The Bose–Einstein condensate (BEC) is a fascinating state of matter predicted to occur for particles obeying Bose statistics. Although the BEC has been observed with bosonic atoms in liquid helium and cold gases, the concept is much more general. We here review analogous states, where excitations in magnetic insulators form the BEC. In antiferromagnets, elementary excitations are magnons, quasiparticles with integer spin and Bose statistics. In certain experiments their density can be controlled by an applied magnetic field leading to the formation of a BEC. Furthermore, interactions between the excitations and the interplay with the crystalline lattice produce very rich physics compared with the canonical BEC. Studies of magnon condensation in a growing number of magnetic materials thus provide a unique window into an exciting world of quantum phase transitions and exotic quantum states, with striking parallels to phenomena studied in ultracold atomic gases in optical lattices

This article reviews static and dynamic interfacial effects in magnetism, focusing on interfacially-driven magnetic effects and phenomena associated with spin-orbit coupling and intrinsic symmetry breaking at interfaces. It provides a historical background and literature survey, but focuses on recent progress, identifying the most exciting new scientific results and pointing to promising future research directions. It starts with an introduction and overview of how basic magnetic properties are affected by interfaces, then turns to a discussion of charge and spin transport through and near interfaces and how these can be used to control the properties of the magnetic layer. Important concepts include spin accumulation, spin currents, spin transfer torque, and spin pumping. An overview is provided to the current state of knowledge and existing review literature on interfacial effects such as exchange bias, exchange spring magnets, spin Hall effect, oxide heterostructures, and topological insulators. The article highlights recent discoveries of interface-induced magnetism and non-collinear spin textures, non-linear dynamics including spin torque transfer and magnetization reversal induced by interfaces, and interfacial effects in ultrafast magnetization processes.

Spin ice, a peculiar thermal state of a frustrated ferromagnet on the pyrochlore lattice, has a finite entropy density and excitations carrying magnetic charge. By combining analytical arguments and Monte Carlo simulations, we show that spin ice on the two-dimensional kagome lattice orders in two stages. The intermediate phase has ordered magnetic charges and is separated from the paramagnetic phase by an Ising transition. The transition to the low-temperature phase is of the three-state Potts or Kosterlitz-Thouless type, depending on the presence of defects in charge order. where D = (µ 0 /4π)µ 2 /r 3 nn is a characteristic strength of dipolar coupling, r i are spin locations,r ij = (r i − r j )/|r i − r j |, and r nn is the distance between nearest neighbors. In the absence of dipolar interactions, D = 0, and for ferromagnetic exchange, J > 0, the system is strongly frustrated because it is impossible to minimize the energy of every bond ij . In a ground state, two spins point into every tetrahedron and two point out, which is reminiscent of proton positions in water ice, where every oxygen has two protons nearby and two farther away. This ice rule is satisfied by a macroscopically large number of microstates, so that both protons in water ice and magnetic moments in spin ice can remain disordered even at low temperatures [6].Large magnetic moments (µ = 10µ B in Ho 2 Ti 2 O 7 ) make magnetic dipolar interactions between nearest neighbors comparable to exchange [7]. Together with the long-distance nature of dipolar interactions, the substantial value of D casts doubt on the usefulness of the shortrange (D = 0) model of spin ice. Yet numerical simulations show that, even after the inclusion of dipolar interactions, energy differences between states obeying the ice rule remain numerically small-so small that magnetic order induced by the dipolar interactions is expected to occur only at a rather low temperature, T ≈ 0.13D [8][9][10]. The persistent near-degeneracy of ice ground states in the presence of dipolar interactions was clarified by Castelnovo et al. [3], who introduced a "dumbbell" version of spin ice, in which magnetic dipoles are stretched into bar magnets of length a such that their poles meet at the centers of tetrahedra. The energy of the resulting model can be represented as a Coulomb interaction of magnetic charges of the dumbbells, q i = ±µ/a [3]:In this expression, Q α = i∈α q i is the sum of magnetic charges at the center of tetrahedron α. In a spin-ice state of the dumbbell model, every tetrahedron has two north and two south poles with a total magnetic charge Q α = 0, minimizing the first term in Eq. (2). As a result, no magnetic field will be generated and the magnetic dipolar energy is strictly zero. A partial cancellation occurs in the original model (1), making the Coulomb energy (2) a very good approximation. The charge of tetrahedron α, expressed in units of µ/a, iswith the plus sign for one sublattice of tetrahedra and minus for the other. Residual interactions, responsible for the fo...

We study the effects of magnetoelastic couplings on pyrochlore antiferromagnets. We employ Landau theory, extending an investigation begun by Yamashita and Ueda for the case of S = 1, and semiclassical analyses to argue that such couplings generate bond order via a spin-Peierls transition. This is followed by, or concurrent with, a transition into one of several possible low-temperature Néel phases, with most simply collinear, but also coplanar or mixed spin patterns. In a collinear Néel phase, a dispersionless string-like magnon mode dominates the resulting excitation spectrum, providing a distinctive signature of the parent geometrically frustrated state. We comment on the experimental situation.Geometrically frustrated magnets [1][2][3] are examples of strongly interacting systems: the vast degeneracy of their classical ground states makes them highly susceptible even to small perturbations. By analogy with quantum Hall systems, where the Landau levels are also macroscopically degenerate, one expects a variety of phases in perturbed frustrated magnets, from Néel states to spin glasses or liquids, with valence-bond solids along the way.Probably the world's most frustrated spin system is the classical Heisenberg antiferromagnet on the pyrochlore lattice ( Fig. 1) where spins reside at vertices of tetrahedra. The number of its classical ground states, which are attained when total spin on each tetrahedron S tot = 4 i=1 S i = 0, is so large that, exceptionally, it does not order at any finite temperature [4]. In real compounds, deviations from the classical Heisenberg model (e. g. dipolar interactions, single-ion anisotropy or quantum fluctuations) determine which ground state is selected at the lowest temperatures. FIG. 1. The pyrochlore latticeIn this note, we discuss an elegant mechanism for lifting the frustration through a coupling between spin and lattice degrees of freedom. The high symmetry of the pyrochlore lattice and the spin degeneracy drive a distortion of tetrahedra via a magnetic Jahn-Teller ("spinTeller") effect. The resulting state exhibits a reduction from cubic to tetragonal symmetry and the development of bond order in the spin system with unequal spin correlations S i · S j on different bonds of a tetrahedron. In the ordered phase, there are 4 strong and 2 weak bonds per tetrahedron-or vice versa. This phenomenon was uncovered by Yamashita and Ueda [5] for pyrochlore antiferromagnets with spins S = 1, for which they described an AKLT-style wavefunction [6] with the requisite bond order. In the following we study this phenomenon in the semiclassical limit with added insight from Landau theory, and discuss its consequences for the excitation spectrum. As many of the candidate systems have moderately large spins and order at finite temperatures, our methods should work well-in particular, they allow us to treat the Néel order that can (and experimentally does) appear in addition to the bond order.We begin by identifying a two-component bond order parameter at the level of a single tetrahedron an...

The dynamics of a vortex in a thin-film ferromagnet resembles the motion of a charged massless particle in a uniform magnetic field. Similar dynamics is expected for other magnetic textures with a nonzero skyrmion number. However, recent numerical simulations revealed that skyrmion magnetic bubbles show significant deviations from this model. We show that a skyrmion bubble possesses inertia and derive its mass from the standard theory of a thin-film ferromagnet. Besides center-ofmass motion, other low energy modes are waves on the edge of the bubble traveling with different speeds in opposite directions.Dynamics of topological defects is a topic of longstanding interest in magnetism. The attention to it stems from rich basic physics as well as from its connection to technological applications [1]. Theory of magnetization dynamics in ferromagnets well below the critical temperature is based on the Landau-Lifshitz equation [2] for the unit vector of magnetization m(r) = M(r)/M ,where γ is the gyromagnetic ratio, α ≪ 1 is a phenomenological damping constant [3], and the effective magnetic field is a functional derivative of the free energy, B(r) = −δU/δM(r). The latter includes local (e.g., exchange and anisotropy) as well as long-range (dipolar) interactions, thus making Eq. (1) a nonlinear and nonlocal partial differential equation with multiple length and time scales solvable in only a few simple cases. For example, translational motion of a rigid texture, m = m(r − R(t)), is fully parametrized by the texture's "center of mass" R. For steady motion, R(t) = Vt, the velocity can be obtained from Thiele's equation [4] expressing the balance of gyrotropic, conservative, and viscous forces:Here G is a gyrocoupling vector, F = −∂U/∂R is the net conservative force, and D is a dissipation tensor. A rigid texture moves like a massless particle with electric charge in a magnetic field and an external potential through a viscous medium. If G = 0, the "Lorentz force" greatly exceeds the viscous drag. We thus ignore dissipation. Although Eq. (2) was derived for steady motion, Thiele anticipated that it could serve as a good first approximation in more general situations. Indeed, his equation describes very well the dynamics of vortices in thin ferromagnetic films [5][6][7][8][9]. In this case, the gyrocoupling vector G = (0, 0, G) is proportional to a topological invariant known as the skyrmion charge q = (1/4π) dx dy m · (∂ x m × ∂ y m), the film thickness t, and the density of angular momentum M/γ; to wit, G = 4πqtM/γ. A vortex has q = ±1/2 and thus G = 0. In a parabolic potential well, U (X, Y ) = K(X 2 + Y 2 )/2, it moves in a circle at a frequency ω = K/G. Similar behavior is expected for other topologically nontrivial textures, e.g., magnetic bubbles in thin films with magnetization normal to the plane of the film [10][11][12]. A bubble is a circular domain with m z < 0 surrounded by a domain with m z > 0, or vice versa, Fig.

We express dynamics of domain walls in ferromagnetic nanowires in terms of collective coordinates generalizing Thiele's steady-state results. For weak external perturbations the dynamics is dominated by a few soft modes. The general approach is illustrated on the example of a vortex wall relevant to recent experiments with flat nanowires. A two-mode approximation gives a quantitatively accurate description of both the steady viscous motion of the wall in weak magnetic fields and its oscillatory behavior in moderately high fields above the Walker breakdown.Dynamics of domain walls in nanosized magnetic wires, strips, rings etc. is a subject of practical importance and fundamental interest [1,2]. Nanomagnets typically have two ground states related to each other by the symmetry of time reversal and thus can serve as a memory bit. Switching between these states proceeds via creation, propagation, and annihilation of domain walls with nontrivial internal structure and dynamics. Although domain-wall (DW) motion in macroscopic magnets has been studied for a long time [3], new phenomena arise on the submicron scale where the local (exchange) and long-range (dipolar) forces are of comparable strengths [4]. In this regime, domain walls are textures with a rich internal structure [2,5]. As a result, they have easily excitable internal degrees of freedom. Providing a description of the domain-wall motion in a nanostrip under an applied magnetic field is the main subject of this paper. We specialize to the experimentally relevant case of thin strips with a thickness-to-width ratio t/w ≪ 1.The dynamics of magnetization is described by the Landau-Lifshitz-Gilbert (LLG) equation [6] Here m = M/|M|, H eff (r) = −δU/δM(r) is an effective magnetic field derived from the free-energy functional U [M(r)], γ = g|e|/2mc is the gyromagnetic ratio, and α ≪ 1 is Gilbert's damping constant [7]. Equation (1) can be solved exactly only in a few simple cases. Walker [8] considered a one-dimensional domain wall m = m(x, t) in a uniform external magnetic field H||x.At a low applied field the wall exhibits steady motion, m = m(x − vt), with the velocity v ≈ γH∆/α, where ∆ is the wall width. Above a critical field H W = αM/2 magnetization starts to precess, the wall motion acquires an oscillatory component and the average speed of the wall drops sharply. Qualitatively similar behavior has been observed in magnetic nanostrips [1], however, numerical studies demonstrate that Walker's theory fails to provide a quantitative account of both the steady and oscillatory regimes [2].We formulate the dynamics of a magnetic texture in terms of collective coordinates ξ(t) = {ξ 0 , ξ 1 , . . .}, so that m(r, t) = m(r, {ξ(t)}). Although a magnetization field has infinitely many modes, its long-time dynamics-most relevant to the motion of domain walls-is dominated by a small subset of soft modes with long relaxation times. Focusing on soft modes and ignoring hard ones reduces complex field equations of magnetization dynamics to a much simpler problem. In Walke...

We provide a simple explanation of complex magnetic patterns observed in ferromagnetic nanostructures. To this end we identify elementary topological defects in the field of magnetization: ordinary vortices in the bulk and vortices with half-integer winding numbers confined to the edge. Domain walls found in experiments and numerical simulations in strips and rings are composite objects containing two or more of the elementary defects.Topological defects [1,2] greatly influence the properties of materials by catalyzing or inhibiting the switching between differently ordered states. In ferromagnetic nanoparticles of various shapes (e.g. strips [3] and rings [4]), the switching process involves creation, propagation, and annihilation of domain walls with complex internal structure [5]. Here we show that these domain walls are composite objects made of two or more elementary defects: vortices with integer winding numbers (n = ±1) and edge defects with fractional winding numbers (n = ±1/2). The simplest domain walls are composed of two edge defects with opposite winding numbers. Creation and annihilation of the defects is constrained by conservation of a topological charge. This framework provides a basic understanding of the complex switching processes observed in ferromagnetic nanoparticles.In ferromagnets the competition between exchange and magnetic dipolar energies creates nonuniform patterns of magnetization in the ground state. Whereas the exchange energy favors a state with uniform magnetization, magnetic dipolar interactions align the vector of magnetization with the surface. In a large magnet a compromise is reached by the formation of uniformly magnetized domains separated by domain walls. In a nanomagnet magnetization varies continuously forming intricate yet highly reproducible textures, which include domain walls and vortices [5,6,7]. Numerical simulations [8, 9] reveal a rich internal structure and complex dynamics of these objects. For example, collisions of two domain walls can have drastically different outcomes: complete annihilation or formation of other stable textures [7]. These puzzling phenomena call for a theoretical explanation.An elementary picture of topological defects in nanomagnets with a planar geometry is presented in this Letter. It is suggested that the elementary defects are vortices with integer winding numbers and edge defects with half-integer winding numbers. All of the intricate textures, including the domain walls, are composite objects made of two or more elementary defects.For simplicity, we consider a ferromagnet without intrinsic anisotropy, which is a good approximation for permalloy. The magnetic energy consists of two parts: the exchange contribution A |∇m| d 3 r, wherê m = M/|M| is the unit vector pointing in the direction of magnetization M, and the magnetostatic energy (µ 0 /2) |H| 2 d 3 r. The magnetic field H is related to the magnetization through Maxwell's equations, ∇ × H = 0 and ∇ · (H + M) = 0. Apart from a few special cases (e.g. an ellipsoidal particle), f...

The spin-lattice coupling plays an important role in strongly frustrated magnets. In ZnCr2O4, an excellent realization of the Heisenberg antiferromagnet on the pyrochlore network, a lattice distortion relieves the geometrical frustration through a spin-Peierls-like phase transition at T(c)=12.5 K. Conversely, spin correlations strongly influence the elastic properties of a frustrated magnet. By using infrared spectroscopy and published data on magnetic specific heat, we demonstrate that the frequency of an optical phonon triplet in ZnCr2O4 tracks the nearest-neighbor spin correlations above T(c). The splitting of the phonon triplet below T(c) provides a way to measure the spin-Peierls order parameter.