Neutron scattering intensity maps magnetic spin correlations in pyrochlore lattices
    Invoked to interpret anisotropic spin-ice correlations and (100) features.

pith-pipeline@v0.9.0 · 5787 in / 1350 out tokens · 35250 ms · 2026-05-18T20:36:01.118661+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Influence of controlled disorder on the dipolar spin ice state of Ho-based pyrochlores

    cond-mat.str-el 2026-01 unverdicted novelty 4.0

    Controlled disorder in Ho2GaSbO7 and Ho2ScSbO7 preserves the dipolar spin-ice state but generates disorder-induced quantum fluctuations via splitting of the non-Kramers ground-state doublet.

Reference graph

Works this paper leans on

54 extracted references · 54 canonical work pages · cited by 1 Pith paper

  1. [1]

    A7(a), we identify a temperature-independent shoulder-like background feature centered aroundℏω∼ 0.1meV , which is absent on the energy-gain side (ℏω <0) at T= 10K

    Subtraction of background In Fig. A7(a), we identify a temperature-independent shoulder-like background feature centered aroundℏω∼ 0.1meV , which is absent on the energy-gain side (ℏω <0) at T= 10K. To remove this feature, we assume the background is temperature independent and there is negligible background at the energy-gain side; this assumption is sup...

  2. [2]

    starfish

    Details of constant-energy slices In Fig. A10(a–h), we present constant-energyQ-maps sim- ilar to those shown in Fig. 5(d–j), but obtained using an inci- dent energy ofE i = 0.99meV . This configuration offers ex- 15 TABLE II. Atomic positions and anisotropic displacement parameters of single-crystal Pr 2Sn2O7 at various temperatures and measurement metho...

  3. [3]

    Broholm and R

    C. Broholm and R. J. Cava and S. A. Kivelson and D. G. No- cera and M. R. Norman and T. Senthil, Quantum spin liquids, Science367, eaay0668 (2020)

  4. [4]

    N. Y . Yao, C. R. Laumann, A. V . Gorshkov, H. Weimer, L. Jiang, J. I. Cirac, P. Zoller, and M. D. Lukin, Topologically protected quantum state transfer in a chiral spin liquid, Nature Communications4, 1585 (2013)

  5. [5]

    Fennell, P

    T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prab- hakaran, A. T. Boothroyd, R. J. Aldus, D. F. McMorrow, and S. T. Bramwell, Magnetic Coulomb phase in the spin ice Ho2Ti2O7, Science326, 415 (2009)

  6. [6]

    B. C. den Hertog and M. J. P. Gingras, Dipolar interactions and origin of spin ice in Ising pyrochlore magnets, Physical Review Letters84, 3430 (2000)

  7. [7]

    Savary and L

    L. Savary and L. Balents, Coulombic quantum liquids in spin- 1/2 pyrochlores, Physical Review Letters108, 037202 (2012)

  8. [8]

    Chen, ”Magnetic monopole” condensation of the pyrochlore ice U(1) quantum spin liquid: Application to Pr 2Ir2O7 and Yb2Ti2O7, Physical Review B94, 205107 (2016)

    G. Chen, ”Magnetic monopole” condensation of the pyrochlore ice U(1) quantum spin liquid: Application to Pr 2Ir2O7 and Yb2Ti2O7, Physical Review B94, 205107 (2016)

  9. [9]

    Chen, Dirac’s ”magnetic monopoles” in pyrochlore ice U(1) spin liquids: Spectrum and classification, Physical Review B 96, 195127 (2017)

    G. Chen, Dirac’s ”magnetic monopoles” in pyrochlore ice U(1) spin liquids: Spectrum and classification, Physical Review B 96, 195127 (2017)

  10. [10]

    K. A. Ross, L. Savary, B. D. Gaulin, and L. Balents, Quantum excitations in quantum spin ice, Physical Review X1, 021002 (2011)

  11. [11]

    Scheie, J

    A. Scheie, J. Kindervater, S. Zhang, H. J. Changlani, G. Sala, G. Ehlers, A. Heinemann, G. S. Tucker, S. M. Koohpayeh, and C. Broholm, Multiphase magnetism in Yb2Ti2O7, Proceedings of the National Academy of Sciences117, 27245 (2020)

  12. [12]

    Gaudet, E

    J. Gaudet, E. M. Smith, J. Dudemaine, J. Beare, C. R. C. Buhariwalla, N. P. Butch, M. B. Stone, D. R. Yahne, K. A. Ross, C. A. Marjerrison, J. D. Garrett, G. M. Luke, A. D. Bianchi, and B. D. Gaulin, Quantum Spin Ice Dynamics in the Dipole–Octupole Pyrochlore Magnet Ce2Zr2O7, Phys. Rev. Lett.122, 187201 (2019)

  13. [13]

    Bhardwaj, S

    A. Bhardwaj, S. Zhang, H. Yan, R. Moessner, A. H. Nevidom- skyy, and H. J. Changlani, Sleuthing out exotic quantum spin liquidity in the pyrochlore magnet Ce 2Zr2O7, npj Quantum Materials7, 51 (2022)

  14. [14]

    B. Gao, T. Chen, D. W. Tam, C. Huang, K. Sasmal, D. T. Adroja, F. Ye, H. Cao, G. Sala, M. B. Stone, C. Baines, J. A. T. Verezhak, H. Hu, J. Chung, X. Xu, S. Cheong, M. Nallaiyan, S. Spagna, M. B. Maple, A. H. Nevidomskyy, E. Morosan, G. Chen, and P. Dai, Experimental signatures of a three-dimensional quantum spin liquid in effective spin-1/2 Ce2Zr2O7 pyro...

  15. [15]

    E. M. Smith, O. Benton, D. R. Yahne, B. Placke, R. Sch ¨afer, J. Gaudet, J. Dudemaine, A. Fitterman, J. Beare, A. R. Wildes, S. Bhattacharya, T. DeLazzer, C. R. C. Buhariwalla, N. P. Butch, R. Movshovich, J. D. Garrett, C. A. Marjerrison, J. P. Clancy, E. Kermarrec, G. M. Luke, A. D. Bianchi, K. A. Ross, and B. D. Gaulin, The case for a U(1) π Quantum Spi...

  16. [16]

    E. M. Smith, R. Sch ¨afer, J. Dudemaine, B. Placke, B. Yuan, Z. Morgan, F. Ye, R. Moessner, O. Benton, A. D. Bianchi, and B. D. Gaulin, Single-Crystal Diffuse Neutron Scattering Study of the Dipole–Octupole Quantum Spin-Ice Candidate Ce2Zr2O7: No Apparent Octupolar Correlations AboveT= 0.05K, Phys. Rev. X15, 021033 (2025)

  17. [17]

    B. Gao, F. Desrochers, D. W. Tam, D. M. Kirschbaum, P. Stef- fens, A. Hiess, D. H. Nguyen, Y . Su, S.-W. Cheong, S. Paschen, et al., Neutron scattering and thermodynamic evidence for emergent photons and fractionalization in a pyrochlore spin ice, Nature Physics , 1 (2025)

  18. [18]

    Sibille, N

    R. Sibille, N. Gauthier, E. Lhotel, V . Por ´ee, V . Pomjakushin, R. A. Ewings, T. G. Perring, J. Ollivier, A. R. Wildes, C. Rit- ter, T. C. Hansen, D. A. Keen, G. J. Nilsen, L. Keller, S. Petit, and T. Fennell, A quantum liquid of magnetic octupoles on the pyrochlore lattice, Nature Physics16, 546 (2020)

  19. [19]

    Por´ee, H

    V . Por´ee, H. Yan, F. Desrochers, S. Petit, E. Lhotel, M. Appel, J. Ollivier, Y . Kim, A. Nevidomskyy, and R. Sibille, Evidence for fractional matter coupled to an emergent gauge field in a quantum spin ice, Nature Physics21, 83 (2025)

  20. [20]

    D. R. Yahne, B. Placke, R. Sch ¨afer, O. Benton, R. Moess- ner, M. Powell, J. W. Kolis, C. M. Pasco, A. F. May, M. D. Frontzek, E. M. Smith, B. D. Gaulin, S. Calder, and K. A. Ross, Dipolar Spin Ice Regime Proximate to an All-in-All-Out N ´eel Ground State in the Dipolar-Octupolar Pyrochlore Ce 2Sn2O7, Phys. Rev. X14, 10.1103/PhysRevX.14.011005 (2024)

  21. [21]

    Por´ee, A

    V . Por´ee, A. Bhardwaj, E. Lhotel, S. Petit, N. Gauthier, H. Yan, V . Pomjakushin, J. Ollivier, J. A. Quilliam, A. H. Nevidom- skyy, H. J. Changlani, and R. Sibille, Dipolar-octupolar corre- lations and hierarchy of exchange interactions in Ce 2Hf2O7, arXiv preprint (2023), arXiv:2305.08261

  22. [22]

    E. M. Smith, A. Fitterman, R. Sch ¨afer, B. Placke, A. Woods, S. Lee, S. H.-Y . Huang, J. Beare, S. Sharma, D. Chatterjee, et al., Two-Peak Heat Capacity Accounts for R ln (2) Entropy and Ground State Access in the Dipole-Octupole Pyrochlore Ce2Hf2O7, Physical Review Letters135, 086702 (2025)

  23. [23]

    Kimura, S

    K. Kimura, S. Nakatsuji, J. Wen, C. Broholm, M. Stone, E. Nishibori, and H. Sawa, Quantum fluctuations in spin-ice- 17 like Pr2Zr2O7, Nature communications4, 1934 (2013)

  24. [24]

    J.-J. Wen, S. M. Koohpayeh, K. A. Ross, B. A. Trump, T. M. McQueen, K. Kimura, S. Nakatsuji, Y . Qiu, D. M. Pajerowski, J. R. D. Copley, and C. L. Broholm, Disordered route to the Coulomb quantum spin liquid: Random transverse fields on spin ice in Pr 2Zr2O7, Physical Review Letters118, 107206 (2017)

  25. [25]

    Petit, E

    S. Petit, E. Lhotel, B. Canals, M. C. Hatnean, G. Balakrishnan, P. Manuel, F. Damay, and R. Ballou, Antiferroquadrupolar cor- relations in the quantum spin ice candidate Pr2Zr2O7, Physical Review B94, 165153 (2016)

  26. [26]

    Martin, P

    N. Martin, P. Bonville, E. Lhotel, S. Guitteny, A. Wildes, C. Decorse, M. C. Hatnean, G. Balakrishnan, I. Mirebeau, and S. Petit, Disorder and quantum spin ice, Physical Review X7, 041028 (2017)

  27. [27]

    Sibille, E

    R. Sibille, E. Lhotel, M. C. Hatnean, G. Balakrishnan, B. F ˚ak, N. Gauthier, T. Fennell, and M. Kenzelmann, Candidate quan- tum spin ice in the pyrochlore Pr 2Hf2O7, Physical Review B 94, 024436 (2016)

  28. [28]

    V . K. Anand, L. Opherden, J. Xu, D. T. Adroja, A. T. M. N. Islam, T. Herrmannsd¨orfer, J. Hornung, R. S. mann, M. Uhlarz, H. C. Walker,et al., Physical properties of the candidate quan- tum spin-ice system Pr2Hf2O7, Physical Review B94, 144415 (2016)

  29. [29]

    Sibille, N

    R. Sibille, N. Gauthier, H. Yan, M. C. Hatnean, J. Ollivier, B. Winn, U. Filges, G. Balakrishnan, M. Kenzelmann, N. Shan- non,et al., Experimental signatures of emergent quantum elec- trodynamics in Pr2Hf2O7, Nature Physics14, 711 (2018)

  30. [30]

    A. J. Princep, D. Prabhakaran, A. T. Boothroyd, and D. T. Adroja, Crystal-field states of Pr 3+ in the candidate quantum spin ice Pr2Sn2O7, Physical Review B88, 104421 (2013)

  31. [31]

    Savary and L

    L. Savary and L. Balents, Disorder-Induced Quantum Spin Liquid in Spin Ice Pyrochlores, Physical Review Letters118, 087203 (2017)

  32. [32]

    Benton, Instabilities of a U(1) Quantum Spin Liquid in Disordered Non-Kramers Pyrochlores, Physical Review Letters 121, 037203 (2018)

    O. Benton, Instabilities of a U(1) Quantum Spin Liquid in Disordered Non-Kramers Pyrochlores, Physical Review Letters 121, 037203 (2018)

  33. [33]

    R ¨ochner, L

    J. R ¨ochner, L. Balents, and K. P. Schmidt, Spin liquid and quantum phase transition without symmetry breaking in a frus- trated three-dimensional Ising model, Physical Review B94, 201111(R) (2016)

  34. [34]

    Pardini, A

    T. Pardini, A. Menon, S. P. Hau-Riege, and R. R. P. Singh, Local entanglement and confinement transitions in the random transverse-field Ising model on the pyrochlore lattice, Physical Review B100, 144437 (2019)

  35. [35]

    R. B. Griffiths, Nonanalytic Behavior Above the Critical Point in a Random Ising Ferromagnet, Physical Review Letters23, 17 (1969)

  36. [36]

    H. D. Zhou, C. R. Wiebe, J. G. Cheng, C. H. Li, L. Balicas, Y . J. Jo, L. N. Pan, J. S. Gardner, Y . Qiu, and J. R. D. Copley, Dynamic Spin Ice: Pr 2Sn2O7, Physical Review Letters101, 227204 (2008)

  37. [37]

    B. R. Ortiz, P. M. Sarte, G. Pokharel, M. J. Knudtson, S. J. G. Alvarado, A. F. May, S. Calder, L. Mangin-Thro, A. R. Wildes, H. Zhou, G. Sala, C. R. Wiebe, S. D. Wilson, J. A. M. Paddi- son, and A. A. Aczel, Revisiting spin ice physics in the ferro- magnetic Ising pyrochlore Pr 2Sn2O7, Physical Review B109, 134420 (2024)

  38. [38]

    Tomasello, C

    B. Tomasello, C. Castelnovo, R. Moessner, and J. Quintanilla, Correlated Quantum Tunneling of Monopoles in Spin Ice, Phys- ical Review Letters123, 067204 (2019)

  39. [39]

    Arnold, J

    O. Arnold, J. C. Bilheux, J. M. Borreguero, A. Buts, S. I. Camp- bell, L. Chapon, M. Doucet, N. Draper, R. F. Leal, M. A. Gigg, V . E. Lynch, A. Markvardsen, D. J. Mikkelson, R. L. Mikkel- son, R. Miller, K. Palmen, P. Parker, G. Passos, T. G. Perring, P. F. Peterson, S. Ren, M. A. Reuter, A. T. Savici, J. W. Taylor, R. J. Taylor, R. Tolchenov, W. Zhou, a...

  40. [40]

    A. A. Coelho, TOPAS and TOPAS-Academic: an optimization program integrating computer algebra and crystallographic ob- jects written in C++, Journal of Applied Crystallography51, 210 (2018)

  41. [41]

    C. L. Farrow, P. Juh ´as, J. W. Liu, D. Bryndin, E. S. Bo ˇzin, J. Bloch, T. Proffen, and S. J. L. Billinge, PDFfit2 and PDFgui: computer programs for studying nanostructure in crystals, Jour- nal of Physics: Condensed Matter19, 335219 (2007)

  42. [42]

    Benton, O

    O. Benton, O. Sikora, and N. Shannon, Seeing the light: Exper- imental signatures of emergent electromagnetism in a quantum spin ice, Physical Review B86, 075154 (2012)

  43. [43]

    J. A. M. Paddison, Spinteract: A Program to Refine Magnetic Interactions to Diffuse Scattering Data, Journal of Physics: Condensed Matter35, 495802 (2023)

  44. [44]

    C. V . Topping and S. J. Blundell, A.C. susceptibility as a probe of low-frequency magnetic dynamics, Journal of Physics: Con- densed Matter31, 015801 (2018)

  45. [45]

    E. S. R. Gopal,Specific Heats at Low Temperatures, Interna- tional Cryogenics Monograph Series (Plenum Press, New York, 1966)

  46. [46]

    N. Tang, Y . Gritsenko, K. Kimura, S. Bhattacharjee, A. Sakai, M. Fu, H. Takeda, H. Man, K. Sugawara, Y . Matsumoto, Y . Shimura, J. Wen, C. Broholm, H. Sawa, M. Takigawa, T. Sakakibara, S. Zherlitsyn, J. Wosnitza, R. Moessner, and S. Nakatsuji, Spin–orbital liquid state and liquid–gas metam- agnetic transition on a pyrochlore lattice, Nature Physics19, 92 (2022)

  47. [47]

    B ¨oni and G

    P. B ¨oni and G. Shirane, Paramagnetic neutron scattering from the Heisenberg ferromagnet EuO, Phys. Rev. B33, 3012 (1986)

  48. [48]

    S. T. Bramwell, M. J. Harris, B. C. den Hertog, M. J. P. Gingras, J. S. Gardner, D. F. McMorrow, A. R. Wildes, A. L. Cornelius, J. D. M. Champion, R. G. Melko, and T. Fennell, Spin correla- tions in Ho2Ti2O7: a dipolar spin ice system, Phys. Rev. Lett. 87, 047205 (2001)

  49. [49]

    Kanada, Y

    M. Kanada, Y . Yasui, Y . Kondo, S. Iikubo, M. Ito, H. Harashina, M. Sato, H. Okumura, K. Kakurai, and H. Kadowaki, Neutron- scattering study of the spin correlation in the spin-ice system Ho2Ti2O7, J. Phys. Soc. Jpn.71, 313 (2002)

  50. [50]

    Yavors’kii, T

    T. Yavors’kii, T. Fennell, M. J. P. Gingras, and S. T. Bramwell, Dy2Ti2O7 spin ice: A test case for emergent clusters in a frus- trated magnet, Physical Review Letters101, 037204 (2008)

  51. [51]

    A. M. Samarakoon, A. Sokolowski, B. Klemke, R. Feyerherm, M. Meissner, R. A. Borzi, F. Ye, Q. Zhang, Z. Dun, H. Zhou, T. Egami, J. N. Hall´en, L. Jaubert, C. Castelnovo, R. Moessner, S. A. Grigera, and D. A. Tennant, Structural magnetic glassi- ness in the spin ice Dy 2Ti2O7, Physical Review Research4, 033159 (2022)

  52. [52]

    Petit, E

    S. Petit, E. Lhotel, B. Canals, M. Ciomaga Hatnean, J. Ollivier, H. Mutka, E. Ressouche, A. R. Wildes, M. R. Lees, and G. Bal- akrishnan, Observation of magnetic fragmentation in spin ice, Nature Physics12, 746 (2016)

  53. [53]

    Fukazawa, R

    H. Fukazawa, R. G. Melko, R. Higashinaka, Y . Maeno, and M. J. P. Gingras, Magnetic anisotropy of the spin-ice compound Dy2Ti2O7, Physical Review B65, 054410 (2002)

  54. [54]

    Bonville, S

    P. Bonville, S. Guitteny, A. Gukasov, I. Mirebeau, B. Ker- marrec, J. Robert, P. Steffens, S. Petit, G. Dhalenne, and A. Revcolevschi, Magnetic properties and crystal field in 18 Pr2Zr2O7, Phys. Rev. B94, 134428 (2016). 19 FIG. A8. (a) Determination of the shoulder-like background feature atℏω∼0.1meV observed in Fig. A7(a), measured withE i = 0.99meV . Th...