Unexpected Phenomenology in Particle-Based Ice Absent in Magnetic Spin Ice

While particle-based ices are often considered essentially equivalent to magnet-based spin ices, the two differ essentially in frustration and energetics. We show that at equilibrium particle-based ices correspond exactly to spin ices coupled to a background field. In trivial geometries, such a field has no effect, and the two systems are indeed thermodynamically equivalent. In other cases, however, the field controls a richer phenomenology, absent in magnetic ices, and still largely unexplored: ice rule fragility, topological charge transfer, radial polarization, decimation induced disorder, and glassiness.

Figure 1

Magnetic force microscopies of hexagonal (a) and square (b) SI show the constitutive degrees of freedom (red rectangles) as dumbbells of positive (white) and negative (black) magnetic charges (from [10]). Optical microscopies of colloidal, hexagonal (c) and square (d) PI. The blue arrows denote the equivalent spins ${\overrightarrow{\sigma }}_{\mathbf{x}}$ (from [11]). Green disks show ice rule obeying vertices of minimal absolute topological charge.

Figure 2

(Top) Schematic illustration of Eq. (1) where a hexagonal PI (here in a random configuration) is decomposed into a SI, with dipolar degrees of freedom, plus a background of positively saturated traps. The energy of the PI (middle) and SI (bottom) vertices, listed in increasing order (from left to right, separated by dotted vertical lines) differ essentially. PI promotes vertices of large negative charge, violating ${\mathbb{Z}}_2$ symmetry. SI promotes vertices of low absolute topological charge (ice rule vertices, circled in green, have the same energy in SI but not in PI).

Figure 3

Same as in Fig. 2, but for the square lattice. Here, however, the degeneracy of the ice rule vertices ($q=0$) is lifted by a difference in interaction strength between perpendicular and collinear traps or dumbbells, and polarized vertices (middle: fourth from left; bottom: second from left) have higher energy. The red $\wedge$ connects two plaquettes where the head-to-toe rule is broken by a monopole (see Fig. 5).

Figure 4

A portion of a decimated, infinitely extended honeycomb PI (top left) in its low energy state can have ice rule violations on $z=2$ vertices ($q=+2$, blue circles) because it is equivalent to a SI stuffed with virtual, negatively saturated traps (red, dashed) in lieu of the removed links (top right) with no violations of the ice rule in the virtual charge ($\tilde{q}=\pm 1$). Indeed, while the SI energetics of the undecimated, $z=3$, vertices (middle) is left unchanged, that of the decimated $z=2$ vertices (bottom) must include virtual negative saturated traps and thus violates the ice rule at lowest energy: the vertex of real charge $q=2$ (virtual charge $\tilde{q}=1$) is degenerate with the vertex of real charge $q=0$ (virtual charge $\tilde{q}=-1$) and with the $z=3$ vertices of real charge $q=\pm 1$ (ice rule vertices circled in dashed green).

Figure 5

(a)–(c) Decimating a square lattice (a) and its antiferromagnetic ground state (decimated traps are replaced with negatively saturated traps, in red) leads to an ordered lowest energy state with $\tilde{q}=-2$ virtual charges on half of the decimated vertices in the SI picture (b). It corresponds to $q=\pm 1$ real charges on decimated vertices in the PI picture (c), and thus, all vertices obey the ice rule. At low decimation, this is the only low energy state. (d)–(f) However, above the decimation threshold corresponding to the percolation of decimated neighboring square plaquettes [e.g., yellow shaded ones neighboring a green one in (a)] the low energy state becomes degenerate. A disordered state can be chosen by connecting (red dotted line) neighboring decimated plaquettes (d) with $q=-2$ monopoles (represented with $\wedge$ connectors as in Fig. 3) on $z=4$ vertices, thus removing the virtual charges from decimated, $z=3$ vertices without increasing the energy [(e) green rectangles frame spins that can be freely flipped]. In the PI picture (f), this corresponds to ice rule violations on the $z=4$ vertices hosting charge $q=-2$: disorder comes from entropic transfer of topological charge.