Unconventional topological Hall effect in skyrmion crystals caused by the topology of the lattice

The hallmark of a skyrmion crystal (SkX) is the topological Hall effect (THE). In this article we predict and explain an unconventional behavior of the topological Hall conductivity in SkXs. In simple terms, the spin texture of the skyrmions causes an inhomogeneous emergent magnetic field whose associated Lorentz force acts on the electrons. By making the emergent field homogeneous, the THE is mapped onto the quantum Hall effect (QHE). Consequently, each electronic band of the SkX is assigned to a Landau level. This correspondence of THE and QHE allows us to explain the unconventional behavior of the THE of electrons in SkXs. For example, a skyrmion crystal on a triangular lattice exhibits a quantized topological Hall conductivity with steps of $2·e^2/h$ below and with steps of $1·e^2/h$ above the van Hove singularity. On top of this, the conductivity shows a prominent sign change $\mathit{at}$ the van Hove singularity. These unconventional features are deeply connected to the topology of the structural lattice.

Figure 1

Core message of the paper. (a) A skyrmion (top hexagon) generates an inhomogeneous emergent magnetic field (central hexagon; blue: positive; white: zero; red: negative). By redistributing this field such that it becomes homogeneous (lower hexagon), the topological Hall effect is mapped onto a quantum Hall effect. (b) Schematic bias dependence of the topological contribution ${\sigma }_{\text{THE}}$ to the Hall conductivity ${\sigma }_{xy}$, exhibiting a sign change at the energy of a van Hove singularity (purple). (c) Magnetic-field dependence of ${\sigma }_{xy}$ for a bias below (blue) and above (red) the van Hove singularity. ${\sigma }_{\text{xy}}$ can show a decrease (blue) as well as an increase (red) in the skyrmion crystal phase which is present for $1\le B\le 3$ in arbitrary units.

Figure 2

Topological Hall effect in SkXs. (a) Electronic band structure of a SkX with skyrmion size $\lambda =3a$ for coupling strength $m=5t$ ($a$ is a lattice constant). The alignment of the electron spin to the magnetic skyrmion texture is indicated by color (parallel: blue; antiparallel: red). (b) Topological Hall conductivity ${\sigma }_{\text{THE}}$ versus Fermi energy for skyrmion sizes $\lambda =3a$ (green), $6a$ (blue), and $9a$ (red). ${\sigma }_{\text{THE}}$ is normalized to the number $n$ of atomic sites per skyrmion unit cell (${\sigma }_0=e^2/h$ conductance quantum). Energy regions with electronlike ($e^{-}$, blue background) and holelike ($h^{+}$, red background) behavior are indicated (see text).

Figure 3

Electronic band structure of a skyrmion crystal and Landau levels. (a) Band structure of a skyrmion crystal (skyrmion size $\lambda =3a$, 12 sites per unit cell). (b) As (a) but with the approximation $cos{\theta }_{ij}/2\to 1$ in Eq. (4). (c) Landau levels for a homogeneous emergent magnetic field. The five topmost bands in (a)–(c) carry Chern number $-1$. The energy of the van Hove singularity is indicated by the purple dashed line.

Figure 4

Quantum Hall effect on the triangular lattice for $n=24$ sites in the unit cell. (a) and (b) The band structure for the triangular lattice without magnetic field is depicted in (b). Cuts at selected energies are labeled $\left(\alpha \right),...,\left(\delta \right)$ and are shown in (a). The numbers ${\zeta }_{\mathrm{e}}$ and ${\zeta }_{\mathrm{h}}$ of enclosed states in the Brillouin zone (black hexagons) has to obey Onsager's quantization scheme. At the van Hove singularity $[E_{\mathrm{VHS}}=-2t$, cut $\left(\gamma \right)]$ the constant-energy contours exhibit a Lifshitz transition. (c) Density of states (DOS) of the band structure in (b) depicted by light smooth curves. The associated Landau levels are shown by dark colors; their Chern numbers $C$ are indicated. (d) Number of enclosed states ${\zeta }_{\mathrm{e}}$ and ${\zeta }_{\mathrm{h}}$ before (light smooth curves) and after Landau quantization (dark). (e) Transverse quantum Hall conductivity in units of ${\sigma }_0=e^2/h$. In all panels, the character of the constant-energy pockets is indicated by color (blue: electronlike; red: holelike).