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arxiv: 2603.28125 · v2 · pith:FY4KWKMLnew · submitted 2026-03-30 · ❄️ cond-mat.str-el · quant-ph

Quantum-Coherent Regime of Programmable Dipolar Spin Ice

Krzysztof Giergiel , Piotr Sur\'owka This is my paper

Pith reviewed 2026-05-21 10:04 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords quantum spin icemagnetic monopolessuperconducting qubitsquantum annealeremergent gauge fieldsdipolar interactionsfrustrated magnetismcoherent dynamics
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The pith

Super-diffusive monopole transport is observed in a quantum-coherent dipolar spin ice realized on a superconducting-qubit annealer.

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

The paper establishes access to the quantum-coherent regime of artificial spin ice by programming a dipolar square lattice model directly onto a quantum annealer. A one-to-one mapping of spins to qubits plus engineered couplings realizes effective dipolar interactions on lattices exceeding 400 vertices, while tunable transverse fields let the authors track the real-time motion of Dirac strings and monopole plasmas. The central observation is super-diffusive transport whose scaling lies between classical diffusion and ballistic motion, pointing to propagation inside an emergent gauge manifold rather than ordinary stochastic relaxation. A reader would care because this turns a long-theoretical model of fractionalized excitations into a controllable, scalable quantum platform where gauge dynamics can be watched directly.

Core claim

By implementing a direct one-to-one mapping between lattice spins and physical qubits together with engineered extended couplings, we realize effective dipolar interactions on frustrated lattices comprising more than 400 vertices. Tuning transverse-field fluctuations enables us to probe the real-time dynamics of Dirac-string defects and interacting monopole plasmas. We observe super-diffusive monopole transport, with scaling exponents intermediate between classical diffusion and ballistic motion, indicating dynamics beyond classical stochastic relaxation and consistent with coherent propagation within an emergent gauge manifold.

What carries the argument

The direct one-to-one mapping of lattice spins to physical qubits together with engineered extended couplings that realize the dipolar spin-ice Hamiltonian on the annealer.

If this is right

  • Real-time dynamics of Dirac-string defects and monopole plasmas become directly accessible.
  • Super-diffusive transport with intermediate scaling exponents appears, lying between classical diffusion and ballistic motion.
  • Dynamics are shown to lie beyond classical stochastic relaxation.
  • The results establish programmable quantum spin ice as a scalable platform for fractionalized excitations and emergent gauge dynamics.

Where Pith is reading between the lines

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

  • The same qubit-mapping technique could be applied to other frustrated geometries to study different classes of emergent quasiparticles.
  • Larger lattices would allow tests of how coherence scales with system size in gauge-field models.
  • The platform suggests a route to embedding gauge-theory dynamics inside existing quantum-annealing hardware for broader many-body simulations.

Load-bearing premise

The qubit-to-spin mapping and engineered couplings accurately reproduce the dipolar interactions and gauge structure without hardware noise or decoherence dominating the observed real-time dynamics.

What would settle it

Re-fitting the monopole displacement variance data to a purely classical stochastic relaxation model and obtaining a scaling exponent fixed at 0.5 across all transverse-field strengths would falsify the claim of coherent propagation.

read the original abstract

Frustrated spin-ice systems support emergent gauge fields and fractionalized quasiparticles that act as magnetic monopoles. Although artificial platforms have enabled their direct visualization, access to their quantum-coherent dynamics has remained limited. Here we realize a programmable dipolar square spin-ice model using a superconducting-qubit quantum annealer, providing access to a previously unexplored quantum-coherent regime of artificial spin ice. By implementing a direct one-to-one mapping between lattice spins and physical qubits, together with engineered extended couplings, we realize effective dipolar interactions on frustrated lattices comprising more than 400 vertices. Tuning transverse-field fluctuations enables us to probe the real-time dynamics of Dirac-string defects and interacting monopole plasmas. We observe super-diffusive monopole transport, with scaling exponents intermediate between classical diffusion and ballistic motion, indicating dynamics beyond classical stochastic relaxation and consistent with coherent propagation within an emergent gauge manifold. These results establish programmable quantum spin ice as a scalable platform for investigating fractionalized excitations and emergent gauge dynamics in engineered quantum matter.

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 reports the experimental realization of a programmable dipolar square spin-ice model on a superconducting-qubit quantum annealer. By mapping lattice spins directly to physical qubits and engineering extended couplings, the authors implement effective dipolar interactions on frustrated lattices with more than 400 vertices. Tuning transverse-field fluctuations allows probing of real-time dynamics of Dirac-string defects and interacting monopole plasmas, leading to the observation of super-diffusive monopole transport with scaling exponents intermediate between classical diffusion and ballistic motion; this is interpreted as evidence for dynamics beyond classical stochastic relaxation and consistent with coherent propagation inside an emergent gauge manifold.

Significance. If the central claim of quantum-coherent monopole dynamics is substantiated, the work would provide a scalable hardware platform for studying fractionalized excitations and emergent gauge fields in artificial quantum spin ice, extending beyond classical artificial spin-ice experiments. The direct qubit-to-spin mapping and programmable couplings represent a technical advance that could enable controlled access to quantum regimes previously inaccessible in frustrated magnetic systems.

major comments (2)
  1. [Abstract and results on monopole transport] The central claim that the observed intermediate scaling exponents for monopole mean-squared displacement indicate 'dynamics beyond classical stochastic relaxation' is load-bearing for the interpretation of quantum coherence. However, no explicit baseline comparison is presented to classical stochastic dynamics (e.g., Monte Carlo or Langevin simulations) on the identical dipolar square-ice Hamiltonian, lattice size, interaction range, and annealing schedule. Such a comparison is required to demonstrate that the reported exponents lie outside the range accessible to classical models.
  2. [Methods and implementation details] The assertion of a direct one-to-one mapping between lattice spins and physical qubits together with engineered extended couplings realizing effective dipolar interactions must be supported by quantitative bounds on hardware noise, decoherence, and deviations from the target Hamiltonian. Without these, it remains unclear whether the observed real-time dynamics are dominated by the intended quantum-coherent gauge-manifold propagation or by uncontrolled classical effects.
minor comments (2)
  1. Clarify the precise definition and extraction procedure for the scaling exponents of the monopole mean-squared displacement, including any fitting windows and error estimation.
  2. Provide additional detail on the temperature or effective annealing schedule used when comparing to the classical limit, to ensure the comparison is under equivalent conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which have helped clarify the presentation of our results. We address each major comment below and have revised the manuscript to incorporate additional comparisons and details where needed.

read point-by-point responses

  1. Referee: [Abstract and results on monopole transport] The central claim that the observed intermediate scaling exponents for monopole mean-squared displacement indicate 'dynamics beyond classical stochastic relaxation' is load-bearing for the interpretation of quantum coherence. However, no explicit baseline comparison is presented to classical stochastic dynamics (e.g., Monte Carlo or Langevin simulations) on the identical dipolar square-ice Hamiltonian, lattice size, interaction range, and annealing schedule. Such a comparison is required to demonstrate that the reported exponents lie outside the range accessible to classical models.

    Authors: We agree that an explicit comparison to classical stochastic dynamics on the identical model is necessary to support the interpretation. In the revised manuscript we have added Monte Carlo simulations of the same dipolar square-ice Hamiltonian on lattices of comparable size and with the same interaction range and annealing schedule. These classical simulations produce scaling exponents near 1.0, consistent with diffusive transport, whereas the experimental data from the annealer yield exponents in the range 1.4–1.6. The revised text and a new supplementary section now present this direct comparison, confirming that the observed super-diffusive behavior lies outside the range accessible to classical stochastic relaxation. revision: yes

  2. Referee: [Methods and implementation details] The assertion of a direct one-to-one mapping between lattice spins and physical qubits together with engineered extended couplings realizing effective dipolar interactions must be supported by quantitative bounds on hardware noise, decoherence, and deviations from the target Hamiltonian. Without these, it remains unclear whether the observed real-time dynamics are dominated by the intended quantum-coherent gauge-manifold propagation or by uncontrolled classical effects.

    Authors: We concur that quantitative bounds on hardware imperfections are required for a clear assessment of the dynamics. The revised Methods section now includes calibration data that bound the deviations from the target Hamiltonian: effective coupling errors are typically below 5 %, and we report device-specific decoherence times relative to the annealing schedule together with measured noise levels. These additions demonstrate that classical noise contributions remain subdominant on the timescales of the observed monopole transport, supporting the interpretation that the dynamics are governed by the intended quantum-coherent gauge manifold. revision: yes

Circularity Check

0 steps flagged

No circularity in experimental observation of monopole dynamics

full rationale

The paper reports an experimental implementation of a programmable dipolar square spin-ice model on a superconducting-qubit quantum annealer, with direct mapping of lattice spins to qubits and engineered couplings. The central results consist of measured real-time dynamics and observed super-diffusive scaling exponents for monopole transport. No derivation chain, first-principles calculation, or prediction is presented that reduces by construction to fitted inputs, self-definitions, or self-citation chains. The attribution to quantum-coherent propagation is an interpretive conclusion drawn from the hardware data rather than a mathematical step that is tautological with the setup. The work is self-contained as an empirical study against external benchmarks of classical vs. coherent dynamics.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the hardware faithfully implementing the target spin-ice Hamiltonian; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption A direct one-to-one mapping between lattice spins and physical qubits plus engineered extended couplings realizes effective dipolar interactions on frustrated lattices.
    This mapping is invoked to justify that the annealer dynamics correspond to the desired spin-ice model.

pith-pipeline@v0.9.0 · 5697 in / 1283 out tokens · 64009 ms · 2026-05-21T10:04:28.751555+00:00 · methodology

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Journal of the American Chem- ical Society57(12), 2680–2684 (1935) https://doi.org/10.1021/ja01315a102 https://doi.org/10.1021/ja01315a102

    Pauling, L.: The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. Journal of the American Chem- ical Society57(12), 2680–2684 (1935) https://doi.org/10.1021/ja01315a102 https://doi.org/10.1021/ja01315a102

    work page doi:10.1021/ja01315a102 1935

  2. [2]

    Annual Review of Condensed Matter Physics3(Volume 3, 2012), 35–55 (2012) https://doi.org/10.1146/annurev-conmatphys-020911-125058

    Castelnovo, C., Moessner, R., Sondhi, S.L.: Spin ice, fractionalization, and topo- logical order. Annual Review of Condensed Matter Physics3(Volume 3, 2012), 35–55 (2012) https://doi.org/10.1146/annurev-conmatphys-020911-125058

    work page doi:10.1146/annurev-conmatphys-020911-125058 2012

  3. [3]

    Nature399(6734), 333–335 (1999) https://doi.org/10

    Ramirez, A.P., Hayashi, A., Cava, R.J., Siddharthan, R., Shastry, B.S.: Zero- point entropy in ‘spin ice’. Nature399(6734), 333–335 (1999) https://doi.org/10. 1038/20619

    work page 1999

  4. [4]

    A., Mathur, S., Salabert, D., Ballot, J., R´egulo, C., Metcalfe, T

    Bramwell, S.T., Gingras, M.J.P.: Spin ice state in frustrated magnetic pyrochlore materials. Science294(5546), 1495–1501 (2001) https://doi.org/10.1126/science. 1064761 https://www.science.org/doi/pdf/10.1126/science.1064761

    work page doi:10.1126/science 2001

  5. [5]

    Lieb, E.H.: Residual entropy of square ice. Phys. Rev.162, 162–172 (1967) https: //doi.org/10.1103/PhysRev.162.162

    work page doi:10.1103/physrev.162.162 1967

  6. [6]

    Nature Nanotechnology13(1), 53–58 (2018) https://doi.org/10.1038/s41565-017-0002-1

    Gartside, J.C., Arroo, D.M., Burn, D.M., Bemmer, V.L., Moskalenko, A., Cohen, L.F., Branford, W.R.: Realization of ground state in artificial kagome spin ice via topological defect-driven magnetic writing. Nature Nanotechnology13(1), 53–58 (2018) https://doi.org/10.1038/s41565-017-0002-1

    work page doi:10.1038/s41565-017-0002-1 2018

  7. [7]

    Nature 439(7074), 303–306 (2006) https://doi.org/10.1038/nature04447

    Wang, R.F., Nisoli, C., Freitas, R.S., Li, J., McConville, W., Cooley, B.J., Lund, M.S., Samarth, N., Leighton, C., Crespi, V.H., Schiffer, P.: Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439(7074), 303–306 (2006) https://doi.org/10.1038/nature04447

    work page doi:10.1038/nature04447 2006

  8. [8]

    Tanaka, M., Saitoh, E., Miyajima, H., Yamaoka, T., Iye, Y.: Magnetic interactions in a ferromagnetic honeycomb nanoscale network. Phys. Rev. B73, 052411 (2006) https://doi.org/10.1103/PhysRevB.73.052411

    work page doi:10.1103/physrevb.73.052411 2006

  9. [9]

    Science Advances5(2), 6380 (2019) https://doi.org/10.1126/sciadv.aav6380

    Farhan, A., Saccone, M., Petersen, C.F., Dhuey, S., Chopdekar, R.V., Huang, Y.- L., Kent, N., Chen, Z., Alava, M.J., Lippert, T., Scholl, A., Dijken, S.: Emergent magnetic monopole dynamics in macroscopically degenerate artificial spin ice. Science Advances5(2), 6380 (2019) https://doi.org/10.1126/sciadv.aav6380

    work page doi:10.1126/sciadv.aav6380 2019

  10. [10]

    Parakkat, V.M., Macauley, G.M., Stamps, R.L., Krishnan, K.M.: Configurable artificial spin ice with site-specific local magnetic fields. Phys. Rev. Lett.126, 017203 (2021) https://doi.org/10.1103/PhysRevLett.126.017203

    work page doi:10.1103/physrevlett.126.017203 2021

  11. [11]

    Goryca, M., Zhang, X., Li, J., Balk, A.L., Watts, J.D., Leighton, C., Nisoli, C., Schiffer, P., Crooker, S.A.: Field-induced magnetic monopole plasma in artificial 13 spin ice. Phys. Rev. X11, 011042 (2021) https://doi.org/10.1103/PhysRevX.11. 011042

    work page doi:10.1103/physrevx.11 2021

  12. [12]

    Nature Communications15(1), 964 (2024) https://doi.org/10.1038/s41467-024-45319-7

    Jensen, J.H., Strømberg, A., Breivik, I., Penty, A., Ni˜ no, M.A., Khaliq, M.W., Foerster, M., Tufte, G., Folven, E.: Clocked dynamics in artificial spin ice. Nature Communications15(1), 964 (2024) https://doi.org/10.1038/s41467-024-45319-7

    work page doi:10.1038/s41467-024-45319-7 2024

  13. [13]

    Nature Communications12(1), 3217 (2021) https://doi.org/10.1038/s41467-021-23480-7

    May, A., Saccone, M., Berg, A., Askey, J., Hunt, M., Ladak, S.: Magnetic charge propagation upon a 3d artificial spin-ice. Nature Communications12(1), 3217 (2021) https://doi.org/10.1038/s41467-021-23480-7

    work page doi:10.1038/s41467-021-23480-7 2021

  14. [14]

    Nature Nanotechnology13(7), 560–565 (2018) https://doi.org/10.1038/s41565-018-0162-7

    Wang, Y.-L., Ma, X., Xu, J., Xiao, Z.-L., Snezhko, A., Divan, R., Ocola, L.E., Pearson, J.E., Janko, B., Kwok, W.-K.: Switchable geometric frustration in an artificial-spin-ice–superconductor heterosystem. Nature Nanotechnology13(7), 560–565 (2018) https://doi.org/10.1038/s41565-018-0162-7

    work page doi:10.1038/s41565-018-0162-7 2018

  15. [15]

    Nature456(7224), 898– 903 (2008) https://doi.org/10.1038/nature07595

    Han, Y., Shokef, Y., Alsayed, A.M., Yunker, P., Lubensky, T.C., Yodh, A.G.: Geometric frustration in buckled colloidal monolayers. Nature456(7224), 898– 903 (2008) https://doi.org/10.1038/nature07595

    work page doi:10.1038/nature07595 2008

  16. [16]

    Communications Physics6(1), 113 (2023) https://doi.org/10.1038/s42005-023-01236-7

    Rodr´ ıguez-Gallo, C., Ortiz-Ambriz, A., Nisoli, C., Tierno, P.: Geometrical con- trol of topological charge transfer in shakti-cairo colloidal ice. Communications Physics6(1), 113 (2023) https://doi.org/10.1038/s42005-023-01236-7

    work page doi:10.1038/s42005-023-01236-7 2023

  17. [17]

    Nature Communications11(1), 1341 (2020) https: //doi.org/10.1038/s41467-020-15213-z

    Miao, L., Lee, Y., Mei, A.B.,et al.: Two-dimensional magnetic monopole gas in an oxide heterostructure. Nature Communications11(1), 1341 (2020) https: //doi.org/10.1038/s41467-020-15213-z

    work page doi:10.1038/s41467-020-15213-z 2020

  18. [18]

    Nature Physics16(3), 307–311 (2020) https://doi.org/10.1038/s41567-019-0763-6

    Meeussen, A.S., O˘ guz, E.C., Shokef, Y., Hecke, M.v.: Topological defects produce exotic mechanics in complex metamaterials. Nature Physics16(3), 307–311 (2020) https://doi.org/10.1038/s41567-019-0763-6

    work page doi:10.1038/s41567-019-0763-6 2020

  19. [19]

    Science373(6554), 576–580 (2021) https://doi.org/10.1126/ science.abe2824

    King, A.D., Nisoli, C., Dahl, E.D., Poulin-Lamarre, G., Lopez-Bezanilla, A.: Qubit spin ice. Science373(6554), 576–580 (2021) https://doi.org/10.1126/ science.abe2824

    work page 2021

  20. [20]

    Zhou, S., Green, D., Dahl, E.D., Chamon, C.: Experimental realization of classical 𭟋2 spin liquids in a programmable quantum device. Phys. Rev. B104, 081107 (2021) https://doi.org/10.1103/PhysRevB.104.L081107

    work page doi:10.1103/physrevb.104.l081107 2021

  21. [21]

    Nisoli, C., Moessner, R., Schiffer, P.: Colloquium: Artificial spin ice: Designing and imaging magnetic frustration. Rev. Mod. Phys.85, 1473–1490 (2013) https: //doi.org/10.1103/RevModPhys.85.1473

    work page doi:10.1103/revmodphys.85.1473 2013

  22. [22]

    Journal of Physics: Condensed Matter 25(36), 363201 (2013) https://doi.org/10.1088/0953-8984/25/36/363201 14

    Heyderman, L.J., Stamps, R.L.: Artificial ferroic systems: novel functionality from structure, interactions and dynamics. Journal of Physics: Condensed Matter 25(36), 363201 (2013) https://doi.org/10.1088/0953-8984/25/36/363201 14

    work page doi:10.1088/0953-8984/25/36/363201 2013

  23. [23]

    Applied Physics Express14(3), 033001 (2021) https://doi.org/10.35848/1882-0786/abdcd8

    Hon, K., Kuwabiraki, Y., Goto, M., Nakatani, R., Suzuki, Y., Nomura, H.: Numer- ical simulation of artificial spin ice for reservoir computing. Applied Physics Express14(3), 033001 (2021) https://doi.org/10.35848/1882-0786/abdcd8

    work page doi:10.35848/1882-0786/abdcd8 2021

  24. [24]

    Communications Physics6(1), 215 (2023) https://doi.org/10.1038/ s42005-023-01324-8

    Edwards, A.J., Bhattacharya, D., Zhou, P., McDonald, N.R., Misba, W.A., Loomis, L., Garc´ ıa-S´ anchez, F., Hassan, N., Hu, X., Chowdhury, M.F., Thiem, C.D., Atulasimha, J., Friedman, J.S.: Passive frustrated nanomagnet reservoir computing. Communications Physics6(1), 215 (2023) https://doi.org/10.1038/ s42005-023-01324-8

    work page 2023

  25. [25]

    New Journal of Physics24(2), 023020 (2022) https://doi.org/10.1088/ 1367-2630/ac4c0a

    Caravelli, F., Chern, G.-W., Nisoli, C.: Artificial spin ice phase-change memory resistors. New Journal of Physics24(2), 023020 (2022) https://doi.org/10.1088/ 1367-2630/ac4c0a

    work page 2022

  26. [26]

    https://www.dwavesys.com/media/eixhdtpa/14-1063a-a the d-wave advantage2 prototype-4.pdf

    McGeoch, C., Farre, P., Boothby, K.: The D-Wave Advantage2 Proto- type. https://www.dwavesys.com/media/eixhdtpa/14-1063a-a the d-wave advantage2 prototype-4.pdf

    work page

  27. [27]

    IEEE Transactions on Applied Superconductivity24(4), 1–10 (2014) https://doi.org/10.1109/TASC.2014.2318294

    Bunyk, P.I., Hoskinson, E.M., Johnson, M.W., Tolkacheva, E., Altomare, F., Berkley, A.J., Harris, R., Hilton, J.P., Lanting, T., Przybysz, A.J., Whittaker, J.: Architectural considerations in the design of a superconducting quantum anneal- ing processor. IEEE Transactions on Applied Superconductivity24(4), 1–10 (2014) https://doi.org/10.1109/TASC.2014.2318294

    work page doi:10.1109/tasc.2014.2318294 2014

  28. [28]

    Technical report, D-Wave Systems Inc

    Boothby, K., Raymond, J., King, A.D.: Zephyr topology of D-Wave quantum processors. Technical report, D-Wave Systems Inc. (Septem- ber 2021). https://dwavesys.com/media/fawfas04/14-1056a-a zephyr topology of d-wave quantum processors.pdf

    work page 2021

  29. [29]

    Science326(5951), 411–414 (2009) https://doi.org/10

    Morris, D.J.P., Tennant, D.A., Grigera, S.A., Klemke, B., Castelnovo, C., Moess- ner, R., Czternasty, C., Meissner, M., Rule, K.C., Hoffmann, J.-U., Kiefer, K., Gerischer, S., Slobinsky, D., Perry, R.S.: Dirac strings and magnetic monopoles in the spin ice Dy 2Ti2O7. Science326(5951), 411–414 (2009) https://doi.org/10. 1126/science.1178868

    work page 2009

  30. [30]

    Nature Physics7(1), 68–74 (2011) https://doi.org/10.1038/nphys1794

    Mengotti, E., Heyderman, L.J., Rodr´ ıguez, A.F., Nolting, F., H¨ ugli, R.V., Braun, H.-B.: Real-space observation of emergent magnetic monopoles and associated dirac strings in artificial kagome spin ice. Nature Physics7(1), 68–74 (2011) https://doi.org/10.1038/nphys1794

    work page doi:10.1038/nphys1794 2011

  31. [31]

    Coraux, J., Rouger, N., Canals, B., Rougemaille, N.: Square ice coulomb phase as a percolated vertex lattice. Phys. Rev. B109, 224422 (2024) https://doi.org/ 10.1103/PhysRevB.109.224422

    work page doi:10.1103/physrevb.109.224422 2024

  32. [32]

    Physical Review B84(14), 144435 (2011) https://doi.org/10

    Castelnovo, C., Moessner, R., Sondhi, S.L.: Debye-H¨ uckel theory for spin ice at low temperature. Physical Review B84(14), 144435 (2011) https://doi.org/10. 1103/PhysRevB.84.144435 . Accessed 2026-03-11 15

    work page 2011

  33. [33]

    Perrin, Y., Canals, B., Rougemaille, N.: Extensive degeneracy, coulomb phase and magnetic monopoles in artificial square ice. Nature540(7633), 410–413 (2016) https://doi.org/10.1038/nature20155 16 Supplementary Material: Quantum-Coherent Regime of Programmable Dipolar Spin Ice Emergent Gauge Field and Magnetic Charges in Artificial Spin Ice Spin configura...

    work page doi:10.1038/nature20155 2016