Chiral Hall Effect in Noncollinear Magnets from a Cyclic Cohomology Approach

We demonstrate the emergence of an anomalous Hall effect in chiral magnetic textures which is neither proportional to the net magnetization nor to the well-known emergent magnetic field that is responsible for the topological Hall effect. Instead, it appears already at linear order in the gradients of the magnetization texture and exists for one-dimensional magnetic textures such as domain walls and spin spirals. It receives a natural interpretation in the language of Alain Connes’ noncommutative geometry. We show that this chiral Hall effect resembles the familiar topological Hall effect in essential properties while its phenomenology is distinctly different. Our findings make the reinterpretation of experimental data necessary, and offer an exciting twist in engineering the electrical transport through magnetic skyrmions.

Figure 1

Angular dependence of the chiral Hall effect in the Rashba model. The shown matrix represents the normalized angular magnetization dependence of the components of the CHE which couple to the real-space derivatives ${\partial }_i{\widehat{n}}_j$, where $i\in \{x,y\}$ is the row index and $j\in \{x,y,z\}$ is the column index of the figure ($\hslash {\alpha }_R=1.41\mathrm{eV}\AA$, ${\mathrm{\Delta }}_{\mathrm{xc}}=0.5\mathrm{eV}$, $\mu =0$, $\mathrm{\Gamma }=50\mathrm{meV}$, $m_{\mathrm{eff}}^{*}=3.81m_e$). All shapes have an inversion center resulting from Onsager reciprocity.

Figure 2

Chiral Hall effect in a chiral Heisenberg magnet. A 2D system of side-length $L$ containing $100\times 100$ spins is simulated at $T=1\mathrm{K}$ using the Spirit code. Starting from the polarized state at $H_z=-4\mathrm{T}$, the magnetic field is increased step by step. (a) At $H_z=0.57\mathrm{T}$ the magnetization starts to switch. Shown is the density of the CHE which displays strong resonances at local conical phases which have no topological charge. (b) At $H_z=2.53\mathrm{T}$, the system is dominated by magnetic skyrmions which exhibit a ring-like CHE signature. (c) Tracing the evolution of the average CHE and the topological charge $\mathcal{Q}$ one finds a correlated signal (the corresponding magnetic state is shown in the inset below). Yet, by its nature, the CHE is not caused by $\mathcal{Q}$.