Emergent Magnetic Monopoles Research Articles

Dichotomous dynamics of magnetic monopole fluids

A recent advance in the study of emergent magnetic monopoles was the discovery that monopole motion is restricted to dynamical fractal trajectories [J. N. Hallén et al., Science 378, 1218 (2022)], thus explaining the characteristics of magnetic monopole noise spectra [R. Dusad et al., Nature 571, 234 (2019); A. M. Samarakoon et al., Proc. Natl. Acad. Sci. U.S.A. 119, e2117453119 (2022)]. Here, we apply this novel theory to explore the dynamics of field-driven monopole currents, finding them composed of two quite distinct transport processes: initially swift fractal rearrangements of local monopole configurations followed by conventional monopole diffusion. This theory also predicts a characteristic frequency dependence of the dissipative loss angle for AC field-driven currents. To explore these novel perspectives on monopole transport, we introduce simultaneous monopole current control and measurement techniques using SQUID-based monopole current sensors. For the canonical material Dy2Ti2O7, we measure [Formula: see text], the time dependence of magnetic flux threading the sample when a net monopole current [Formula: see text] is generated by applying an external magnetic field [Formula: see text] These experiments find a sharp dichotomy of monopole currents, separated by their distinct relaxation time constants before and after t ~[Formula: see text] from monopole current initiation. Application of sinusoidal magnetic fields [Formula: see text] generates oscillating monopole currents whose loss angle [Formula: see text] exhibits a characteristic transition at frequency [Formula: see text] over the same temperature range. Finally, the magnetic noise power is also dichotomic, diminishing sharply after t ~[Formula: see text]. This complex phenomenology represents an unprecedented form of dynamical heterogeneity generated by the interplay of fractionalization and local spin configurational symmetry.

Magnetoresistive Evidence of Degeneracy in Nanomagnets Obtained by Electrodeposition Technique

Inspired in pyrochlore materials presenting residual entropy and featuring collective excitation behaving like emergent monopoles, geometrically frustrated arrays of nanomagnets, denominated artificial spin ices (ASIs), were proposed as an interesting platform to investigate such excitation at room temperature. However, in such artificial systems, emergent magnetic monopoles lack the same freedom present their natural counterpart, once energetic strings connecting opposite magnetic charges arise. In this work, we aim to experimentally investigate the proposed degeneracy obtained in connected square arrays of ASIs, a characteristic that allows a reduction in the string connecting monopole–antimonopole pairs in regular non-connected ASIs and could represent an important development for technological applications of connected nanomagnets. As in general those systems are developed by nanofabrication processes involving expensive and time-consuming physical vapour deposition techniques, we also present a new nanofabrication route using an electrodeposition technique for permalloy growth in different lattice geometries as an alternative for fast and low-cost ASI system production.

Macroscopic magnetic monopoles in a 3D-printed mechano-magnet

The notion of magnetic monopoles has puzzled physicists since the introduction of Maxwell’s Equations. Famously, Dirac had hypothesized them in the context of quantum mechanics. While they have proved experimentally elusive as elementary particles, the concept has come to describe excitations or topological defects in various material systems, from liquid crystals, to Hall systems, skyrmion lattices, and Bose–Einstein condensates. Perhaps the most versatile manifestation of magnetic monopoles as quasiparticles in matter has been in so-called spin ice materials. There, they represent violations of the ice rule, carry a magnetic charge, and can move freely. Spin ice emergent magnetic monopoles appear at the atomic scale in rare earth pyrochlores or at the nano-scale in lithographically patterned systems. Here we demonstrate for the first time that the notion of magnetic monopoles can be transported at the macroscopic scale. We have built a mechano-magnet realized via 3D-printing, that consists of mechanical rotors on which macroscopic magnets can pivot. By controlling the relative height of the rotors we can achieve different regimes for magnetic monopoles, including the unbound monopole state. We then explore and characterize their field-driven dynamics in different regimes. Despite of several investigation that can be performed in such a macroscopic system, such as insertion of defects, manipulation of rotor friction or dynamics investigation, in the future, integration of our proof of principle in an elastic matrix can lead to novel macroscopic mechano-magnetic materials, to explore unusual piezomagnetism and magnetostriction, with possible applications to actuators and soft-robotics.

Micromagnetic simulations of emergent monopole defects and magnetization reversal in connected and dipolar Square Artificial Spin Ice

Artificial spin ice (ASI) structures have garnered significant attention due to their unique properties including emergent magnetic monopoles, phase transitions, and collective switching dynamics. In this study, we employ micromagnetic simulations to investigate the magnetic microstates and magnetization reversal dynamics of square ASI with varying connectivity. In particular, we explore an intermediate or a quasi-connected system where the connectivity is altered without significantly changing the strength of dipolar interactions through the introduction of narrow gaps at the vertices. Our results reveal distinct differences in the populations of different vertex types during magnetic reversal for lattices with varying connectivity. These results suggest the possibility of engineering ASI systems in which specific vertex types can be repeatably accessed through adjustments in lattice connectivity.

Micromagnetic Simulations of Emergent Monopole Defects and Magnetization Reversal in Connected and Dipolar Kagome Artificial Spin Ice

Artificial spin ice (ASI) materials, created by lithographically patterning nanoscale magnets in desired geometries, have shown a number of interesting phenomena, such as emergent magnetic monopoles (monopole defects), collective dynamics and phase transitions. The control of monopole defects with external stimuli such as magnetic and electric fields, strain, electric currents etc. would be of much interest in fabrication of future devices. In this work, we investigate the magnetization reversal as a function of the connectivity of the Kagome ASI, using micromagnetic simulations. The domain wall motion is expected to play an important role in the connected spin ice, whereas the dipolar interactions dominate for the unconnected case. The magnetic microstate of each lattice is uniquely determined by the vertex configuration, with the Kagome lattice supporting a six-fold-degenerate vertex state, obeying either a two-in/one-out (vertex charge +q) or one-in/two-out (vertex charge q) ice rule. We analyze the fraction of each vertex type to highlight changes in the magnetic microstate as a function of applied magnetic field. Our results could be correlated with magneto-transport measurements and direct imaging of vertex states in ASI using magnetic force microscopy.

Hidden topological transitions in emergent magnetic monopole lattices

Topological defects, called magnetic hedgehogs, realize emergent magnetic monopoles, which are not allowed in the ordinary electromagnetism described by Maxwell's equations. Such monopoles were experimentally discovered in magnets in two different forms: tetrahedral $4Q$ and cubic $3Q$ hedgehog lattices. The spin textures are modulated by the chemical composition, an applied magnetic field, and temperature, leading to quantum transport and optical phenomena through movement and pair annihilation of magnetic monopoles, but the theoretical understanding remains elusive, especially in the regions where different types of hedgehog lattices are competing. Here we propose a theoretical model that can stabilize both tetrahedral and cubic hedgehog lattices, and perform a thorough investigation of the phase diagram while changing the interaction parameters, magnetic field, and temperature, by using a recently developed method that delivers exact solutions in the thermodynamic limit. We find that the model exhibits various types of topological transitions with changes of the density of monopoles and antimonopoles, some of which are accompanied by singularities in the thermodynamic quantities, while the others are hidden with less or no anomaly. We also find another hidden topological transition with pair annihilation of two-dimensional vortices in the three-dimensional system. These results not only provide useful information for understanding the existing experimental data but also challenge the identification of hidden topological transitions and the exploration of emergent electromagnetism in magnetic monopole lattices.

Real-Space Observations of Three-Dimensional Antiskyrmions and Skyrmion Strings.

Nanometric topological spin textures, such as skyrmions (Sks) and antiskyrmions (antiSks), have attracted much attention recently. However, most studies have focused on two-dimensional spin textures in films with inherent or synthetic antisymmetric spin-exchange interaction, termed Dzyaloshinskii-Moriya interaction, although three-dimensional (3D) topological spin textures, such as antiSks composed of alternating Bloch- and Néel-type spin spirals, chiral bobbers carrying emergent magnetic monopoles, and deformed Sk strings, are ubiquitous. To elucidate these textures, we have developed a 3D nanometric magnetic imaging technique, tomographic Lorentz transmission electron microscopy (TEM). The approach enables the visualization of the 3D shape of magnetic objects and their 3D vector field mapping. Here we report 3D vector field maps of deformed Sk-strings and antiSk using the technique. This research approach will lead to discoveries and understanding of fertile 3D magnetic structures in a broad class of magnets, providing insight into 3D topological magnetism.

Probing emergent QED in quantum spin ice via Raman scattering of phonons: Shallow inelastic scattering and pair production

We present an unconventional mechanism for Raman scattering of phonons, which is based on the linear magnetoelastic coupling present in non-Kramers magnetic ions. This provides a direct coupling of Raman-active phonons to the magnet's quasiparticles. We propose to use this mechanism to probe the emergent magnetic monopoles, electric charges, and photons of the emergent quantum electrodynamics (eQED) of the U(1) quantum spin liquid known as quantum spin ice. Detecting this eQED in candidate rare-earth pyrochlore materials, or indeed signatures of topological magnetic phases more generally, is a challenging task. We show that the Raman scattering cross section of the phonons directly yields relevant information, with the broadening of the phonon linewidth, which we compute, exhibiting a characteristic frequency dependence reflecting the two-particle density of states of the emergent excitations. Remarkably, we find that the Raman linewidth is sensitive to the details of the symmetry fractionalization and hence can reveal information about the projective implementation of symmetry in the quantum spin liquid, thereby providing a diagnostic for a $\ensuremath{\pi}$-flux phase. The Raman scattering of the phonons thus provides a useful experimental tool to probe the fractionalization in quantum spin liquids that turns out to closely mirror pair production in quantum electrodynamics and the deep inelastic scattering of quantum chromodynamics. Indeed, the difference to the latter is conceptual more than technical: the partons (quarks) emerge from the hadrons at high energies due to asymptotic freedom, while those in eQED arise from fractionalization of the spins at low energies.

Thermal conductivity of square ice

We investigate thermal transport in square ice, a two-dimensional analogue of spin ice, exploring the role played by emergent magnetic monopoles in transporting energy. Using kinetic Monte Carlo simulations based on energy preserving extensions of single-spin-flip dynamics, we explicitly compute the (longitudinal) thermal conductivity, $\kappa$, over a broad range of temperatures. We use two methods to determine $\kappa$: a measurement of the energy current between thermal baths at the boundaries, and the Green-Kubo formula, yielding quantitatively consistent values for the thermal conductivity. We interpret these results in terms of transport of energy by diffusion of magnetic monopoles. We relate the thermal diffusivity, $\kappa/C$ where $C$ is the heat capacity, to the diffusion constant of an isolated monopole, showing that the subdiffusive monopole implies $\kappa/C$ vanishes at zero temperature. Finally, we discuss the implications of these results for thermal transport in three-dimensional spin ice, in spin ice materials such as Dy$_2$Ti$_2$O$_7$ and Ho$_2$Ti$_2$O$_7$, and outline some open questions for thermal transport in highly frustrated magnets.

Spin moiré engineering of topological magnetism and emergent electromagnetic fields

A superposition of spin helices can yield topological spin textures, such as skyrmion and hedgehog lattices. Based on the analogy with the moir\'e in optics, we study the magnetic and topological properties of such superpositions in a comprehensive way by modulating the interference pattern continuously. We find that the control of the angles between the superposed helices and the net magnetization yields successive topological transitions associated with pair annihilation of hedgehogs and antihedgehogs. Accordingly, emergent electromagnetic fields, magnetic monopoles and antimonopoles, and Dirac strings arising from the noncoplanar spin textures show systematic evolution. In addition, we also show how the system undergoes the magnetic transitions with dimensional reduction from the three-dimensional hedgehog lattice to a two-dimensional skyrmion lattice or a one-dimensional conical state. The results indicate that the concept of "spin moir\'{e}" provides an efficient way of engineering the emergent electromagnetism and topological nature in magnets.

Control of emergent magnetic monopole currents in artificial spin ice

The control of emergent magnetic monopoles for the generation of monopole currents in artificial spin ice is essential for their use in nanomagnet-based device applications. Here we present a scheme to inject monopole currents into an artificial square ice at specific locations, which provides a means to control the propagation of the generated emergent monopole currents. Specifically, we modify an artificial square ice by populating two of its edges with different vertex configurations consisting of two, three, and four nanomagnets meeting at a common point. After setting an initial state with a global magnetic field, injection of monopoles occurs at one of the edges where the vertices have higher switching probability. We experimentally observe this vertex-specific nucleation of emergent magnetic monopoles using x-ray photoemission electron microscopy. Additionally, we demonstrate that a lateral shift in the reversal of the magnets, leading to the formation of large domains, is consistent with theoretical simulations incorporating higher-order contributions in the magnetic Hamiltonian. Finally, we find a strong correlation between the location of emergent monopole injection and the film thickness, which is a result of the switching probability associated with the different vertex configurations.

Direct Observation of the Statics and Dynamics of Emergent Magnetic Monopoles in a Chiral Magnet.

In the three-dimensional (3D) Heisenberg model, topological point defects known as spin hedgehogs behave as emergent magnetic monopoles, i.e., quantized sources and sinks of gauge fields that couple strongly to conduction electrons, and cause unconventional transport responses such as the gigantic Hall effect. We observe a dramatic change in the Hall effect upon the transformation of a spin hedgehog crystal in a chiral magnet MnGe through combined measurements of magnetotransport and small-angle neutron scattering (SANS). At low temperatures, well-defined SANS peaks and a negative Hall signal are each consistent with expectations for a static hedgehog lattice. In contrast, a positive Hall signal takes over when the hedgehog lattice fluctuates at higher temperatures, with a diffuse SANS signal observed upon decomposition of the hedgehog lattice. Our approach provides a simple way to both distinguish and disentangle the roles of static and dynamic emergent monopoles on the augmented Hall motion of conduction electrons.

Real-Space Observation of Topological Defects in Extended Skyrmion-Strings

Exotic topological spin textures such as emergent magnetic monopole/anti-monopoles (hedgehog/anti-hedgehog) in the metastable extended skyrmion-strings attract much attention to the fundamental physics owing to their novel electromagnetic properties. However, the direct imaging of such spin textures is lacking. Here, we report the real-space observation of emergent magnetic monopoles involved in extended skyrmion-strings by Lorentz transmission electron microscopy (TEM) in combination with micromagnetic simulations. The in-plane extended skyrmion-strings are observed directly by Lorentz TEM to accompany the topological hedgehog-like defect, where the skyrmion-string terminates or merges with another skyrmion-string, as well as the surface-related defects where skyrmion-string bends 90° and ends on the surface. We also demonstrate the transformation of a metastabilized lattice of out-of-plane short skyrmion-strings into an in-plane array of extended skyrmion-strings by tuning the magnitude of oblique fields in a room-temperature helimagnet, revealing the stability of such topological spin textures and the possibility to control them.

Two-dimensional magnetic monopole gas in an oxide heterostructure

Magnetic monopoles have been proposed as emergent quasiparticles in pyrochlore spin ice compounds. However, unlike semiconductors and two-dimensional electron gases where the charge degree of freedom can be actively controlled by chemical doping, interface modulation, and electrostatic gating, there is as of yet no analogue of these effects for emergent magnetic monopoles. To date, all experimental investigations have been limited to large ensembles comprised of equal numbers of monopoles and antimonopoles in bulk crystals. To address these issues, we propose the formation of a two-dimensional magnetic monopole gas (2DMG) with a net magnetic charge, confined at the interface between a spin ice and an isostructural antiferromagnetic pyrochlore iridate and whose monopole density can be controlled by an external field. Our proposal is based on Monte Carlo simulations of the thermodynamic and transport properties. This proposed 2DMG should enable experiments and devices which can be performed on magnetic monopoles, akin to two-dimensional electron gases in semiconductor heterostructures.

Engineering skyrmions and emergent monopoles in topological spin crystals

Spin structures with a non-trivial topology can emerge through the complex interplay of underlying magnetic interactions. Representative examples are magnetic skyrmions and hedgehogs observed in various materials. Although the most typical size of a skyrmion is 10–100 nm, there has been remarkable progress in the discovery of ultra-small (<3 nm) skyrmions and hedgehogs in the last few years. The dense topological spin crystals not only hold promise for technological applications but also provide a good arena to explore gigantic responses from emergent electromagnetic fields or Berry curvature. Here, we review design principles as well as electronic functions of versatile topological spin crystals, highlighting the distinct properties between skyrmion- and hedgehog-lattice states. Among them, unconventional outcomes from hedgehog-lattice states, such as their formation mechanisms and transport properties induced by the emergent magnetic monopoles, are discussed. The manipulation of such topological spin crystals, based on the strong couplings between topology and spin-charge-lattice degrees of freedom, may pave the way for electronics emerging in the near future.

Advances in artificial spin ice

Artificial spin ices consist of nanomagnets arranged on the sites of various periodic and aperiodic lattices. They have enabled the experimental investigation of a variety of fascinating phenomena such as frustration, emergent magnetic monopoles and phase transitions that have previously been the domain of bulk spin crystals and theory, as we discuss in this Review. Artificial spin ices also show promise as reprogrammable magnonic crystals and, with this in mind, we give an overview of the measurements of fast dynamics in these magnetic metamaterials. We survey the variety of geometries that have been implemented, in terms of both the form of the nanomagnets and the lattices on which they are placed, including quasicrystalline systems and artificial spin systems in 3D. Different magnetic materials can also be incorporated to modify anisotropies and blocking temperatures, for example. With this large variety of systems, the way is open to discover new phenomena, and we complete this Review with possible directions for the future. Artificial spin ices are metamaterials displaying fascinating phenomena arising from the collective behaviour of nanoscale magnets. We review recent developments in terms of emergent magnetic monopoles, phase transitions, dynamics and geometries, and discuss future directions for research and potential applications.

Thermally and field-driven mobility of emergent magnetic charges in square artificial spin ice

Designing and constructing model systems that embody the statistical mechanics of frustration is now possible using nanotechnology. We have arranged nanomagnets on a two-dimensional square lattice to form an artificial spin ice, and studied its fractional excitations, emergent magnetic monopoles, and how they respond to a driving field using X-ray magnetic microscopy. We observe a regime in which the monopole drift velocity is linear in field above a critical field for the onset of motion. The temperature dependence of the critical field can be described by introducing an interaction term into the Bean-Livingston model of field-assisted barrier hopping. By analogy with electrical charge drift motion, we define and measure a monopole mobility that is larger both for higher temperatures and stronger interactions between nanomagnets. The mobility in this linear regime is described by a creep model of zero-dimensional charges moving within a network of quasi-one-dimensional objects.

Role of defects in determining the magnetic ground state of ytterbium titanate

Pyrochlore systems are ideally suited to the exploration of geometrical frustration in three dimensions, and their rich phenomenology encompasses topological order and fractional excitations. Classical spin ices provide the first context in which it is possible to control emergent magnetic monopoles, and anisotropic exchange leads to even richer behaviour associated with large quantum fluctuations. Whether the magnetic ground state of Yb2Ti2O7 is a quantum spin liquid or a ferromagnetic phase induced by a Higgs transition appears to be sample dependent. Here we have determined the role of structural defects on the magnetic ground state via the diffuse scattering of neutrons. We find that oxygen vacancies stabilise the spin liquid phase and the stuffing of Ti sites by Yb suppresses it. Samples in which the oxygen vacancies have been eliminated by annealing in oxygen exhibit a transition to a ferromagnetic phase, and this is the true magnetic ground state.

Dipolar Spin Ice States with a Fast Monopole Hopping Rate in CdEr_{2}X_{4} (X=Se, S).

Excitations in a spin ice behave as magnetic monopoles, and their population and mobility control the dynamics of a spin ice at low temperature. CdEr_{2}Se_{4} is reported to have the Pauling entropy characteristic of a spin ice, but its dynamics are three orders of magnitude faster than the canonical spin ice Dy_{2}Ti_{2}O_{7}. In this Letter we use diffuse neutron scattering to show that both CdEr_{2}Se_{4} and CdEr_{2}S_{4} support a dipolar spin ice state-the host phase for a Coulomb gas of emergent magnetic monopoles. These Coulomb gases have similar parameters to those in Dy_{2}Ti_{2}O_{7}, i.e., dilute and uncorrelated, and so cannot provide three orders faster dynamics through a larger monopole population alone. We investigate the monopole dynamics using ac susceptometry and neutron spin echo spectroscopy, and verify the crystal electric field Hamiltonian of the Er^{3+} ions using inelastic neutron scattering. A quantitative calculation of the monopole hopping rate using our Coulomb gas and crystal electric field parameters shows that the fast dynamics in CdEr_{2}X_{4} (X=Se, S) are primarily due to much faster monopole hopping. Our work suggests that CdEr_{2}X_{4} offer the possibility to study alternative spin ice ground states and dynamics, with equilibration possible at much lower temperatures than the rare earth pyrochlore examples.

Statistics of Fluctuations of Magnetic Monopole Concentration in Spin Ice

In this paper, we theoretically investigate the fluctuations in the number of emergent magnetic monopoles in spin ice. The Langevin equation for these fluctuations is deduced. This allows us to calculate the spectrum of relative fluctuations, which can be measured experimentally.