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Field-induced phase coexistence in an artificial spin ice


Artificial spin-ice systems are magnetic metamaterials consisting of nanomagnet arrays that can be designed to study exotic magnetic states not found in natural materials. Typically, these arrays are modelled as interacting binary macrospins that can only be in an up or down state and are described by the Ising model. These materials have demonstrated ordering transitions, but only via a spontaneous symmetry-breaking mechanism. We have designed and studied a quadrupole artificial spin-ice system consisting of interacting plaquettes of coupled single-domain nanomagnets that can be interpreted as a composite, ternary variable. After annealing this system in an external magnetic field, we observe both a ferroquadrupolar and an antiferroquadrupolar phase, with an apparent first-order phase boundary and a coexistence regime. The phase diagram of this material is reminiscent of a model used to describe phase coexistence in the superfluid transition of 4He with 3He impurities. These results illustrate how composite magnetic objects realize exotic statistical physics models beyond the Ising model.

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Fig. 1: Artificial quadrupole lattice.
Fig. 2: Two ordered Potts states.
Fig. 3: Phase coexistence in the quadrupole lattice.
Fig. 4: Phase diagram and neighbour correlations.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon request.


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The work of J.S., Y.L. and P.S. was funded by the US Department of Energy, Office of Basic Energy Sciences, Materials Sciences and Engineering Division under grant no. DE-SC0010778. The work of C.N. was carried out under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under contract no. DE-AC52-06NA25396. Work at the University of Minnesota was supported by the National Science Foundation under DMR-1507048.

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Authors and Affiliations



J.S. conceived the quadrupole geometry and experiment. J.S. and P.S. designed the field annealing capability. Y.L. and J.S. prepared the lithographic patterns and configured the annealing apparatus. A.A. and J.D.W. prepared the permalloy deposition for the samples. J.S. measured and processed the data. G.-W.C. developed connections to statistical physics models, and C.N. developed the theoretical Potts model formalism. Under supervision from G.-W.C. and C.N., J.S. performed Monte Carlo simulations. J.S., G.-W.C., C.N. and P.S wrote the manuscript and all authors read and edited it. P.S. supervised the experimental work and coordinated the entire project.

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Correspondence to Joseph Sklenar.

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The authors declare no competing interests.

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Supplementary information

Supplementary Information

Supplementary theoretical details, Supplementary References 1–4, Supplementary Figures 1–14

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Sklenar, J., Lao, Y., Albrecht, A. et al. Field-induced phase coexistence in an artificial spin ice. Nature Phys 15, 191–195 (2019).

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