Letter | Published:

Extensive degeneracy, Coulomb phase and magnetic monopoles in artificial square ice

Nature volume 540, pages 410413 (15 December 2016) | Download Citation


Artificial spin-ice systems are lithographically patterned arrangements of interacting magnetic nanostructures that were introduced as way of investigating the effects of geometric frustration in a controlled manner1,2,3,4. This approach has enabled unconventional states of matter to be visualized directly in real space5,6,7,8,9,10,11,12,13,14,15,16,17,18, and has triggered research at the frontier between nanomagnetism, statistical thermodynamics and condensed matter physics. Despite efforts to create an artificial realization of the square-ice model—a two-dimensional geometrically frustrated spin-ice system defined on a square lattice—no simple geometry based on arrays of nanomagnets has successfully captured the macroscopically degenerate ground-state manifold of the model19. Instead, square lattices of nanomagnets are characterized by a magnetically ordered ground state that consists of local loop configurations with alternating chirality1,20,21,22,23,24,25,26. Here we show that all of the characteristics of the square-ice model are observed in an artificial square-ice system that consists of two sublattices of nanomagnets that are vertically separated by a small distance. The spin configurations we image after demagnetizing our arrays reveal unambiguous signatures of a Coulomb phase and algebraic spin-spin correlations, which are characterized by the presence of ‘pinch’ points in the associated magnetic structure factor. Local excitations—the classical analogues of magnetic monopoles27—are free to evolve in an extensively degenerate, divergence-free vacuum. We thus provide a protocol that could be used to investigate collective magnetic phenomena, including Coulomb phases28 and the physics of ice-like materials.

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This work was supported by the Agence Nationale de la Recherche through project number ANR12-BS04-009 ‘Frustrated’. We acknowledge support from the Nanofab team at the Institut NÉEL and thank S. Le-Denmat and O. Fruchart for technical help during atomic force microscope and magnetic force microscope measurements.

Author information


  1. CNRS, Institut NÉEL, F-38000 Grenoble, France

    • Yann Perrin
    • , Benjamin Canals
    •  & Nicolas Rougemaille
  2. Université Grenoble Alpes, Institut NÉEL, F-38000 Grenoble, France

    • Yann Perrin
    • , Benjamin Canals
    •  & Nicolas Rougemaille


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B.C. and N.R. conceived the project. Y.P. was in charge of the sample fabrication and characterization, the magnetic imaging measurements and the analysis of the data. All authors contributed to the preparation of the manuscript.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Nicolas Rougemaille.

Reviewer Information Nature thanks C. Nisoli, A. Ramirez and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Extended data

Supplementary information


  1. 1.

    Avalanche Process

    Video showing how magnetization reverses during a modelled field demagnetization protocol. The applied external magnetic field is represented by a rotating black arrow. Small black and white arrows represent point dipole spins on a square lattice, and red/blue/green squares code for type-II, type-I and type-III vertices, respectively.

  2. 2.

    Full Demagnetization shifted array

    Video showing how magnetization reverses during a modelled field demagnetization protocol. The applied external magnetic field is represented by a rotating black arrow. Small black and white arrows represent point dipole spins on a square lattice, and red/blue/green squares code for type-II, type-I and type-III vertices, respectively.

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