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We derive a generalized set of Ward identities that captures the effects of topological charge on Hall transport. The Ward identities follow from the 2+1 dimensional momentum algebra, which includes a central extension proportional to the topological charge density. In the presence of topological objects like Skyrmions, we observe that the central term leads to a direct relation between the thermal Hall conductivity and the topological charge density. We extend this relation to incorporate the effects of a magnetic field and an electric current. The topological charge density produces a distinct signature in the electric Hall conductivity, which is identified in existing experimental data, and yields further novel predictions. For insulating materials with translation invariance, the Hall viscosity can be directly determined from the Skyrmion density and the thermal Hall conductivity to be measured as a function of momentum.
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The Shastry-Sutherland model and its generalizations have been shown to capture emergent complex magnetic properties from geometric frustration in several quasi-two-dimensional quantum magnets. Using an $sd$ exchange model, we show here that metallic Shastry-Sutherland magnets can exhibit topological Hall effect driven by magnetic skyrmions under realistic conditions. The magnetic properties are modelled with competing symmetric Heisenberg and asymmetric Dzyaloshinskii-Moriya exchange interactions, while a coupling between the spins of the itinerant electrons and the localized moments describes the magnetotransport behavior. Our results, employing complementary Monte Carlo simulations and a novel machine learning analysis to investigate the magnetic phases, provide evidence for field-driven skyrmion crystal formation for extended range of Hamiltonian parameters. By constructing an effective tight-binding model of conduction electrons coupled to the skyrmion lattice, we clearly demonstrate the appearance of topological Hall effect. We further elaborate on effects of finite temperatures on both magnetic and magnetotransport properties.
Motivated by recent experiments on the phonon contribution to the thermal Hall effect in the cuprates, we present an analysis of chiral phonon transport. We assume the chiral behavior arises from a non-zero phonon Hall vicosity, which is likely induced by the coupling to electrons. Phonons with a non-zero phonon Hall viscosity have an intrinsic thermal Hall conductivity, but Chen et al. (Phys. Rev. Lett. 124, 167601 (2020)) have argued that a significantly larger thermal Hall conductivity can arise from an extrinsic contribution which is inversely proportional to the density of impurities. We solve the Boltzmann equation for phonon transport and compute the temperature ($T$) dependence of the thermal Hall conductivity originating from skew scattering off point-like impurities. We find that the dominant source for thermal Hall transport is an interference between impurity skew scattering channels with opposite parity. The thermal Hall conductivity $sim T^{d+2}$ at low $T$ in $d$ dimensions, and has a window of $T$-independent behavior for $T > T_{rm imp}$, where $T_{rm imp}$ is determined by the ratio of scattering potentials with opposite parity. We also consider the role of non-specular scattering off the sample boundary, and find that it leads to negligible corrections to thermal Hall transport at low $T$.
We present a novel approach to understanding the extraordinary diversity of magnetic skyrmion solutions. Our approach combines a new classification scheme with efficient analytical and numerical methods. We introduce the concept of chiral kinks to account for regions of disfavoured chirality in spin textures, and classify two-dimensional magnetic skyrmions in terms of closed domain walls carrying such chiral kinks. In particular, we show that the topological charge of magnetic skyrmions can be expressed in terms of the constituent closed domain walls and chiral kinks. Guided by our classification scheme, we propose a method for creating hitherto unknown magnetic skyrmions which involves initial spin configurations formulated in terms of holomorphic functions and subsequent numerical energy minimization. We numerically study the stability of the resulting magnetic skyrmions for a range of external fields and anisotropy parameters, and provide quantitative estimates of the stability range for the whole variety of skyrmions with kinks. We show that the parameters limiting this range can be well described in terms of the relative energies of particular skyrmion solutions and isolated stripes with and without chiral kinks.
The Skyrme-particle, the $skyrmion$, was introduced over half a century ago and used to construct field theories for dense nuclear matter. But with skyrmions being mathematical objects - special types of topological solitons - they can emerge in much broader contexts. Recently skyrmions were observed in helimagnets, forming nanoscale spin-textures that hold promise as information carriers. Extending over length-scales much larger than the inter-atomic spacing, these skyrmions behave as large, classical objects, yet deep inside they are of quantum origin. Penetrating into their microscopic roots requires a multi-scale approach, spanning the full quantum to classical domain. By exploiting a natural separation of exchange energy scales, we achieve this for the first time in the skyrmionic Mott insulator Cu$_2$OSeO$_3$. Atomistic ab initio calculations reveal that its magnetic building blocks are strongly fluctuating Cu$_4$ tetrahedra. These spawn a continuum theory with a skyrmionic texture that agrees well with reported experiments. It also brings to light a decay of skyrmions into half-skyrmions in a specific temperature and magnetic field range. The theoretical multiscale approach explains the strong renormalization of the local moments and predicts further fingerprints of the quantum origin of magnetic skyrmions that can be observed in Cu$_2$OSeO$_3$, like weakly dispersive high-energy excitations associated with the Cu$_4$ tetrahedra, a weak antiferromagnetic modulation of the primary ferrimagnetic order, and a fractionalized skyrmion phase.
At high magnetic fields, where the Fermi level lies in the N=0 lowest Landau level (LL), a clean two-dimensional electron system (2DES) exhibits numerous incompressible liquid phases which display the fractional quantized Hall effect (FQHE) (Das Sarma and Pinczuk, 1997). These liquid phases do not break rotational symmetry, exhibiting resistivities which are isotropic in the plane. In contrast, at lower fields, when the Fermi level lies in the $Nge2$ third and several higher LLs, the 2DES displays a distinctly different class of collective states. In particular, near half filling of these high LLs the 2DES exhibits a strongly anisotropic longitudinal resistance at low temperatures (Lilly et al., 1999; Du et al., 1999). These stripe phases, which do not exhibit the quantized Hall effect, resemble nematic liquid crystals, possessing broken rotational symmetry and orientational order (Koulakov et al., 1996; Fogler et al., 1996; Moessner and Chalker, 1996; Fradkin and Kivelson, 1999; Fradkin et al, 2010). Here we report a surprising new observation: An electronic configuration in the N=1 second LL whose resistivity tensor simultaneously displays a robust fractionally quantized Hall plateau and a strongly anisotropic longitudinal resistance resembling that of the stripe phases.
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