Anisotropic interactions induce dynamical arrest in artificial colloidal ice

Technical Summary: Anisotropic Interactions Induce Dynamical Arrest in Artificial Colloidal Ice

Problem Statement
Artificial Colloidal Ice (ACI) serves as a tunable model system for studying geometric frustration and spin glass behavior, typically characterized by isotropic repulsive interactions under perpendicular magnetic fields. While spin freezing in bulk materials is often attributed to quenched disorder (random coupling energies) or lattice distortions, macroscopic signatures of freezing have also been observed in pure systems. This paper investigates whether dynamical arrest can occur in a disorder-free ACI system solely through the introduction of anisotropic interactions. Specifically, the authors examine the system's behavior when an in-plane magnetic field is applied, creating a landscape with both attractive and repulsive regions, and whether the system can reach its theoretically defined ground state or becomes trapped in metastable configurations.

Methodology
The study employs Brownian Dynamics simulations to model superparamagnetic colloidal particles confined within a 2D square lattice of bistable optical traps. The system is governed by the overdamped equation of motion, incorporating three primary forces:

1. Trap Forces ($F_{trap}$): A bistable potential confining particles to specific sites, preventing inter-trap hopping.
2. Magnetic Dipole-Dipole Interactions ($F_{dd}$): Anisotropic interactions induced by an external magnetic field applied along one of the lattice basis vectors. This creates a "magic angle" ($\phi = 54.7^\circ$) separating regions of attraction and repulsion.
3. Thermal Noise: Represented by a random variable with zero mean, simulating thermal fluctuations at $T = 300$ K.

The simulations utilize a system size of $30 \times 30$ vertices (1800 particles) with periodic boundary conditions. The magnetic field is increased from zero to 5 mT using two distinct quench protocols: a "fast" rate ($\dot{B} = 33.3 \, \mu\text{T/s}$) and a "slow" rate ($\dot{B} = 0.333 \, \mu\text{T/s}$). Following the quench, the system is allowed to relax for one hour at a constant field to observe temporal evolution. Theoretical predictions for the ground state are derived by calculating the effective energy of vertex types (defined by their topological charge) and applying Boltzmann statistics.

Key Contributions and Results

• Theoretical Ground State: Under an in-plane field, the system's symmetry is modified such that the energetically favored ground state consists of a checkerboard pattern of fully charged vertices (types I and VI, with topological charges $q = \pm 4$). This configuration represents an antiferromagnetic ordering of maximum charge.
• Dynamical Arrest: Despite the existence of a well-defined, low-energy ground state and the absence of quenched disorder in the lattice, the simulations fail to reach this equilibrium. Instead, the system becomes arrested in metastable, disordered states.
  • Fast Quench: Results in a highly disordered state where domains of the ground state are comparable in size to defect regions.
  • Slow Quench: Produces a more ordered state with scattered defects on a background of ground-state vertices, yet the system still fails to achieve the theoretical equilibrium fraction of ground-state vertices (reaching only ~60% even after 4 hours).
• Mechanism of Arrest: The arrest is driven by the local enhancement of potential barriers. As the magnetic field increases, the potential landscape for individual particles changes, increasing the energy barrier required to transition between equilibrium positions.
  • In single-vertex analysis, the barrier height ($\tilde{h}$) increases linearly with the square of the magnetic field ($B^2$).
  • In coupled vertices, the barrier separating metastable states (e.g., $q=\pm 2$) from the ground state ($q=\pm 4$) becomes sufficiently high to prevent thermal exploration of the phase space, effectively freezing the dynamics.
• Characterization of Freezing:
  • Spatial Correlation: The correlation length ($l_c$) grows initially but saturates at approximately 2 mT. For fast quenches, $l_c$ remains below one lattice site; for slow quenches, it reaches up to three lattice sites.
  • Temporal Correlation: The Edwards-Anderson parameter and relaxation times ($\tau$) indicate that while particles retain some local mobility (jumping within traps), the collective domain structure stops evolving. The relaxation time diverges by several orders of magnitude as the field increases, following a stretched exponential decay.

Significance and Claims
The paper claims to demonstrate a distinct mechanism for spin freezing that does not rely on quenched disorder or lattice distortions, which are the conventional explanations for freezing in pure spin glass materials. Instead, the freezing arises intrinsically from the magnetic interactions themselves. As the system is "cooled" (by increasing the field), the potential barriers between equilibrium states grow, eventually arresting the dynamics.

The authors emphasize that this system, with only one degree of freedom per particle, allows for the precise observation of the potential barrier rise that causes the arrest while maintaining the collective nature of the phenomenon. They suggest this mechanism is distinct from other colloidal models where glassiness is induced by short-range anisotropic interactions or confinement effects.

Finally, the paper notes that because the system freezes and stops evolving (unlike isotropic colloidal ice where domains continue to coarsen), it may serve as a suitable model for observing Kibble-Zurek scaling mechanisms, overcoming the difficulties associated with time-dependent length scales in other systems. However, the authors explicitly state that investigating this scaling is left for future work.