Spin-momentum Locking and Topological Vector Charge Response with Conserved Spin

TL;DR

The paper shows that spin–momentum locking can coexist with conserved, commuting pseudospins constructed from spin and orbitals, leading to a 2D CSR mode with anomalous mixed spin–charge responses. To resolve the intrinsic anomaly, the CSR mode is realized as a boundary state of a 3D Weyl semimetal whose bulk hosts a Weyl spin–momentum quadrupole, producing a mixed Chern–Simons response that cancels the surface anomaly and yields a giant 3D spin Hall effect. The key result is that the bulk quadrupole moment $Q_{iA}$ determines the spin–charge transport coefficients via a term $S_{CS} \propto \epsilon^{\mu\nu\rho\sigma} Q_{\mu A} A_\nu a^A_\rho \partial_\sigma A$, linking surface anomalies to bulk multipole structure. This quasi-topological framework links spintronics concepts to nodal semimetal physics and suggests realizations in metamaterials and engineered quantum materials.

Abstract

Spin-momentum locking plays a fundamental role in spintronics and, more broadly, is an important concept in condensed matter physics. In 2D and 3D, spin-momentum locking typically does not allow spin-conservation because the spin-1/2 operators of electrons anticommute. Instead, here we study spin-momentum locking terms with conserved, commuting pseudospins built from a combination of spin and orbitals. We find that 2D spin-momentum locking terms with conserved pseudospins generally lead to linearly dispersing modes at low-energy with anomalous charge and pseudospin currents. To cure the anomaly we show that such anomalous modes can be realized on the surface of a 3D Weyl semimetal (or an associated weak topological insulator) with a nonzero mixed spin-momentum quadrupole moment, which is determined by the momentum location and pseudospin eigenvalues of Weyl points at the Fermi level. Crucially, this mixed quadrupole moment captures a mixed pseudospin-charge bulk response that cancels the anomaly of surface modes, and can generate a giant 3D spin Hall effect, among other phenomena.

Figures (3)

• Figure 1: Energy spectrum and anomalous responses of $h_{{\mathbf k}}$. (a) A portion of the energy spectrum for eigenstates of $h_{{\mathbf k}}$ with different pseudospins $(s_X,s_Y)$. (b) FS contours of $h_{{\mathbf k}}$. [(c) and (d)] Low-energy spectra and fillings as a function of $k_x$ at a fixed $k_y \,{>}\, 0$ when (c) the gauge fields are zero, (d) the charge $U(1)_{\rm EM}$ gauge field is adiabatically shifted by $2\pi/(eL_x)$, and (e) the spin $U(1)_Y$ gauge field is adiabatically shifted by $2\pi/L_x$. Filled (empty) circles represent occupied (unoccupied) states. The pseudospin sectors $(\,{-}\,,\,{+}\,)$, $(\,{+}\,,\,{+}\,)$, $(\,{+}\,,\,{-}\,)$, and $(\,{-}\,,\,{-}\,)$ are presented by blue, yellow, red, and gray, respectively.
• Figure 2: Tight-binding model $H_{3D}({\mathbf k})$ for 3D semimetal with nonzero $Q_{iA}$. (a) Energy spectra in the $k_z \,{=}\, 0$ plane for $m \,{=}\,-1$. Eight Weyl points are located in pairs at $(\pm \pi/3, \pm \pi/3, 0)$. The representative point $(\pi/3,\pi/3,0)$ is indicated by the filled black dots. (b) [001] surface energy spectrum for a 20-layer slab. The CSR modes (red color) are realized as the surface states of 3D semimetal, [cf. Fig. \ref{['fig1']}(a)]. [(c) and (d)] A pair of (pseudospin) magnetic flux tubes with opposite signs is inserted along the translationally invariant $x$ direction, centered at $(y,z)=(16.5,9.5)$ and $(16.5,24.5)$, in a $32 \times 32$ system along the $y$ and $z$ directions. (c) Pseudospin density $\mathcal{J}^0_X$ induced by magnetic flux $\Phi \,{=}\, \pi/2$. (d) Charge density $J^0$ induced by pseudospin magnetic flux $\varphi_X \,{=}\, \pi/2$. Note that $\widetilde{\mathcal{J}}^0_X \,{=}\, \frac{4\pi^2}{Q_0} \mathcal{J}^0_X$ and $\widetilde{J}^0 \,{=}\, \frac{4\pi^2}{Q_0} J^0$.
• Figure 3: Pair of flux tubes. Flux tubes with magnitudes $-\Phi$ and $\Phi$ are inserted at $(y,z)={\mathbf r}_-$ and ${\mathbf r}_+$, respectively. These flux tubes are stretched along $x$ direction. The system is made with $(N_y,N_z)$ unit cells along the $y$ and $z$ directions where the periodic boundary conditions are imposed. The red line $L$ connects two flux tubes. The Peierls substitution is implemented such that a hopping amplitude $t$ is substituted by $t e^{\mp i\Phi}$ when it passes through $L$ depending on the direction of hopping process $\hat{t}$; $te^{-i\Phi}$ for $\hat{t} \cdot \hat{y}>0$ and $te^{i\Phi}$ for $\hat{t} \cdot \hat{y}<0$. Pseudospin magnetic flux $\mathcal{B}^A_i$ can be implemented in a similar way. For example, a pair of flux tubes for $\mathcal{B}^X_x$ can be described by the following Peierls substitution. For hopping amplitudes pass through $L$, $t$ is substituted by $t e^{i S_X \varphi_X}$ if $\hat{t} \cdot \hat{y}>0$ and $t e^{-i S_X \varphi_X}$ if $\hat{t} \cdot \hat{y}<0$. Here, $\varphi_X$ is the flux of $\mathcal{B}^X_x$. In our numerical calculation in Fig. \ref{['fig2']}, $(N_y,N_z)=(32,32)$, ${\mathbf r}_- = (16.5,9.5)$, and ${\mathbf r}_+ = (16.5,24.5)$.