We present direct, real-time observations of emergent magnetic monopole dynamics in highly frustrated artificial spin ice.

Geometrical frustration occurs when entities in a system, subject to given lattice constraints, are hindered to simultaneously minimize their local interactions. In magnetism, systems incorporating geometrical frustration are fascinating, as their behavior is not only hard to predict, but also leads to the emergence of exotic states of matter. Here, we provide a first look into an artificial frustrated system, the dipolar trident lattice, where the balance of competing interactions between nearest-neighbor magnetic moments can be directly controlled, thus allowing versatile tuning of geometrical frustration and manipulation of ground state configurations. Our findings not only provide the basis for future studies on the low-temperature physics of the dipolar trident lattice, but also demonstrate how this frustration-by-design concept can deliver magnetically frustrated metamaterials.

We explore the thermodynamics in two-dimensional arrays consisting of Ising-type nanomagnets lithographically arranged onto random sites and angular orientations. Introducing these basic spin-glass ingredients, we study the characteristic features of the low-energy states achieved, following thermal-annealing protocols. From direct visualization of real-time dynamics, we record relaxation timescales together with magnetic susceptibility variations over temperature, revealing trends towards short-range order as randomness is increased, but falling short of pure spin-glass behavior. Our work provides a route towards the realization of artificial Ising spin-glass systems.

Magnetic charge propagation in spin-ice materials has yielded a paradigm-shift in science, allowing the symmetry between electricity and magnetism to be studied. Recent work is now suggesting the spin-ice surface may be important in mediating the ordering and associated phase space in such materials. Here, we detail a 3D artificial spin-ice, which captures the exact geometry of bulk systems, allowing magnetic charge dynamics to be directly visualized upon the surface. Using magnetic force microscopy, we observe vastly different magnetic charge dynamics along two principal directions. For a field applied along the surface termination, local energetics force magnetic charges to nucleate over a larger characteristic distance, reducing their magnetic Coulomb interaction and producing uncorrelated monopoles. In contrast, applying a field transverse to the surface termination yields highly correlated monopole-antimonopole pairs. Detailed simulations suggest it is the difference in effective chemical potential as well as the energy landscape experienced during dynamics that yields the striking differences in monopole transport.

We have studied low-energy configurations in two-dimensional arrays consisting of Ising-type dipolar coupled nanomagnets lithographically defined onto a two-dimensional Cairo lattice, thus dubbed the dipolar Cairo lattice. Employing synchrotron-based photoemission electron microscopy (PEEM), we perform real-space imaging of moment configurations achieved after thermal annealing. These states are then characterized in terms of vertex populations, spin-and emergent magnetic charge correlations, and a topology-enforced emergent ice rule. The results reveal a strong dominance of short-range correlations and the absence of long-range order, reflecting the high degree of geometrical spin frustration present in this example of an artificial frustrated spin system.

Spin glasses, generally defined as disordered systems with randomized competing interactions that result in an extensively degenerate ground state 1, 2 , are a widely investigated complex system. Theoretical models describing spin glasses are broadly used in other complex systems, such as those describing brain function 3,4 , error-correcting codes 5 , or stock-market dynamics 6 . This wide interest in spin glasses provides strong motivation to generate an artificial spin glass within the framework of artificial spin ice systems [7][8][9] . Here, we present the first experimental realization of an artificial spin glass, consisting of dipolar coupled single-domain Ising-type nanomagnets arranged onto an interaction network that replicates the aspects of a Hopfield neural network 10 . Using cryogenic x-ray photoemission electron microscopy (XPEEM), we performed temperature dependent imaging of thermally driven moment fluctuations within these networks and observed characteristic features of a two-dimensional Ising spin glass. Specifically, the temperature dependence of the spin glass correlation function follows a power law trend predicted from theoretical models on two-dimensional spin glasses 11 . Furthermore, we observe clear signatures of the hard to observe rugged spin glass free

Tree like interaction structures increase effective dimension in this significant step towards the first ever artificial spin glass system.

We present a realization of highly frustrated planar triangular antiferromagnetism achieved in a quasi-three-dimensional artificial spin system consisting of monodomain Ising-type nanomagnets lithographically arranged onto a deep-etched silicon substrate. We demonstrate how the three-dimensional spin architecture results in the first direct observation of long-range ordered planar triangular antiferromagnetism, in addition to a highly disordered phase with short-range correlations, once competing interactions are perfectly tuned. Our work demonstrates how escaping two-dimensional restrictions can lead to new types of magnetically frustrated metamaterials.