Torsional chiral magnetic effect due to skyrmion textures in a Weyl superfluid ${}^3\mathrm{He}\text{−}\mathrm{A}$

We investigate torsional chiral magnetic effect (TCME) induced by skyrmion-vortex textures in the A phase of the superfluid ${}^3\mathrm{He}$. In ${}^3\mathrm{He}\text{−}\mathrm{A}$, Bogoliubov quasiparticles around point nodes behave as Weyl fermions, and the nodal direction represented by the $\ell$ vector may form a spatially modulated texture. $\ell$ textures generate a chiral gauge field and a torsion field directly acting on the chirality of Weyl-Bogoliubov quasiparticles. It has been clarified by Volovik [Pi'sma Zh. Eksp. Teor. Fiz. 43, 428 (1986)] that, if the $\ell$ vector is twisted as $\widehat{\mathbf{\ell }}·\left(\mathrm{curl}\widehat{\mathbf{\ell }}\right)\ne 0$, the chiral gauge field is responsible for the chiral anomaly, leading to an anomalous current along $\mathbf{\ell }$. Here we show that, even if $\widehat{\mathbf{\ell }}·\left(\mathrm{curl}\widehat{\mathbf{\ell }}\right)=0$, a torsion arising from $\ell$ textures brings about contributions to the equilibrium currents of Weyl-Bogoliubov quasiparticles along $\mathrm{curl}\mathbf{\ell }$. This implies that while the anomalous current appears only for the twisted (Bloch-type) skyrmion of the $\ell$ vector, the extra mass current due to TCME always exists regardless of the skyrmion type. Solving the Bogoliubov–de Gennes equation, we demonstrate that both Bloch-type and Néel-type skyrmions induce chiral fermion states with spectral asymmetry, and possess spatially inhomogeneous structures of Weyl bands in the real coordinate space. Furthermore, we discuss the contributions of Weyl-Bogoliubov quasiparticles and continuum states to the mass current density in the vicinity of the topological phase transition. In the weak-coupling limit, continuum states give rise to backflow to the mass current generated by Weyl-Bogoliubov quasiparticles, which makes a non-negligible contribution to the orbital angular momentum. As the topological transition is approached, the mass current density is governed by the contribution of continuum states.

Figure 1

Real-space skyrmion $\ell$ textures and “torsional” magnetic fields acting on Wely-Bogoliubov quasiparticles, ${\mathit{T}}^{\overline{3}}=\mathrm{curl}\widehat{\mathbf{\ell }}$: (a) Néel-type skyrmion vortex with $\alpha =0$ and (b) Bloch-type skyrmion with $\alpha =\pi /2$ [see Eq. (27)]. The former texture has ${\mathit{j}}^{\mathrm{TCME}}\propto {\mathit{T}}^{\overline{3}}$ and ${\mathit{j}}^{\mathrm{an}}=\mathit{0}$, while the latter has nonvanishing ${\mathit{j}}^{\mathrm{TCME}}\propto {\mathit{T}}^{\overline{3}}$ and ${\mathit{j}}^{\mathrm{an}}\propto \widehat{\mathbf{\ell }}$.

Figure 2

Bogoliubov quasiparticle spectra for Néel-type skyrmion ($\alpha =0$) at $E_n(m,k=0)$ (a) and $E_n(m=0,k)$ (b). (c) $k$-resolved zero-energy local density of states, $N(k,r,E=0)$, representing the spatial distribution of the chiral fermion. We fix $R=200k_{\mathrm{F}}^{-1}$ and $p_{\mathrm{F}}\xi =20$.

Figure 3

Bogoliubov quasiparticle spectra for twisted skyrmion $\left[\alpha =\pi /2(1-r/R)\right]$ at $E_n(m,k=0)$ (a) and $E_n(m=0,k)$ (b). We fix $R=200k_{\mathrm{F}}^{-1}$ and $p_{\mathrm{F}}\xi =20$.

Figure 4

(a) Current density in the Néel-type skyrmion vortex, where $n$ is the particle density. (b) $E$-resolved current density profile, $j_{\theta }(r,E)$, for $k_{\mathrm{F}}\xi =20$. Here we fix $T=0$, $R=200k_{\mathrm{F}}^{-1}$, and $\mu =E_{\mathrm{F}}$.

Figure 5

$p_{\mathrm{F}}\xi$ dependence of the total angular momentum at $T=0$: $L_z/N$ (solid line), $L_z^{\mathrm{Weyl}}/N$ (dashed line), and $L_z^{\mathrm{cont}}/N$ (dashed-dotted line). $N$ is the total particle number, $N=\int n\left(\mathit{r}\right)d\mathit{r}$, and $L_z^{\mathrm{Weyl}}$ stands for the contributions of low-lying Weyl-Bogoliubov quasiparticles within $|E_n(m,k)|<{\mathrm{\Delta }}_{\mathrm{A}}$ to the total angular momentum.

Figure 6

(a),(b) $T$ dependence of total angular momentum $L_z\left(T\right)$ for the Néel-type skyrmion vortex (red line): (a) $p_{\mathrm{F}}\xi =20$ and (b) 2.0. (c),(d) $T$ dependence of Weyl-Bogoliubov quasiparticle contributions, $L_z^{\mathrm{Weyl}}\left(T\right)$ (red line), for $p_{\mathrm{F}}\xi =10$ (a) and 2.0 (b). In all figures, the solid lines are the components of the superfluid density tensor parallel and perpendicular to $\widehat{\mathbf{\ell }}$, ${\rho }_{\parallel }$ and ${\rho }_{\perp }$. The dashed line corresponds to ${\rho }_{\mathrm{ave}}$, where ${\rho }_{\mathrm{ave}}=L_z^{\mathrm{Cross}}\left(T\right)/L_z^{\mathrm{Cross}}\left(0\right)$ for $p_{\mathrm{F}}\xi \gg 1$.

Figure 7

Current densities in the twisted skyrmion-vortex state at $T=0$ and $R=200k_{\mathrm{F}}^{-1}$: (a) azimuthal current $j_{\theta }\left(r\right)/n$ and (b) axial current $j_z\left(r\right)/n$. The particle number density $n$ is obtained from the eigenstates of the BdG equation. (c),(d) Rescaled current densities, where $n^{*}$ is a fitting parameter to rescale $j_{\mu }\left(r\right)$ to $j_{\mu }^{\mathrm{MM}}\left(R\right)=j_{\mu }^{\mathrm{IMU}}\left(R\right)$.