Approaching the Topological Low-Energy Physics of the $F$ Model in a Two-Dimensional Magnetic Lattice

V. Schánilec, O. Brunn, M. Horáček, S. Krátký, P. Meluzín, T. Šikola, B. Canals, and N. Rougemaille

Phys. Rev. Lett. 129, 027202 – Published 7 July 2022

Abstract

We demonstrate that the physics of the $F$ model can be approached very closely in a two-dimensional artificial magnetic system. Faraday lines spanning across the lattice and carrying a net polarization, together with chiral Faraday loops characterized by a zero magnetic susceptibility, are imaged in real space using magnetic force microscopy. Our measurements reveal the proliferation of Faraday lines and Faraday loops as the system is brought from low- to high-energy magnetic configurations. They also reveal a link between the Faraday loop density and icelike spin-spin correlations in the magnetic structure factor. Key for this Letter, the density of topological defects remains small, on the order of 1% or less, and negligible compared to the density of Faraday loops. This is made possible by replacing the spin degree of freedom used in conventional lattices of interacting nanomagnets by a micromagnetic knob, which can be finely tuned to adjust the vertex energy directly, rather than modifying the two-body interactions.

Received 10 February 2022
Revised 11 April 2022
Accepted 15 June 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.027202

Physics Subject Headings (PhySH)

Frustrated magnetism Magnetism Spin ice

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

V. Schánilec ^1,2, O. Brunn ^1,3,4, M. Horáček ³, S. Krátký³, P. Meluzín³, T. Šikola^2,4, B. Canals¹, and N. Rougemaille ¹

¹Université Grenoble Alpes, CNRS, Grenoble INP, Institut NEEL, 38000 Grenoble, France
²Central European Institute of Technology, CEITEC BUT, Brno University of Technology, Purkyňova 123, Brno 612 00, Czech Republic
³Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic
⁴Institute of Physical Engineering, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic

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Vol. 129, Iss. 2 — 8 July 2022

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Images

Figure 1
Schematics showing an excited configuration of the $F$ model. Type I domains are colored in green and blue, whereas type II vertices appear in red. Only type II vertices carry a magnetic moment that are joined by the Faraday lines (red and blue oriented thick lines). Faraday lines have an assigned parity depending on the $A$ or $B$ plaquettes they bridge (the red Faraday line has $A$ parity and separates neighboring $B$ plaquettes).
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Figure 2
(a) Electron micrograph of one lattice. Scale bar is 500 nm. (b) Micromagnetic energy of the four vertex types as a function of the hole diameter. (c) Micromagnetic configurations of type I, II, III, and IV vertices. Black arrows represent the local direction of magnetization, while the blue and red contrast codes for the divergence of the magnetization vector.
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Figure 3
Real space (a) and reciprocal space (b) analyses of the magnetic configurations obtained after demagnetizing six lattices having different hole diameters $ϕ$ . (a) The two degenerate antiferromagnetic domains are shaded in blue and green. These type I domains are separated by Faraday loops made of type II vertices and represented by directed lines colored in blue and red depending on their parity. Faraday lines having different parities never intersect, except in the presence of an ice-rule defect (shown by a colored open circle). (b) The magnetic structure factors are averaged over four real space configurations (i.e., 3600 vertices and 7440 spins) for each hole diameter to improve statistics and are calculated for wave vectors covering $\pm 6$ reciprocal lattice units (RLUs). Intensity scale is the same in all six MSFs.
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Figure 4
(a) Average number of closed loops for 24 measured lattices (for each hole diameter, four lattices are analyzed). The loops are characterized by a parity and a chirality. Both quantities are equally populated, showing that our demagnetization protocol does not induce any bias. (b) Average number of the smallest possible closed loops. (c) Number of times a loop changes direction on average.
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Figure 5
(a) Two Faraday lines bridged by a pair of ice-rule defects. The schematics on the right side illustrate the local environment of the topological defects and the energy cost associated with a spin flip event. (b) Specific case when propagating the white defect into the type I domain: once the defect has moved into the domain, it is statistically likely that it will be pushed back into the domain wall.
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