Ground state lost but degeneracy found: the effective thermodynamics of artificial spin ice

Clocked dynamics in artificial spin ice

Artificial spin ice (ASI) are nanomagnetic metamaterials with a wide range of emergent properties. Through local interactions, the magnetization of the nanomagnets self-organize into extended magnetic domains. However, controlling when, where and how domains change has proven difficult, yet is crucial for technological applications. Here, we introduce astroid clocking, which offers significant control of ASI dynamics in both time and space. Astroid clocking unlocks a discrete, step-wise and gradual dynamical process within the metamaterial. Notably, our method employs global fields to selectively manipulate local features within the ASI. Sequences of these clock fields drive domain dynamics. We demonstrate, experimentally and in simulations, how astroid clocking of pinwheel ASI enables ferromagnetic domains to be gradually grown or reversed at will. Richer dynamics arise when the clock protocol allows both growth and reversal to occur simultaneously. With astroid clocking, complex spatio-temporal behaviors of magnetic metamaterials become easily controllable with high fidelity.

Searching for the ground state of complex spin-ice systems using deep learning techniques

Searching for the ground state of a given system is one of the most fundamental and classical questions in scientific research fields. However, when the system is complex and large, it often becomes an intractable problem; there is essentially no possibility of finding a global energy minimum state with reasonable computational resources. Recently, a novel method based on deep learning techniques was devised as an innovative optimization method to estimate the ground state. We apply this method to one of the most complicated spin-ice systems, aperiodic Penrose P3 patterns. From the results, we discover new configurations of topologically induced emergent frustrated spins, different from those previously known. Additionally, a candidate of the ground state for a still unexplored type of Penrose P3 spin-ice system is first proposed through this study. We anticipate that the capabilities of the deep learning techniques will not only improve our understanding on the physical properties of artificial spin-ice systems, but also bring about significant advances in a wide range of scientific research fields requiring computational approaches for optimization.

Crystallizing Kagome Artificial Spin Ice

Artificial spin ices are engineered arrays of dipolarly coupled nanobar magnets. They enable direct investigations of fascinating collective phenomena from their diverse microstates. However, experimental access to ground states in the geometrically frustrated systems has proven difficult, limiting studies and applications of novel properties and functionalities from the low energy states. Here, we introduce a convenient approach to control the competing diploar interactions between the neighboring nanomagnets, allowing us to tailor the vertex degeneracy of the ground states. We achieve this by tuning the length of selected nanobar magnets in the spin ice lattice. We demonstrate the effectiveness of our method by realizing multiple low energy microstates in a kagome artificial spin ice, particularly the hardly accessible long range ordered ground state-the spin crystal state. Our strategy can be directly applied to other artificial spin systems to achieve exotic phases and explore new emergent collective behaviors.

Reconfigurable training and reservoir computing in an artificial spin-vortex ice via spin-wave fingerprinting

Strongly interacting artificial spin systems are moving beyond mimicking naturally occurring materials to emerge as versatile functional platforms, from reconfigurable magnonics to neuromorphic computing. Typically, artificial spin systems comprise nanomagnets with a single magnetization texture: collinear macrospins or chiral vortices. By tuning nanoarray dimensions we have achieved macrospin-vortex bistability and demonstrated a four-state metamaterial spin system, the 'artificial spin-vortex ice' (ASVI). ASVI can host Ising-like macrospins with strong ice-like vertex interactions and weakly coupled vortices with low stray dipolar field. Vortices and macrospins exhibit starkly differing spin-wave spectra with analogue mode amplitude control and mode frequency shifts of Δf = 3.8 GHz. The enhanced bitextural microstate space gives rise to emergent physical memory phenomena, with ratchet-like vortex injection and history-dependent non-linear fading memory when driven through global magnetic field cycles. We employed spin-wave microstate fingerprinting for rapid, scalable readout of vortex and macrospin populations, and leveraged this for spin-wave reservoir computation. ASVI performs non-linear mapping transformations of diverse input and target signals in addition to chaotic time-series forecasting.

On the degeneracy of spin ice graphs, and its estimate via the Bethe permanent

The concept of spin ice can be extended to a general graph. We study the degeneracy of spin ice graph on arbitrary interaction structures via graph theory. We map spin ice graphs to the Ising model on a graph and clarify whether the inverse mapping is possible via a modified Krausz construction. From the gauge freedom of frustrated Ising systems, we derive exact, general results about frustration and degeneracy. We demonstrate for the first time that every spin ice graph, with the exception of the one-dimensional Ising model, is degenerate. We then study how degeneracy scales in size, using the mapping between Eulerian trails and spin ice manifolds, and a permanental identity for the number of Eulerian orientations. We show that the Bethe permanent technique provides both an estimate and a lower bound to the frustration of spin ices on arbitrary graphs of even degree. While such a technique can also be used to obtain an upper bound, we find that in all finite degree examples we studied, another upper bound based on Schrijver inequality is tighter.

Comparison of Spin-Wave Modes in Connected and Disconnected Artificial Spin Ice Nanostructures Using Brillouin Light Scattering Spectroscopy

Artificial spin ice systems have seen burgeoning interest due to their intriguing physics and potential applications in reprogrammable memory, logic, and magnonics. Integration of artificial spin ice with functional magnonics is a relatively recent research direction, with a host of promising results. As the field progresses, direct in-depth comparisons of distinct artificial spin systems are crucial to advancing the field. While studies have investigated the effects of different lattice geometries, little comparison exists between systems comprising continuously connected nanostructures, where spin-waves propagate via dipole-exchange interaction, and systems with nanobars disconnected at vertices, where spin-wave propagation occurs via stray dipolar field. Gaining understanding of how these very different coupling methods affect both spin-wave dynamics and magnetic reversal is key for the field to progress and provides crucial system-design information including for future systems containing combinations of connected and disconnected elements. Here, we study the magnonic response of two kagome spin ices via Brillouin light scattering, a continuously connected system and a disconnected system with vertex gaps. We observe distinct high-frequency dynamics and magnetization reversal regimes between the systems, with key distinctions in spin-wave localization and mode quantization, microstate trajectory during reversal and internal field profiles. These observations are pertinent for the fundamental understanding of artificial spin systems and broader design and engineering of reconfigurable functional magnonic crystals.

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.

Modal Frustration and Periodicity Breaking in Artificial Spin Ice

Here, an artificial spin ice lattice is introduced that exhibits unique Ising and non-Ising behavior under specific field switching protocols because of the inclusion of coupled nanomagnets into the unit cell. In the Ising regime, a magnetic switching mechanism that generates a uni- or bimodal distribution of states dependent on the alignment of the field is demonstrated with respect to the lattice unit cell. In addition, a method for generating a plethora of randomly distributed energy states across the lattice, consisting of Ising and Landau states, is investigated through magnetic force microscopy and micromagnetic modeling. It is demonstrated that the dispersed energy distribution across the lattice is a result of the intrinsic design and can be finely tuned through control of the incident angle of a critical field. The present manuscript explores a complex frustrated environment beyond the 16-vertex Ising model for the development of novel logic-based technologies.

Energetic analysis of disorder effects in an artificial spin ice with dipolar interactions

We study the effect of quenched disorder in square artificial spin ice by means of numerical simulations. We introduce disorder in the length of magnetic islands using two kinds of distributions: Gaussian and uniform. As the system behavior depends on its geometrical parameters, we focus on studying it in the proximity of the ice regime which is quite difficult to thermalize both in experiments and simulations. We show how length disorder affect the antiferromagnetic and (locally) ferromagnetic ordering, by inducing the system, in the case of weak disorder, to intermediate or mix states. Moreover, in the case of strong disorder, ferromagnetic plaquettes prevail regardless of whether the mean length of the islands corresponds to an antiferromagnetic ordering.

Magnetization dynamics in artificial spin ice

In this topical review, we present key results of studies on magnetization dynamics in artificial spin ice (ASI), which are arrays of magnetically interacting nanostructures. Recent experimental and theoretical progress in this emerging area, which is at the boundary between research on frustrated magnetism and high-frequency studies of artificially created nanomagnets, is reviewed. The exploration of ASI structures has revealed fascinating discoveries in correlated spin systems. Artificially created spin ice lattices offer unique advantages as they allow for a control of the interactions between the elements by their geometric properties and arrangement. Magnonics, on the other hand, is a field that explores spin dynamics in the gigahertz frequency range in magnetic micro- and nanostructures. In this context, magnonic crystals are particularly important as they allow the modification of spin-wave properties and the observation of band gaps in the resonance spectra. Very recently, there has been considerable progress, experimentally and theoretically, in combining aspects of both fields-artificial spin ice and magnonics-enabling new functionalities in magnonic and spintronic applications using ASI, as well as providing a deeper understanding of geometrical frustration in the gigahertz range. Different approaches for the realization of ASI structures and their experimental characterization in the high-frequency range are described and the appropriate theoretical models and simulations are reviewed. Special attention is devoted to linking these findings to the quasi-static behavior of ASI and dynamic investigations in magnonics in an effort to bridge the gap between both areas further and to stimulate new research endeavors.

Magnetization dynamics of weakly interacting sub-100 nm square artificial spin ices

Artificial Spin Ice (ASI), consisting of a two dimensional array of nanoscale magnetic elements, provides a fascinating opportunity to observe the physics of out-of-equilibrium systems. Initial studies concentrated on the static, frozen state, whilst more recent studies have accessed the out-of-equilibrium dynamic, fluctuating state. This opens up exciting possibilities such as the observation of systems exploring their energy landscape through monopole quasiparticle creation, potentially leading to ASI magnetricity, and to directly observe unconventional phase transitions. In this work we have measured and analysed the magnetic relaxation of thermally active ASI systems by means of SQUID magnetometry. We have investigated the effect of the interaction strength on the magnetization dynamics at different temperatures in the range where the nanomagnets are thermally active. We have observed that they follow an Arrhenius-type Néel-Brown behaviour. An unexpected negative correlation of the average blocking temperature with the interaction strength is also observed, which is supported by Monte Carlo simulations. The magnetization relaxation measurements show faster relaxation for more strongly coupled nanoelements with similar dimensions. The analysis of the stretching exponents obtained from the measurements suggest 1-D chain-like magnetization dynamics. This indicates that the nature of the interactions between nanoelements lowers the dimensionality of the ASI from 2-D to 1-D. Finally, we present a way to quantify the effective interaction energy of a square ASI system, and compare it to the interaction energy computed with micromagnetic simulations.

Spontaneous Magnetic Superdomain Wall Fluctuations in an Artificial Antiferromagnet

Collective dynamics often play an important role in determining the stability of ground states for both naturally occurring materials and metamaterials. We studied the temperature dependent dynamics of antiferromagnetically ordered superdomains in a square artificial spin lattice using soft x-ray photon correlation spectroscopy. We observed an exponential slowing down of superdomain wall motion below the antiferromagnetic onset temperature, similar to the behavior of typical bulk antiferromagnets. Using a continuous time random walk model we show that these superdomain walls undergo low-temperature ballistic and high-temperature diffusive motions.

Realization of ground state in artificial kagome spin ice via topological defect-driven magnetic writing

Arrays of non-interacting nanomagnets are widespread in data storage and processing. As current technologies approach fundamental limits on size and thermal stability, enhancing functionality through embracing the strong interactions present at high array densities becomes attractive. In this respect, artificial spin ices are geometrically frustrated magnetic metamaterials that offer vast untapped potential due to their unique microstate landscapes, with intriguing prospects in applications from reconfigurable logic to magnonic devices or hardware neural networks. However, progress in such systems is impeded by the inability to access more than a fraction of the total microstate space. Here, we demonstrate that topological defect-driven magnetic writing-a scanning probe technique-provides access to all of the possible microstates in artificial spin ices and related arrays of nanomagnets. We create previously elusive configurations such as the spin-crystal ground state of artificial kagome dipolar spin ices and high-energy, low-entropy 'monopole-chain' states that exhibit negative effective temperatures.

The theory of spin noise spectroscopy: a review

Direct measurements of spin fluctuations are becoming the mainstream approach for studies of complex condensed matter, molecular, nuclear, and atomic systems. This review covers recent progress in the field of optical spin noise spectroscopy (SNS) with an additional goal to establish an introduction into its theoretical foundations. Various theoretical techniques that have been recently used to interpret results of SNS measurements are explained alongside examples of their applications.

Specific Heat of Square Spin Ice in Finite Point Ising-Like Dipoles Model

In the model of finite number (up to 24) of point Ising-like magnetic dipoles with magnetostatic interaction on square 2D lattice within the framework of statistical physics, with using Gibbs formalism and by the means of Metropolis algorithm the heating dependence of temperature has been evaluated. The temperature dependence of the heat capacity on finite number of point dipoles has the finite value of maximum. Together with increase of the system in size the heating peak grows and moves to the area with higher temperature. The obtained results are useful in experimental verification of statistical models, as well as in development and testing of approximate calculation methods of systems with great number of particles.

Topological solitons in helical strings

The low-energy physics of (quasi)degenerate one-dimensional systems is typically understood as the particle-like dynamics of kinks between stable, ordered structures. Such dynamics, we show, becomes highly nontrivial when the ground states are topologically constrained: a dynamics of the domains rather than on the domains which the kinks separate. Motivated by recently reported observations of charged polymers physio-adsorbed on nanotubes, we study kinks between helical structures of a string wrapping around a cylinder. While their motion cannot be disentangled from domain dynamics, and energy and momentum is not concentrated in the solitons, the dynamics of the domains can be folded back into a particle-like description of the local excitations.

Physics in ordered and disordered colloidal matter composed of poly(N-isopropylacrylamide) microgel particles

This review collects and describes experiments that employ colloidal suspensions to probe physics in ordered and disordered solids and related complex fluids. The unifying feature of this body of work is its clever usage of poly(N-isopropylacrylamide) (PNIPAM) microgel particles. These temperature-sensitive colloidal particles provide experimenters with a 'knob' for in situ control of particle size, particle interaction and particle packing fraction that, in turn, influence the structural and dynamical behavior of the complex fluids and solids. A brief summary of PNIPAM particle synthesis and properties is given, followed by a synopsis of current activity in the field. The latter discussion describes a variety of soft matter investigations including those that explore formation and melting of crystals and clusters, and those that probe structure, rearrangement and rheology of disordered (jammed/glassy) and partially ordered matter. The review, therefore, provides a snapshot of a broad range of physics phenomenology which benefits from the unique properties of responsive microgel particles.

Degeneracy and criticality from emergent frustration in artificial spin ice

Although initially introduced to mimic the spin-ice pyrochlores, no artificial spin ice has yet exhibited the expected degenerate ice phase with critical correlations similar to the celebrated Coulomb phase in the pyrochlore lattice. Here we study a novel artificial spin ice based on a vertex-frustrated rather than pairwise frustrated geometry and show that it exhibits a quasicritical ice phase of extensive residual entropy and, significantly, algebraic correlations. Interesting in its own regard as a novel realization of frustration in a vertex system, our lattice opens new pathways to study defects in a critical manifold and to design degeneracy in artificial magnetic nanoarrays, a task so far elusive.

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.

Perpendicular magnetization and generic realization of the Ising model in artificial spin ice

We have studied frustrated kagome arrays and unfrustrated honeycomb arrays of magnetostatically interacting single-domain ferromagnetic islands with magnetization normal to the plane. The measured pairwise spin correlations of both lattices can be reproduced by models based solely on nearest-neighbor correlations. The kagome array has qualitatively different magnetostatics but identical lattice topology to previously studied artificial spin ice systems composed of in-plane moments. The two systems show striking similarities in the development of moment pair correlations, demonstrating a universality in artificial spin ice behavior independent of specific realization in a particular material system.

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.

Emerging chirality in artificial spin ice

Artificial spin ice, made up of planar nanostructured arrays of simple ferromagnetic bars, is a playground for rich physics associated with the spin alignment of the bars and spin texture associated with the magnetic frustration at the bar vertices. The phase diagram is exotic, showing magnetic monopole-like defects and liquid and solid phases of spins arranged in loop states with predicted chiral order. We show that magnetotransport measurements in connected honeycomb structures yield the onset of an anomalous Hall signal at 50 kelvin. The temperature scale can be attributed to the long-range dipolar ice phase. The topological Hall signal arises because chiral loops form at the sample edges, indicating a generic route to exotic states via nanoarray edge structure.

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.

Thermodynamic limits of dynamic cooling

We study dynamic cooling, where an externally driven two-level system is cooled via reservoir, a quantum system with initial canonical equilibrium state. We obtain explicitly the minimal possible temperature T(min)>0 reachable for the two-level system. The minimization goes over all unitary dynamic processes operating on the system and reservoir and over the reservoir energy spectrum. The minimal work needed to reach T(min) grows as 1/T(min). This work cost can be significantly reduced, though, if one is satisfied by temperatures slightly above T(min). Our results on T(min)>0 prove unattainability of the absolute zero temperature without ambiguities that surround its derivation from the entropic version of the third law. We also study cooling via a reservoir consisting of N≫1 identical spins. Here we show that T(min)∝1/N and find the maximal cooling compatible with the minimal work determined by the free energy. Finally we discuss cooling by reservoir with an initially microcanonic state and show that although a purely microcanonic state can yield the zero temperature, the unattainability is recovered when taking into account imperfections in preparing the microcanonic state.

Dynamics of magnetic charges in artificial spin ice

Artificial spin ice has been recently implemented in two-dimensional arrays of mesoscopic magnetic wires. We propose a theoretical model of magnetization dynamics in artificial spin ice under the action of an applied magnetic field. Magnetization reversal is mediated by domain walls carrying two units of magnetic charge. They are emitted by lattice junctions when the local field exceeds a critical value Hc required to pull apart magnetic charges of opposite sign. Positive feedback from Coulomb interactions between magnetic charges induces avalanches in magnetization reversal.

Effective temperature in an interacting vertex system: theory and experiment on artificial spin ice

Frustrated arrays of interacting single-domain nanomagnets provide important model systems for statistical mechanics, as they map closely onto well-studied vertex models and are amenable to direct imaging and custom engineering. Although these systems are manifestly athermal, we demonstrate that an effective temperature, controlled by an external magnetic drive, describes their microstates and therefore their full statistical properties.

Vertex dynamics in finite two-dimensional square spin ices

Local magnetic ordering in artificial spin ices is discussed from the point of view of how geometrical frustration controls dynamics and the approach to steady state. We discuss the possibility of using a particle picture based on vertex configurations to interpret the time evolution of magnetic configurations. Analysis of possible vertex processes allows us to anticipate different behaviors for open and closed edges and the existence of different field regimes. Numerical simulations confirm these results and also demonstrate the importance of correlations and long-range interactions in understanding particle population evolution. We also show that a mean-field model of vertex dynamics gives important insights into finite size effects.

Creating artificial ice states using vortices in nanostructured superconductors

We demonstrate that it is possible to realize vortex ice states that are analogous to square and kagome ice. With numerical simulations, we show that the system can be brought into a state that obeys either global or local ice rules by applying an external current according to an annealing protocol. We explore the breakdown of the ice rules due to disorder in the nanostructure array and show that in square ice, topological defects appear along grain boundaries, while in kagome ice, individual defects appear. We argue that the vortex system offers significant advantages over other artificial ice systems.