Topology Restricts Quasidegeneracy in Sheared Square Colloidal Ice

Recovery of ground-state degeneracy in two-dimensional square ice is a significant challenge in the field of geometric frustration with far-reaching fundamental implications, such as realization of vertex models and understanding the effect of dimensionality reduction. We combine experiments, theory, and numerical simulations to demonstrate that sheared square colloidal ice partially recovers the ground-state degeneracy for a wide range of field strengths and lattice shear angles. Our method could inspire engineering a novel class of frustrated microstructures and nanostructures based on sheared magnetic lattices in a wide range of soft- and condensed-matter systems.

Figure 1

(a) Schematic of one of the lowest-energy configurations of water ice $I_h$ (top) and the corresponding projection on a 2D plane to give the square lattice (bottom). (b) In 2D square colloidal ice (top), vertex type 3 has lower energy than type 4, in which the two in-particles are closer to each other. Upon shearing the lattice (bottom), type 4 splits so that the energy of type 4a approaches that of type 3, as the shear angle $\theta$ increases. (c) Experimental realization of a sheared square colloidal ice with lattice constant $a$, trap length $\ell$, and shear angle $\theta =25°$ under a magnetic field $B=7.9\mathrm{mT}$ which is applied perpendicular to the plane. Scale bar is $30\mu \mathrm{m}$.

Figure 2

(a) Evolution of the energy spectrum of the different types of a single vertex with increasing shear angle, ranging from the unsheared square lattice ($\theta =0$) on the left, to the maximally sheared lattice that we studied experimentally ($\theta =45°$) on the right. Vertex types 2 and 5 and types 1 and 6 are clumped to particle-balanced effective vertices (2,5) and (1,6) with the average energies of these unbalanced types (see text). (b) Ratio $G=G_2/G_1$ between the second and the first gaps in the energy spectrum depends weakly on the ratio $R=\ell /a$ of trap length to lattice constant, and reaches considerable values of 30–60 already at moderate shear of $\theta =45°$.

Figure 3

Simulation results (symbols) versus single-vertex mean-field theory (solid lines) and combined with 1D Ising theory for line defects (black dashed lines) for $\theta =25°$ (a) and 45° (b). (c),(d) Snapshots from simulation at $\theta =45°$, showing how flipping the positions of a sequence of particles in the type 3 ground state generates a double row of type 4a vertices with type 2 and type 5 vertices surrounding this line defect. The Voronoi parallelogram around every vertex is colored according to the vertex type, following the color scheme in (a) and (b).

Figure 4

Fractions of different types of vertices versus magnetic field in experiment (symbols) and single-vertex mean-field theory (lines) for shear angles of 25° (left) and 45° (right). Results are plotted versus the measured field (a),(b) and versus the effective field deduced from the occupancy of type 3 vertices (c),(d). Black dashed lines show the exact solution of the 1D Ising model for our line defect theory. At high fields, we predict that all vertices are of type 3, namely, $p_3=1$, and all other $p_i=0$.