Emergent charge crystallization and frustration in a particle anti-spin Ice

Abstract

Artificial spin ices have transcended their origins in frustrated rare-earth pyrochlores to become a versatile platform for engineering exotic states of matter. Across diverse implementations, from nanomagnets and superconducting vortices to colloids, quantum annealers, liquid crystals, and metamaterials, they are unified by the ice rule, which often leads to degeneracy and constrained disorder by enforcing minimization of the local topological charge. Here, we report the first realization of an "anti-spin ice" in which not only the ice rule does not hold, but its opposite is true as the system seeks to maximize, rather than minimize, spin ice charges. Using fast-rotating, in-plane magnetic fields to generate isotropic attraction between colloidal particles, we invert the conventional paradigm of repulsive interactions in colloidal spin ices. Combining experiments and simulations across standard square and honeycomb lattices as well as novel pentaheptite geometries, we establish rules for order and disorder in the anti-spin ice. With the pentaheptite lattice, we demonstrate that the anti-spin ice system can also exhibit frustration, but of a new kind. This topological charge frustration arises from the lattice connectivity, where networks of unequal, odd-sided polygons suppress charge crystallization at high interaction strength.

Figures (5)

• Figure 1: Repulsive and attractive interactions in the particle ice. (a) Schematic showing an ensemble of paramagnetic colloidal particles confined by gravity within a square lattice of cylindrical wells and subjected to an external magnetic field $\bm{B}$. The field induces an equal dipole moment $\bm{m}$ in each particle. (b) Top: Equilibrium states in a vertex of a square lattice for colloids subjected to an out-of-plane (left), an in-plane (middle) and a rotating (right) magnetic field. Bottom: Magnetic dipolar force $F_{dd}$ acting on a particle, with repulsion (attraction) highlighted in blue (red). Here $F_0= 0.26$ pN. (c) Measurements of the attractive interactions between two particles under a rotating field with frequency $f=50$Hz and different field amplitudes. Top: Separation distance $\Delta r$ versus time interval $\Delta t=t-t_{\sigma}$, with $t_{\sigma}= t(\Delta r = \sigma)$ and $\sigma=2.8 \rm{\mu m}$. Inset shows microscope images of two particles, see Supplementary Video 1. Bottom: re-scaled separation, being $D=0.156 \, \rm{\mu m^2 s^{-1}}$ the particle diffusion coefficient and $B_{c}=12\sqrt{2\eta D\mu_0 }/ (\chi \sigma)=0.106$ mT the field amplitude required to keep a pair of approaching particles in close contact. Here calculate $B_{c}$ by comparing the typical Brownian speed $v_B=D/\sigma$ to the speed of approach of a magnetic particle $v$ due to the dipolar repulsion with a neighbor at $\Delta r=\sigma$. In both graphs, scattered symbols are experimental data, shaded regions indicate numerical simulations results with their standard deviation.
• Figure 2: Charge crystals in bipartite lattices: Numerical results. (a,b) Snapshots of the low energy states of the square, $z=4$ (Supplementary Video 2) and the honeycomb, $z=3$ (Supplementary Video 3) particle anti-spin ice ($f=50$ Hz, $\alpha = 1.2\cdot 10^{-3} \, \rm{mT \, s^{-1}}$, $B_{\rm{max}}=1.2$ mT). Legends on the side show the normalized topological charge $Q=q/z$ for each vertex. Ice rule vertices have minimal absolute charge which is $Q=0$ ($Q=\pm 1/3$) for the square (honeycomb), but the ground state in the anti-spin ice features crystallization of high charges ( $Q=\pm1$) for both lattices. These charges are highlighted in the central region of the two images. The vertices with $Q=\pm 1/2$ and one of the two $Q=0$ vertex type in the square as the $Q=\pm 1/3$ for the honeycomb present a net moment shown by a small arrow. (c,d) Fraction of topological charges for the square (c) and the honeycomb (d) lattices versus rotating field amplitude $B$. For both cases, charge crystallization at high field is evidenced by the similar fraction of $Q=\pm 1$ defects (continuous lines) and disappearance of the low charged vertices (dashed lines). (e,g) Charge-charge correlation $\langle Q_i\cdot Q_j \rangle$ and (f,h) mean chirality $\bar{\chi}$ for the square (e,f) and the honeycomb (g,h) anti-spin ice. Small insets in (f,h) show the spins associated to each particle in a plaquette which gives $\bar{\chi}_{4,6}=0$.
• Figure 3: Frustrated pentaheptite anti-spin ice.a Simulation snapshot of the pentaheptite anti-spin ice ($f=50$ Hz, $\alpha = 2\cdot 10^{-3} \, \rm{mT \, s^{-1}}$, $B_{\rm{max}}=2$ mT). Overlaid to the particle positions are topological charges classified following the legend at the bottom (Supplementary Video 4). b Experimental realization of the Pentaheptite anti-spin ice using paramagnetic colloidal particles trapped within lithographic elliptical wells. Supplementary Video 5 shows the lattice subjected to a rotating magnetic field ($f=1$ Hz, $\alpha = 6.25\cdot 10^{-3} \, \rm{mT \, s^{-1}}$, $B_{\rm{max}}=3.75$ mT). c Vertex fraction versus rotating field amplitude $B$ showing the increase of $Q=\pm 1$ high topological charges and the decrease of ice rules $Q=\pm 1/3$ which did not vanishes at large amplitudes due to the topological frustration. d-e Charge-charge correlation $-\langle Q_i\cdot Q_j \rangle$ (d) and mean chirality (e) of pentagonal $\chi_5$ (top) and heptagonal $\chi_7$ (bottom) plaquettes versus rotating field amplitude $B$ ($f=50$ Hz) for the pentaheptite anti-spin ice. The dashed lines in (e) represents the predicted value for the lowest energy states. In (c-e) symbols are experimental data, and continuous line result from numerical simulations. The good agreement was obtained by dividing the experimental values of the field to $2.4$ (see text).
• Figure 4: Construction of the Pentaheptite lattice.a Schematic showing the angles and distances that result when constructing the pentaheptite lattice imposing an equal distance between the vertices. (b) Configurations of the four vertex types that results from the presence of the different angles when constructing the pentaheptite lattice. Top row shows the mean charge difference between the different configurations, and bottom the distribution of occurrence of these configurations in a lattice.(c,d) Schematic showing the low energy configuration predicted for the triangular (c) and the kagome (d) anti-spin ice in presence of strong attractive interactions.
• Figure 5: Pentaheptite unit cells.a,b Two low energy configurations of the pentaheptite anti-spin ice obtained from repeating $6$ times the basic unit cell of the lattice made of $2$ pentagons and $2$ heptagons. The difference between both structures is that in (a) the $2$ frustrated vertices are adjacent, while in (b) they are separated by one vertex. The unit cell highlighted in the central region has $8$ vertices : the $2$ vertices at center, $8$ vertices shared between $2$ unit cells and $6$ vertices shared by $3$ unit cells: $2+8/4+6/3=8$. Using this arrangement it is possible to compute the different statistical quantities measured at high field in the manuscript. The calculated spin-spin correlation value is $\langle Q_i \cdot Q_j \rangle = -128/216=-0.593$ for (a) and $\langle Q_i \cdot Q_j \rangle = -126/216=-0.583$ for (b), very close to the values obtained in experiments and simulations, Fig. \ref{['figure3']}(d).