Creating artificial ice states using vortices in nanostructured superconductors

Toroidic phase transitions in a direct-kagome artificial spin ice

Ferrotoroidicity-the fourth form of primary ferroic order-breaks both space and time-inversion symmetry. So far, direct observation of ferrotoroidicity in natural materials remains elusive, which impedes the exploration of ferrotoroidic phase transitions. Here we overcome the limitations of natural materials using an artificial nanomagnet system that can be characterized at the constituent level and at different effective temperatures. We design a nanomagnet array as to realize a direct-kagome spin ice. This artificial spin ice exhibits robust toroidal moments and a quasi-degenerate ground state with two distinct low-temperature toroidal phases: ferrotoroidicity and paratoroidicity. Using magnetic force microscopy and Monte Carlo simulation, we demonstrate a phase transition between ferrotoroidicity and paratoroidicity, along with a cross-over to a non-toroidal paramagnetic phase. Our quasi-degenerate artificial spin ice in a direct-kagome structure provides a model system for the investigation of magnetic states and phase transitions that are inaccessible in natural materials.

Kagome qubit ice

Topological phases of spin liquids with constrained disorder can host a kinetics of fractionalized excitations. However, spin-liquid phases with distinct kinetic regimes have proven difficult to observe experimentally. Here we present a realization of kagome spin ice in the superconducting qubits of a quantum annealer, and use it to demonstrate a field-induced kinetic crossover between spin-liquid phases. Employing fine control over local magnetic fields, we show evidence of both the Ice-I phase and an unconventional field-induced Ice-II phase. In the latter, a charge-ordered yet spin-disordered topological phase, the kinetics proceeds via pair creation and annihilation of strongly correlated, charge conserving, fractionalized excitations. As these kinetic regimes have resisted characterization in other artificial spin ice realizations, our results demonstrate the utility of quantum-driven kinetics in advancing the study of topological phases of spin liquids.

Generalized Gibbs Phase Rule and Multicriticality Applied to Magnetic Systems

A generalization of the original Gibbs phase rule is proposed in order to study the presence of single phases, multiphase coexistence, and multicritical phenomena in lattice spin magnetic models. The rule is based on counting the thermodynamic number of degrees of freedom, which strongly depends on the external fields needed to break the ground state degeneracy of the model. The phase diagrams of some spin Hamiltonians are analyzed according to this general phase rule, including general spin Ising and Blume–Capel models, as well as q-state Potts models. It is shown that by properly taking into account the intensive fields of the model in study, the generalized Gibbs phase rule furnishes a good description of the possible topology of the corresponding phase diagram. Although this scheme is unfortunately not able to locate the phase boundaries, it is quite useful to at least provide a good description regarding the possible presence of critical and multicritical surfaces, as well as isolated multicritical points.

Topological Boundary Constraints in Artificial Colloidal Ice

The effect of boundaries and how these can be used to influence the bulk behavior in geometrically frustrated systems are both long-standing puzzles, often relegated to a secondary role. Here, we use numerical simulations and "proof of concept" experiments to demonstrate that boundaries can be engineered to control the bulk behavior in a colloidal artificial ice. We show that an antiferromagnetic frontier forces the system to rapidly reach the ground state (GS), as opposed to the commonly implemented open or periodic boundary conditions. We also show that strategically placing defects at the corners generates novel bistable states, or topological strings, which result from competing GS regions in the bulk. Our results could be generalized to other frustrated micro- and nanostructures where boundary conditions may be engineered with lithographic techniques.

Reconfigurable magnonic mode-hybridisation and spectral control in a bicomponent artificial spin ice

Strongly-interacting nanomagnetic arrays are finding increasing use as model host systems for reconfigurable magnonics. The strong inter-element coupling allows for stark spectral differences across a broad microstate space due to shifts in the dipolar field landscape. While these systems have yielded impressive initial results, developing rapid, scaleable means to access a broad range of spectrally-distinct microstates is an open research problem. We present a scheme whereby square artificial spin ice is modified by widening a 'staircase' subset of bars relative to the rest of the array, allowing preparation of any ordered vertex state via simple global-field protocols. Available microstates range from the system ground-state to high-energy 'monopole' states, with rich and distinct microstate-specific magnon spectra observed. Microstate-dependent mode-hybridisation and anticrossings are observed at both remanence and in-field with dynamic coupling strength tunable via microstate-selection. Experimental coupling strengths are found up to g/2π = 0.16 GHz. Microstate control allows fine mode-frequency shifting, gap creation and closing, and active mode number selection.

Skyrmion Spin Ice in Liquid Crystals

We propose the first skyrmion spin ice, realized via confined, interacting liquid crystal skyrmions. Skyrmions in a chiral nematic liquid crystal behave as quasiparticles that can be dynamically confined, bound, and created or annihilated individually with ease and precision. We show that these quasiparticles can be employed to realize binary variables that interact to form ice-rule states. Because of their unique versatility, liquid crystal skyrmions can open entirely novel avenues in the field of frustrated systems. More broadly, our findings also demonstrate the viability of liquid crystal skyrmions as elementary degrees of freedom in the design of collective complex behaviors.

Self-assembled vortex crystals induced by inhomogeneous magnetic textures

We investigate the self-assembly of vortices in a type-II superconducting disk subjected to highly nonuniform confining potentials produced by inhomogeneous magnetic textures. Using a series of numerical experiments performed within the Ginzburg-Landau theory, we show that vortices can arrange spontaneously in highly nonuniform, defect-free crystals, reminiscent of conformal lattices, even though the strict conditions for the conformal crystal are not fulfilled. These results contradict continuum-limit theory, which predicts that the order of a nonuniform crystal is unavoidably frustrated by the presence of topological defects. By testing different cooling routes of the superconductor, we observed several different self-assembled configurations, each of which corresponding to one in a set of allowed conformal transformations, which depends on the magnetic and thermal histories of the system.

Ice rule fragility via topological charge transfer in artificial colloidal ice

Artificial particle ices are model systems of constrained, interacting particles. They have been introduced theoretically to study ice-manifolds emergent from frustration, along with domain wall and grain boundary dynamics, doping, pinning-depinning, controlled transport of topological defects, avalanches, and memory effects. Recently such particle-based ices have been experimentally realized with vortices in nano-patterned superconductors or gravitationally trapped colloids. Here we demonstrate that, although these ices are generally considered equivalent to magnetic spin ices, they can access a novel spectrum of phenomenologies that are inaccessible to the latter. With experiments, theory and simulations we demonstrate that in mixed coordination geometries, entropy-driven negative monopoles spontaneously appear at a density determined by the vertex-mixture ratio. Unlike its spin-based analogue, the colloidal system displays a "fragile ice" manifold, where local energetics oppose the ice rule, which is instead enforced through conservation of the global topological charge. The fragile colloidal ice, stabilized by topology, can be spontaneously broken by topological charge transfer.

Tunable and switchable magnetic dipole patterns in nanostructured superconductors

Design and manipulation of magnetic moment arrays have been at the focus of studying the interesting cooperative physical phenomena in various magnetic systems. However, long-range ordered magnetic moments are rather difficult to achieve due to the excited states arising from the relatively weak exchange interactions between the localized moments. Here, using a nanostructured superconductor, we investigate a perfectly ordered magnetic dipole pattern with the magnetic poles having the same distribution as the magnetic charges in an artificial spin ice. The magnetic states can simply be switched on/off by applying a current flowing through nanopatterned area. Moreover, by coupling magnetic dipoles with the pinned vortex lattice, we are able to erase the positive/negative poles, resulting in a magnetic dipole pattern of only one polarity, analogous to the recently predicted vortex ice. These switchable and tunable magnetic dipole patterns open pathways for the study of exotic ordering phenomena in magnetic systems.

Switchable geometric frustration in an artificial-spin-ice-superconductor heterosystem

Geometric frustration emerges when local interaction energies in an ordered lattice structure cannot be simultaneously minimized, resulting in a large number of degenerate states. The numerous degenerate configurations may lead to practical applications in microelectronics¹, such as data storage, memory and logic². However, it is difficult to achieve very high degeneracy, especially in a two-dimensional system^3,4. Here, we showcase in situ controllable geometric frustration with high degeneracy in a two-dimensional flux-quantum system. We create this in a superconducting thin film placed underneath a reconfigurable artificial-spin-ice structure⁵. The tunable magnetic charges in the artificial-spin-ice strongly interact with the flux quanta in the superconductor, enabling switching between frustrated and crystallized flux quanta states. The different states have measurable effects on the superconducting critical current profile, which can be reconfigured by precise selection of the spin-ice magnetic state through the application of an external magnetic field. We demonstrate the applicability of these effects by realizing a reprogrammable flux quanta diode. The tailoring of the energy landscape of interacting 'particles' using artificial-spin-ices provides a new paradigm for the design of geometric frustration, which could illuminate a path to control new functionalities in other material systems, such as magnetic skyrmions⁶, electrons and holes in two-dimensional materials^7,8, and topological insulators⁹, as well as colloids in soft materials^10-13.

Unexpected Phenomenology in Particle-Based Ice Absent in Magnetic Spin Ice

While particle-based ices are often considered essentially equivalent to magnet-based spin ices, the two differ essentially in frustration and energetics. We show that at equilibrium particle-based ices correspond exactly to spin ices coupled to a background field. In trivial geometries, such a field has no effect, and the two systems are indeed thermodynamically equivalent. In other cases, however, the field controls a richer phenomenology, absent in magnetic ices, and still largely unexplored: ice rule fragility, topological charge transfer, radial polarization, decimation induced disorder, and glassiness.

Inner Phases of Colloidal Hexagonal Spin Ice

Using numerical simulations that mimic recent experiments on hexagonal colloidal ice, we show that colloidal hexagonal artificial spin ice exhibits an inner phase within its ice state that has not been observed previously. Under increasing colloid-colloid repulsion, the initially paramagnetic system crosses into a disordered ice regime, then forms a topologically charge ordered state with disordered colloids, and finally reaches a threefold degenerate, ordered ferromagnetic state. This is reminiscent of, yet distinct from, the inner phases of the magnetic kagome spin ice analog. The difference in the inner phases of the two systems is explained by their difference in energetics and frustration.

Freezing and thawing of artificial ice by thermal switching of geometric frustration in magnetic flux lattices

The problem of an ensemble of repulsive particles on a potential-energy landscape is common to many physical systems and has been studied in multiple artificial playgrounds. However, the latter usually involve fixed energy landscapes, thereby impeding in situ investigations of the particles' collective response to controlled changes in the landscape geometry. Here, we experimentally realize a system in which the geometry of the potential-energy landscape can be switched using temperature as the control knob. This realization is based on a high-temperature superconductor in which we engineer a nanoscale spatial modulation of the superconducting condensate. Depending on the temperature, the flux quanta induced by an applied magnetic field see either a geometrically frustrated energy landscape that favours an ice-like flux ordering, or an unfrustrated landscape that yields a periodic flux distribution. This effect is reflected in a dramatic change in the superconductor's magneto-transport. The thermal switching of the energy landscape geometry opens new opportunities for the study of ordering and reorganization in repulsive particle manifolds.

Artificial ferroic systems: novel functionality from structure, interactions and dynamics

Lithographic processing and film growth technologies are continuing to advance, so that it is now possible to create patterned ferroic materials consisting of arrays of sub-1 μm elements with high definition. Some of the most fascinating behaviour of these arrays can be realised by exploiting interactions between the individual elements to create new functionality. The properties of these artificial ferroic systems differ strikingly from those of their constituent components, with novel emergent behaviour arising from the collective dynamics of the interacting elements, which are arranged in specific designs and can be activated by applying magnetic or electric fields. We first focus on artificial spin systems consisting of arrays of dipolar-coupled nanomagnets and, in particular, review the field of artificial spin ice, which demonstrates a wide range of fascinating phenomena arising from the frustration inherent in particular arrangements of nanomagnets, including emergent magnetic monopoles, domains of ordered macrospins, and novel avalanche behaviour. We outline how demagnetisation protocols have been employed as an effective thermal anneal in an attempt to reach the ground state, comment on phenomena that arise in thermally activated systems and discuss strategies for selectively generating specific configurations using applied magnetic fields. We then move on from slow field and temperature driven dynamics to high frequency phenomena, discussing spinwave excitations in the context of magnonic crystals constructed from arrays of patterned magnetic elements. At high frequencies, these arrays are studied in terms of potential applications including magnetic logic, linear and non-linear microwave optics, and fast, efficient switching, and we consider the possibility to create tunable magnonic crystals with artificial spin ice. Finally, we discuss how functional ferroic composites can be incorporated to realise magnetoelectric effects. Specifically, we discuss artificial multiferroics (or multiferroic composites), which hold promise for new applications that involve electric field control of magnetism, or electric and magnetic field responsive devices for high frequency integrated circuit design in microwave and terahertz signal processing. We close with comments on how enhanced functionality can be realised through engineering of nanostructures with interacting ferroic components, creating opportunities for novel spin electronic devices that, for example, make use of the transport of magnetic charges, thermally activated elements, and reprogrammable nanomagnet systems.

Realization of artificial ice systems for magnetic vortices in a superconducting MoGe thin film with patterned nanostructures

We report an anomalous matching effect in MoGe thin films containing pairs of circular holes arranged in such a way that four of those pairs meet at each vertex point of a square lattice. A remarkably pronounced fractional matching was observed in the magnetic field dependences of both the resistance and the critical current. At the half matching field the critical current can be even higher than that at zero field. This has never been observed before for vortices in superconductors with pinning arrays. Numerical simulations within the nonlinear Ginzburg-Landau theory reveal a square vortex ice configuration in the ground state at the half matching field and demonstrate similar characteristic features in the field dependence of the critical current, confirming the experimental realization of an artificial ice system for vortices for the first time.

Hysteresis and return-point memory in colloidal artificial spin ice systems

Using computer simulations, we investigate hysteresis loops and return-point memory for artificial square and kagome spin ice systems by cycling an applied bias force and comparing microscopic effective spin configurations throughout the hysteresis cycle. Return-point memory loss is caused by motion of individual defects in kagome ice or of grain boundaries in square ice. In successive cycles, return-point memory is recovered rapidly in kagome ice. Memory is recovered more gradually in square ice due to the extended nature of the grain boundaries. Increasing the amount of quenched disorder increases the defect density but also enhances the return-point memory since the defects become trapped more easily.

Disorder strength and field-driven ground state domain formation in artificial spin ice: experiment, simulation, and theory

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.

Orbital ice: an exact Coulomb phase on the diamond lattice

We demonstrate the existence of an orbital Coulomb phase as the exact ground state of a p-orbital exchange Hamiltonian on the diamond lattice. The Coulomb phase is an emergent state characterized by algebraic dipolar correlations and a gauge structure resulting from local constraints (ice rules) of the underlying lattice models. For most ice models on the pyrochlore lattice, these local constraints are a direct consequence of minimizing the energy of each individual tetrahedron. On the contrary, the orbital ice rules are emergent phenomena resulting from the quantum orbital dynamics. We show that the orbital ice model exhibits an emergent geometrical frustration by mapping the degenerate quantum orbital ground states to the spin-ice states obeying the 2-in-2-out constraints on the pyrochlore lattice. We also discuss possible realization of the orbital ice model in optical lattices with p-band fermionic cold atoms.

Diversity enabling equilibration: disorder and the ground state in artificial spin ice

We report a novel approach to the question of whether and how the ground state can be achieved in square artificial spin ices where frustration is incomplete. We identify two sources of randomness that affect the approach to ground state: quenched disorder in the island response to fields and randomness in the sequence of driving fields. Numerical simulations show that quenched disorder can lead to final states with lower energy, and randomness in the sequence of driving fields always lowers the final energy attained by the system. We use a network picture to understand these two effects: disorder in island responses creates new dynamical pathways, and a random sequence of driving fields allows more pathways to be followed.