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Direct observation of a dynamical glass transition in a nanomagnetic artificial Hopfield network


Spin glasses, generally defined as disordered systems with randomized competing interactions1,2, are a widely investigated complex system. Theoretical models describing spin glasses are broadly used in other complex systems, such as those describing brain function3,4, error-correcting codes5 or stock-market dynamics6. This wide interest in spin glasses provides strong motivation to generate an artificial spin glass within the framework of artificial spin ice systems7,8,9. Here we present the experimental realization of an artificial spin glass consisting of dipolar coupled single-domain Ising-type nanomagnets arranged onto an interaction network that replicates the aspects of a Hopfield neural network10. Using cryogenic X-ray photoemission electron microscopy (XPEEM), we performed temperature-dependent imaging of thermally driven moment fluctuations within these networks and observed characteristic features of a two-dimensional Ising spin glass. Specifically, the temperature dependence of the spin glass correlation function follows a power-law trend predicted from theoretical models on two-dimensional spin glasses11. Furthermore, we observe clear signatures of the hard-to-observe rugged spin glass free energy1 in the form of sub-aging, out-of-equilibrium autocorrelations12 and a transition from stable to unstable dynamics1,13.

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Fig. 1: Nanomagnetic artificial Hopfield networks.
Fig. 2: Imaging low-energy moment configurations in a nanomagnetic artificial Hopfield network.
Fig. 3: Temperature-dependent inverse susceptibility and correlation length derived from real-space observations.
Fig. 4: Dynamical behaviour of a nanomagnetic Hopfield network.

Data availability

The data that support the findings in this study are available in the public repository at Source data are provided with this paper.

Code availability

The code used in this study is available from the authors upon request.


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We thank A.P. Young for fruitful discussions on estimating the spin glass correlation function’s unbiased estimator. A.F. and K.H. acknowledge support from the Swiss National Science Foundation (projects nos. 174306 and 172774, respectively). Funding was also received from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 737093, FEMTOTERABYTE. S.v.D. acknowledges support from the Academy of Finland (project no. 316857). M.S. acknowledges the support of the Centre for Nonlinear Studies and Theory Division at Los Alamos (grants nos. PRD20190195 and LA-UR-21-27055). F.C. was also financed via DOE-LDRD grants nos. PRD20170660 and PRD20190195. Part of this project was performed at the SIM Beamline of the Swiss Light Source, Paul Scherrer Institute, Switzerland. Samples were fabricated at the Molecular Foundry, Lawrence Berkeley National Laboratory, USA. The Molecular Foundry is supported by the Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under contract no. DE-AC02-05CH11231.

Author information

Authors and Affiliations



M.S. and A.F. conceived the project. A.F. designed and performed the experiments with support from K.H., S.P. and A.K. S.D. fabricated the samples. K.H. fabricated the samples for SQUID magnetometry and Y.A.B. performed the X-ray reflectivity measurements and fitting. M.S. analysed and interpreted the data with support from F.C. and A.F. M.S. and A.F. wrote the manuscript with input from all other authors. S.v.D., C.N. and A.F. supervised the project.

Corresponding authors

Correspondence to Michael Saccone or Alan Farhan.

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Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–5. Supplementary Video captions 1–3.

Supplementary Video 1

XMCD image sequence of thermally-activated nanomagnetic Hopfield network recorded at 168 K.

Supplementary Video 2

XMCD image sequence of thermally-activated nanomagnetic Hopfield network recorded at 181 K.

Supplementary Video 3

Schematic representation of low- and high-temperature dynamics within free energy landscapes.

Source data

Source Data Fig. 1

A high-resolution SEM image of the patterned nanomagnetic thin film.

Source Data Fig. 2

High-resolution XMCD images that comprise Fig. 2.

Source Data Fig. 3

The x and y coordinates, along with the error in y, that are plotted in Fig. 3.

Source Data Fig. 4

The x and y coordinates, along with the error in y, that are plotted in Fig. 4.

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Saccone, M., Caravelli, F., Hofhuis, K. et al. Direct observation of a dynamical glass transition in a nanomagnetic artificial Hopfield network. Nat. Phys. 18, 517–521 (2022).

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