Macroscopic Magnetic Frustration

Although geometrical frustration transcends scale, it has primarily been evoked in the micro- and mesoscopic realm to characterize such phases as spin ice, liquids, and glasses and to explain the behavior of such materials as multiferroics, high-temperature superconductors, colloids, and copolymers. Here we introduce a system of macroscopic ferromagnetic rotors arranged in a planar lattice capable of out-of-plane movement that exhibit the characteristic honeycomb spin ice rules studied and seen so far only in its mesoscopic manifestation. We find that a polarized initial state of this system settles into the honeycomb spin ice phase with relaxation on multiple time scales. We explain this relaxation process using a minimal classical mechanical model that includes Coulombic interactions between magnetic charges located at the ends of the magnets and viscous dissipation at the hinges. Our study shows how macroscopic frustration arises in a purely classical setting that is amenable to experiment, easy manipulation, theory, and computation, and shows phenomena that are not visible in their microscopic counterparts.

Figure 1

(a) A triad of magnetic rotors (lying in one of the sublattices indicated as 1, 2, and 3) having length $L=2a=1.9\times {10}^{-2}\mathrm{m}$, diameter $d=1.5\times {10}^{-3}\mathrm{m}$, mass $M=0.28\times {10}^{-3}\mathrm{kg}$, and saturation magnetization $M_s=1.2\times {10}^6\mathrm{A}{\mathrm{m}}^{-1}$ are located at $\theta =120°$ with respect to each other. The out-of-plane degree of freedom is denoted by the polar angle $\alpha$. Painted in black, the magnet south pole (S) is distinguished from its north pole (N). (b) Picture of the lattice with its centers located at distance $l=\sqrt{3}(a+\Delta )$ with the $n$ rods lying in the $x\mathrm{\text{−}}y$ plane fulfilling the honeycomb spin ice rules. Inset shows the Fourier transform of the lattice. (c) The numerical equivalent lattice having the same experimental parameters. In this case the point of the arrow denotes the S pole of the magnet. (d) Top: Histograms taken from 10 experiments and simulations showing the experimental (red or light bars) and numerical (black or dark bars) distribution of vertices. Bottom: Local energy of the eight vertex configurations possible in the honeycomb lattice, in units of $D={10}^{-5}\mathrm{J}$ (Supplemental Material, S6 [15]).

Figure 2

Lattice dynamics characterized by nearest neighbor spin correlations, $⟨S_{\alpha }{S}_{\beta }⟩$. (a) Stage I: once $B_z$ is turned off, the rotors originally pointing along $\widehat{z}$ break their axial symmetry. (b) The image showing the end of stage I and the onset of stage II when rods rotate with respect to their center of mass yielding a plateau in $⟨S_{\alpha }{S}_{\beta }⟩$. (c) A snapshot of the rods oscillating in stage III. (d) In red (dark dots), experimental data obtained via image processing; in blue (light squares), molecular dynamics simulation results from the numerical solution of equation (S4) [15] where the full coulomb contributions from all neighbors is taken into account. At $t=0$ all S poles point along $\widehat{z}$. Stage I is dominated by Coulomb interactions between rods and characterized by the Coulomb time scale $t_c$. In Stage II, all rotors spin until dissipation damps out the spin in favor of oscillations, leading to Stage III where they exhibit damped oscillations. After relaxation the rods lie in the $x\mathrm{\text{−}}y$ plane in a honeycomb spin ice magnetic configuration, with its characteristic nearest neighbor spin correlations $⟨S_{\alpha }{S}_{\beta }⟩=1/3$ (solid line). Inset: Experimentally measured value of $⟨S_{\alpha }{S}_{\beta }⟩$ during the initial explosive evolution (red or dots) compared with $\cos (\alpha )$ where $\alpha$ is the solution of Eq. (S2) in the Supplemental Material [15] for one rotor interacting with two neighbors, in the absence of damping and external torques (green or continuous curve).

Figure 3

Phase diagram of the lattice dynamical response to an external perturbation. The horizontal axis shows the dimensionless ratio of the kinetic and phononic time scales with $v$ the speed of an external dipole, while the vertical axis shows the dimensionless ratio of the internal and the external magnetic forces due to an external dipole of strength $Q$ located at distance $h$ from the lattice (see text for details). Experimental and numerical data shown in squares and circles, respectively, and colors define the nature of the lattice dynamical response to the external perturbation. We see that the dynamics may be broken up into a frictionally dominated, interaction dominated, or inertially dominated regime as a function of the relative magnitude and rate of external forcing.