The thermally driven formation and evolution of vertex domains is studied for square artificial spin ice. A self-consistent mean-field theory is used to show how domains of ground state ordering form spontaneously, and how these evolve in the presence of disorder. The role of fluctuations is studied using Monte Carlo simulations and analytical modelling. Domain wall dynamics are shown to be driven by a biasing of random fluctuations towards processes that shrink closed domains, and fluctuations within domains are shown to generate isolated small excitations, which may stabilize as the effective temperature is lowered. Domain dynamics and fluctuations are determined by interaction strengths, which are controlled by inter-element spacing. The role of interaction strength is studied via experiments and Monte Carlo simulations. Our mean-field model is applicable to ferroelectric 'spin' ice, and we show that features similar
Quenched disorder affects how nonequilibrium systems respond to driving. In the context of artificial spin ice, an athermal system comprised of geometrically frustrated classical Ising spins with a twofold degenerate ground state, we give experimental and numerical evidence of how such disorder washes out edge effects and provide an estimate of disorder strength in the experimental system. We prove analytically that a sequence of applied fields with fixed amplitude is unable to drive the system to its ground state from a saturated state. These results should be relevant for other systems where disorder does not change the nature of the ground state.
We have studied the magnetic microstates arising from single-shot thermalization processes that occur during growth in artificial square spin ices. The populations of different vertex types can be controlled by the system's lattice constant, as well as by depositing different material underlayers.The statistics of these populations are well-described by a simple model based on the canonical ensemble, which is used to infer an effective temperature for an arrested microstate. The normalized energy level spacings of the different magnetic vertex configurations are found to be very close to those predicted for a point dipole model: this is shown to be a very good approximation to energy level spacings calculated for finite-sized cuboid magnetic bodies. States prepared with a rotating field (an athermal method commonly used to lower the energy of these systems) cannot be described by this model, showing that such a method does not induce a near-equilibrium state. PACS numbers: 75.50.Lk, 75.10.Hk, 1 Theories of magnetism have long provided models for more complex statistical mechanical systems in physics and beyond. For instance, the venerable Ising model of a strongly anisotropic ferromagnet (famously exactly solved in two dimensional systems by Onsager, 1 and infamously unsolved in three dimensions) has been used in the context of describing the phase stability of ordered alloys, 2 the unbinding of DNA, 3 the structure of surfactant solutions, 4 and the behavior of neural networks. 5 Understanding systems that depart from equilibrium remains a challenge across all these fields.The approach of constructing model magnetic systems that are comparatively easy to understand can be extended from theory to experiment in order to address this challenge. This is accomplished in the designer metamaterials known as artificial spin ices. They comprise an array of single-domain ferromagnetic islands, built using nanolithography, 6 that replicates much of the physics of pyrochlore crystal spin ice systems, 7 which in turn replicate the geometrical frustration of the proton disorder in water ice.8 As all the parameters of the array may be engineered during fabrication, they allow for much wider exploration of phase space than the mere handful of naturally-occurring spin ices allow. Moreover, they are embodiments of statistical mechanical vertex models where the exact magnetic configuration (microstate) may be directly observed using advanced magnetic microscopy techniques. 9-11Inspecting the microstate allows for such important statistical mechanical properties as the effective temperature of the system, 12,13 and its entropy, 14 to be directly determined from magnetic images, once the appropriate theoretical apparatus is in place.The artificial square ice system we study here is depicted in Fig. 1 out(in) arrangement of the moments, hence it possesses both a magnetic charge and dipole moment, analogous to the monopole excitation of Castelnovo et al. 19 Type 4 has the highest energy, with all the moments pointing either ...
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