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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 ...

In our Letter, uncalibrated field amplitudes were mistakenly used in Fig. 3(a). The calibration is a simple rescaling by a factor of 0.866, so the forms of the vertex populations are unaffected. A corrected figure is shown below. The field values reported in the text of the Letter are correct, as are the field values in Fig. 2, and none of the results or conclusions of the Letter are affected. FIG. 3 (color online). Vertex populations vs hold field for (a) experiment and (b) theory. Symbols represent vertex types as shown in the legend, with open (closed) symbols for open (closed) edge arrays. Each data point is the average over several runs; error bars represent the standard error.

We have studied experimentally the states formed in artificial square ice nanomagnet systems following demagnetization in a rotating in-plane applied magnetic field that reduces to zero in a manner that is linear in time. The final states are found to be controlled via the system's lattice constant, which determines the strength of the magnetostatic interactions between the elements, as well as the field ramping rate. We understand these effects as a requirement that the system undergoes a sufficiently large number of active rotations within the critical field window in which elements may be reversed, such that the interactions are allowed to locally exert their influence if the ground state is to be approached. On the other hand, if quenched disorder is too strong when compared to the interaction strength, any close approach to the ground state is impossible. These results show that it is not necessary for there to be any ac component to the field amplitude that is applied to the system during demagnetization, which is the method almost exclusively employed in field protocols reported to date. Furthermore, by optimizing the parameters of our linear demagnetization protocol, the largest field-generated ground state domains yet reported are found.