We derive the spectrum of low-frequency spin waves in skyrmion crystals observed recently in noncentrosymmetric ferromagnets. We treat the skyrmion crystal as a superposition of three helices whose wavevectors form an equilateral triangle. The low-frequency spin waves are Goldstone modes associated with displacements of skyrmions. Their dispersion is determined by the elastic properties of the skyrmion crystal and by the kinetic terms of the effective Lagrangian, which include both kinetic energy and a Berry-phase term reflecting a non-trivial topology of magnetization. The Berryphase term acts like an effective magnetic field, mixing longitudinal and transverse vibrations into a gapped cyclotron mode and a twist wave with a quadratic dispersion.

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.

Artificial spin ice has become a valuable tool for understanding magnetic interactions on a microscopic level. The strength in the approach lies in the ability of a synthetic array of nanoscale magnets to mimic crystalline materials, composed of atomic magnetic moments. Unfortunately, these nanoscale magnets, patterned from metal alloys, can show substantial variation in relevant quantities such as coercive field, with deviations up to 16%. By carefully studying the reversal process of artificial kagome ice, we can directly measure the distribution of coercivities, and by switching from disconnected islands to a connected structure, we find that the coercivity distribution can achieve a deviation of only 3.3%. These narrow deviations should allow the observation of behavior that mimics canonical spin-ice materials more closely.Water ice and spin ice are classic examples of geometrically frustrated systems [1,2], both with residual low-T entropy [3]. In water ice, thermodynamic phases with ordered protons were discovered after decades of experiments [4]. In contrast, no dipole-ordered phase has been observed in spin ice even at the lowest accessible temperatures, contrary to a theoretical prediction [5]. Divergent relaxation times and quenched disorder in samples have been cited as possible explanations. Artificial spin ice has been proposed to help address these questions [6], as it allows the direct control of the geometry of the lattice, with the combined ability to directly image the resulting microstate. Here, samples are composed of lattices of nanoscale ferromagnetic islands, where the magnetization of each element points along its longitudinal axis. At the vertices of the lattice, the ferromagnetic elements interact, and because of the geometry of the system, their magnetic configurations are frustrated [6][7][8][9][10][11]. This allows the study of frustration in systems where crystalline imperfections can be completely removed by design, or introduced in a controlled way. Unfortunately, current lithographic techniques are limited by unintended roughness at edges and interfaces, creating inadvertent disorder. This diminishes the ability to compare observations from artificial spin ice materials with studies of spin ice oxides, where magnetic atoms are presumed to be identical.Edge roughness of nanomagnetic elements is known to substantially influence the coercive field, by creating nucleation sites that can initiate the magnetic reversal [12]. In some artificial spin ice geometries, this edge roughness can create a large variability in the behavior of the artificial "atoms" (magnetic nano-islands). In recent studies of artificial kagome ice, the variations in coercivity were found to be substantial-up to 16% of the average coercive value [8,9,13]. This variability can easily be reduced by choosing materials with low crystal anisotropy [9], but we here show substantial further reduction with a geometry with connected magnetic islands. In a connected geometry, nontrivial spin textures (domain walls) alread...

We model the dynamics of magnetization in an artificial analogue of spin ice specializing to the case of a honeycomb network of connected magnetic nanowires. The inherently dissipative dynamics is mediated by the emission and absorption of domain walls in the sites of the lattice, and their propagation in its links. These domain walls carry two natural units of magnetic charge, whereas sites of the lattice contain a unit magnetic charge. Magnetostatic Coulomb forces between these charges play a major role in the physics of the system, as does quenched disorder caused by imperfections of the lattice. We identify and describe different regimes of magnetization reversal in an applied magnetic field determined by the orientation of the applied field with respect to the initial magnetization. One of the regimes is characterized by magnetic avalanches with a 1/n distribution of lengths.

We study the gapped phase of Kitaev's honeycomb model (a Z2 spin liquid) on a lattice with topological defects. We find that some dislocations and string defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a Z2 gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a Z2 vortex around one of the dislocations. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes. To facilitate these theoretical developments, we recast the degenerate perturbation theory for spins in the language of Majorana fermions.

We consider the effect of adding quantum dynamics to a classical topological spin liquid, with particular view to how best to detect its presence in experiment. For the Coulomb phase of spin ice, we find quantum effects to be most visible in the gauge-charged monopole excitations. In the presence of weak dilution with nonmagnetic ions we find a particularly crisp phenomenon, namely the emergence of hydrogenic excited states in which a magnetic monopole is bound to a vacancy at various distances. Via a mapping to an analytically tractable single particle problem on the Bethe lattice, we obtain an approximate expression for the dynamic neutron scattering structure factor.Comment: 4 pages, 4 figures; supplemental material: 3 pages, 2 figure

We study the gapped phase of Kitaev's honeycomb model (a Z2 spin liquid) in the presence of lattice defects. We find that some dislocations and bond defects carry unpaired Majorana fermions. Physical excitations associated with these defects are (complex) fermion modes made out of two (real) Majorana fermions connected by a Z2 gauge string. The quantum state of these modes is robust against local noise and can be changed by winding a Z2 vortex around a dislocation. The exact solution respects gauge invariance and reveals a crucial role of the gauge field in the physics of Majorana modes.In three dimensions all particles can be divided into two categories: bosons and fermions. In two dimensions, particles can exhibit statistics that interpolates continuously between Bose and Fermi's, hence the name anyons. When two Abelian anyons are exchanged, the system's wavefunction picks up a phase that is not restricted to integer multiples of π as it is in three dimensions. NonAbelian statistics arises when the ground state of a system is degenerate and winding one particle around another amounts to a unitary transformation in the space of degenerate ground states. The nonlocal nature of the winding process makes the evolution of the quantum ground state insensitive to local noise. Therefore non-Abelian anyons could provide a pathway to faulttolerant quantum computing. One of the main obstacles is the scarcity of systems with non-Abelian excitations. In this Letter, we discuss a scenario in which the addition of topological defects to a system with Abelian anyons can give rise to non-Abelian statistics.An exactly solvable spin model with Abelian anyon excitations was constructed by Kitaev [1]. The model has one gapless and three gapped phases at zero temperature. The gapped phases have a Z 2 topological order, which yields Abelian anyonic statistics. Kitaev sketched a heuristic argument that a dislocation in his model should harbor an unpaired Majorana mode. Two such modes can be combined into a zero-energy fermion. The ground state of a system with a pair of distant dislocations is then doubly degenerate and can be manipulated by winding a vortex around one of the dislocations [2]. In this Letter, we characterize the properties of dislocations in Kitaev's spin model and confirm his conjecture by an explicit construction of the Majorana modes.Kitaev's model has spins 1/2 living on sites of a honeycomb lattice. They interact with their nearest neighbors through anisotropic exchange whose nature depends on the direction of the bond:

We present a three-dimensional cubic lattice spin model, anisotropic in theẑ direction, that exhibits fracton topological order. The latter is a novel type of topological order characterized by the presence of immobile pointlike excitations, named fractons, residing at the corners of an operator with two-dimensional support. As other recent fracton models, ours exhibits a subextensive ground state degeneracy: On an Lx × Ly × Lz three-torus, it has a 2 2Lz topological degeneracy, and an additional non-topological degeneracy equal to 2 LxLy −2 . The fractons can be combined into composite excitations that move either in a straight line along theẑ direction, or freely in the xy plane at a given height z. While our model draws inspiration from the toric code, we demonstrate that it cannot be adiabatically connected to a layered toric code construction. Additionally, we investigate the effects of imposing open boundary conditions on our system. We find zero energy modes on the surfaces perpendicular to either thex orŷ directions, and their absence on the surfaces normal toẑ. This result can be explained using the properties of the two kinds of composite twofracton mobile excitations. arXiv:1709.10094v1 [cond-mat.str-el]