Phase-space networks of geometrically frustrated systems

Topological Boundary Constraints in Artificial Colloidal Ice

The effect of boundaries and how these can be used to influence the bulk behavior in geometrically frustrated systems are both long-standing puzzles, often relegated to a secondary role. Here, we use numerical simulations and "proof of concept" experiments to demonstrate that boundaries can be engineered to control the bulk behavior in a colloidal artificial ice. We show that an antiferromagnetic frontier forces the system to rapidly reach the ground state (GS), as opposed to the commonly implemented open or periodic boundary conditions. We also show that strategically placing defects at the corners generates novel bistable states, or topological strings, which result from competing GS regions in the bulk. Our results could be generalized to other frustrated micro- and nanostructures where boundary conditions may be engineered with lithographic techniques.

Ground-state phase-space structures of two-dimensional ±J spin glasses: A network approach

We illustrate a complex-network approach to study the phase spaces of spin glasses. By mapping the whole ground-state phase spaces of two-dimensional Edwards-Anderson bimodal (±J) spin glasses exactly into networks for analysis, we discovered various phase-space properties. The Gaussian connectivity distribution of the phase-space networks demonstrates that both the number of free spins and the visiting frequency of all microstates follow the Gaussian distribution. The spectra of phase-space networks are Gaussian, which is proven to be exact when the system is infinitely large. The phase-space networks exhibit community structures. By coarse graining to the community level, we constructed a network representing the entropy landscape of the ground state and discovered its scale-free property. The phase-space networks exhibit fractal structures, as a result of the rugged entropy landscape. Moreover, we show that the connectivity distribution, community structures, and fractal structures change drastically at the ferromagnetic-to-glass phase transition. These quantitative measurements of the ground states provide new insight into the study of spin glasses. The phase-space networks of spin glasses share a number of common features with those of lattice gases and geometrically frustrated spin systems and form a new class of complex networks with unique topology.

Diversity enabling equilibration: disorder and the ground state in artificial spin ice

We report a novel approach to the question of whether and how the ground state can be achieved in square artificial spin ices where frustration is incomplete. We identify two sources of randomness that affect the approach to ground state: quenched disorder in the island response to fields and randomness in the sequence of driving fields. Numerical simulations show that quenched disorder can lead to final states with lower energy, and randomness in the sequence of driving fields always lowers the final energy attained by the system. We use a network picture to understand these two effects: disorder in island responses creates new dynamical pathways, and a random sequence of driving fields allows more pathways to be followed.

Self-similarity of phase-space networks of frustrated spin models and lattice gas models

We studied the self-similar properties of the phase spaces of two frustrated spin models and two lattice gas models. The two frustrated spin models were (1) the antiferromagnetic Ising model on a two-dimensional triangular lattice (1a) at the ground states and (1b) above the ground states and (2) the six-vertex model. The two lattice gas models were (3) the one-dimensional lattice gas model and (4) the two-dimensional lattice gas model. Their phase spaces were mapped to networks so that the fractal analysis of complex networks can be applied. These phase spaces, in turn, establish new classes of networks with unique self-similar properties. Models 1a, 2, and 3 with long-range power-law correlations in real space exhibit fractal phase spaces, while models 1b and 4 with short-range exponential correlations in real space exhibit nonfractal phase spaces. This behavior agrees with one of the untested assumptions in Tsallis nonextensive statistics. All the phase spaces have power-law "mass" -radius relations that reflect the local self-similar structures.

Phase-space networks of the six-vertex model under different boundary conditions

The six-vertex model is mapped to three-dimensional sphere stacks and different boundary conditions corresponding to different containers. The shape of the container provides a qualitative visualization of the boundary effect. Based on the sphere-stacking picture, we map the phase spaces of the six-vertex models to discrete networks. A node in the network represents a state of the system, and an edge between two nodes represents a zero-energy spin flip, which corresponds to adding or removing a sphere. The network analysis shows that the phase spaces of systems with different boundary conditions share some common features. We derived a few formulas for the number and the sizes of the disconnected phase-space subnetworks under the periodic boundary conditions. The sphere stacking provides new challenges in combinatorics and may cast light on some two-dimensional models.