No abstract

Systems of interacting nanomagnets known as artificial spin ice 1-4 have allowed the design, realization and study of geometrically frustrated exotic collective states 5-10 that are absent in natural magnets. We have experimentally measured 11,12 the thermally induced moment fluctuations in the Shakti geometry of artificial spin ice. We show that its disordered moment configuration is a topological phase described by an emergent dimer-cover model 13 with excitations that can be characterized as topologically charged defects. Examination of the lowenergy dynamics of the system confirms that these effective topological charges have long lifetimes associated with their topological protection, that is, they can be created and annihilated only as charge pairs with opposite sign and are kinetically constrained. This manifestation of classical topological order 14-19 demonstrates that geometrical design in nanomagnetic systems can lead to emergent, topologically protected kinetics that can limit pathways to equilibration and ergodicity. Artificial spin ices are lithographically fabricated systems of interacting single-domain nanomagnets. These systems can be used to investigate the collective magnetic behaviour of interacting moments as effective models for understanding the complex phenomena of frustration. Each nanomagnet moment aligns along the edges of a lattice and points towards or away from the lattice vertices. In their low-energy collective states, the moments enter a so-called ice-manifold; an ensemble in which, at each vertex, the difference between the number of moments pointing in and out is minimized, leading to the ice-rule (2-in/2-out 20 at vertices where four moments meet or 1-in/2-out, 2-in/1-out at vertices where three moments meet). Originally inspired by rare-earth pyrochlore spin ice materials, these artificial spin ice systems evolved towards new geometries 5,6 , with exotic phases absent in natural magnets 2,3,7,8,21. Recent experimental works have characterized the thermal fluctuations of the individual magnetic moments, opening new vistas in the real-time, real-space analysis of frustration 11,12,22-25. The Shakti lattice geometry 5-7 (Fig. 1) is a decimation of the square ice lattice geometry. In Fig. 1e, we show the possible moment configurations at vertices and label them by the number of islands at each vertex (the coordination number, z) and by their relative energy hierarchy. The collective ground state is a configuration in which the z = 2 and z = 4 vertices are all in their lowest energy state (that is, type I 4 for the four-island vertices and type I 2 for the twoisland vertices) while only half of the z = 3 vertices lie in their lowest

No abstract

We experimentally demonstrate that arrays of interacting nanoscale ferromagnetic islands, known as artificial spin ice, develop reproducible microstates upon cycling an applied magnetic field. The onset of this memory effect is determined by the strength of the applied field relative to the array coercivity. Specifically, when the applied field strength is almost exactly equal to the array coercivity, several training cycles are required before the array achieves a nearly completely repeatable microstate, whereas when the applied field strength is stronger or weaker than the array coercivity, a repeatable microstate is achieved after the first minor loop. We show through experiment and simulation that this memory exhibited by artificial spin ice is due to a ratchet effect on interacting, magnetically-charged defects in the island moment configuration and to the complexity of the network of strings of reversed moments that forms during magnetization reversal.2

The influence of an electrical current on the propagation of magnetostatic surface waves is investigated in a relatively thick (40 nm) permalloy film both experimentally and theoretically. Contrary to previously studied thinner films where the dominating effect is the current-induced spin-wave Doppler shift, the magnetic field generated by the current (Oersted field) is found to induce a strong non-reciprocal frequency shift which overcompensates the Doppler shift. The measured current induced frequency shift is in agreement with the developed theory. The theory relates the sign of of the frequency shift to the spin wave modal profiles. The good agreement between the experiment and the theory confirms a recent prediction of a counter-intuitive mode localization for magnetostatic surface waves in the dipole-exchange regime.

We have performed a detailed study of thermal annealing of the moment configuration in artificial spin ice. Permalloy (Ni 80 Fe 20 ) artificial spin ice samples were examined in the prototypical square ice geometry, studying annealing as a function of island thickness, island shape, and annealing temperature and duration. We also measured the Curie temperature as a function of film thickness, finding that thickness has a strong effect on the Curie temperature in regimes of relevance to many studies of the dynamics of artificial spin ice systems. Increasing the interaction energy between island moments and reducing the energy barrier to flipping the island moments allows the system to more closely approach the collective low energy state of the moments upon annealing, suggesting new channels for understanding the thermalization processes in these important model systems.Artificial spin ice systems are two-dimensional arrays of nanoscale elements, typically composed of single domain ferromagnetic islands 1 . These systems have been the subject of extensive study and have provided models for the study of a range of novel collective behaviors 2 . Certain artificial spin ice geometries have well-defined collective magnetic ground states, such as the square lattice 1 , while others have intrinsically disordered and complex ground states, such as the Shakti lattice 3-5 . These low-energy collective states have sparked considerable interest in attempting to realize the lowest energy state of different artificial spin ice lattices 6-10 . One successful approach to collective energy minimization involves annealing the arrays by heating them to temperatures near or above the Curie temperature (T C ) of the ferromagnetic material 11,12 . Upon cooling, the island moments arrange themselves into a low energy state via magnetostatic interactions. Using this method, both long-range-ordered 11-13 and intrinsically disordered ground states 4,14 have been achieved, both in permalloy (Ni 80 Fe 20 ) and in other alloys 9,10 . Notably, the method works well even for geometries known to exhibit slow relaxation toward the low energy state 4 . Given the high T C of permalloy, and its importance as a model material for these systems, we investigated thermal annealing of permalloy artificial spin ice by varying the annealing conditions and the geometry of the islands, with the goal of understanding how to improve the effectiveness of annealing.We fabricated our artificial square spin ice samples on Si wafers coated with a 200-nm-thick layer of Si-N deposited a) Electronic mail: peter.schiffer@yale.edu by low pressure chemical vapor deposition. The nanoislands, with varied lateral dimensions and inter-island gaps indicated below, were produced by electron beam lithography and liftoff as described previously 12 . The total area of all nanoislands in each square ice sample was about 200×200 µm 2 . In order to keep uniformity of all nanoislands, the write field for lithography was set to cover the whole sample. We deposited our samples with va...

One-dimensional strings of local excitations are a fascinating feature of the physical behavior of strongly correlated topological quantum matter. Here we study strings of local excitations in a classical system of interacting nanomagnets, the Santa Fe Ice geometry of artificial spin ice. We measured the moment configuration of the nanomagnets, both after annealing near the ferromagnetic Curie point and in a thermally dynamic state. While the Santa Fe Ice lattice structure is complex, we demonstrate that its disordered magnetic state is naturally described within a framework of emergent strings. We show experimentally that the string length follows a simple Boltzmann distribution with an energy scale that is associated with the system’s magnetic interactions and is consistent with theoretical predictions. The results demonstrate that string descriptions and associated topological characteristics are not unique to quantum models but can also provide a simplifying description of complex classical systems with non-trivial frustration.

Tensor network states (TNS) methods combined with the Monte Carlo (MC) technique have been proven a powerful algorithm for simulating quantum many-body systems. However, because the ground state energy is a highly non-linear function of the tensors, it is easy to get stuck in local minima when optimizing the TNS of the simulated physical systems. To overcome this difficulty, we introduce a replica-exchange molecular dynamics optimization algorithm to obtain the TNS ground state, based on the MC sampling technique, by mapping the energy function of the TNS to that of a classical mechanical system. The method is expected to effectively avoid local minima. We make benchmark tests on a 1D Hubbard model based on matrix product states (MPS) and a Heisenberg J1-J2 model on square lattice based on string bond states (SBS). The results show that the optimization method is robust and efficient compared to the existing results.