Link of the Zitterbewegung with the spin conductivity and the spin-textures of multiband systems

TL;DR

This work links Zitterbewegung amplitudes in multiband systems to spin transport properties, showing that frequency-dependent spin conductivity and intrinsic spin Hall conductivity are governed by interband ZB amplitudes. By deriving a spin-Kubo framework and a texture-based criterion, the authors provide a practical method to predict which ZB channels are allowed or suppressed based on spin/pseudospin/valley textures, without full dynamical simulations. Applying the approach to three Dirac-type models—the Rashba–Dresselhaus system, Kek-Y graphene, and the α–T3 dice lattice—reveals controlled suppression of specific ZB frequencies and clarifies the role of flat bands and texture symmetries in determining the observable Zitterbewegung. This has potential implications for designing spintronic materials where ZB-related dynamics can be engineered or muted to tailor spin currents and spin-orbit torques.

Abstract

The Zitterbewegung phenomenon in multiband electronic systems is known to be subtly related to the charge conductivity, Berry curvature and the Chern number. Here we show that some spin-dependent properties as the optical spin conductivity, and intrinsic spin Hall conductivity are also entangled with the Zitterbewegung amplitudes. We also show that in multiband Dirac-type Hamiltonians, a direct link between the Zitterbewegung and the spin textures and spin transition amplitudes can be established. The later allow us to discern the presence or not of the Zitterbewegung oscillations by simply analyzing the spin or pseudo-spin textures. We provide examples of the applicability of our approach for Hamiltonian models that show the suppression of specific Zitterbewegung oscillations.

Figures (3)

• Figure 1: (Colour online) (Above) Energy dispersions of the Rashba and Dresselhaus Hamiltonian. (Top left-hand) with different spin-orbit coupling strengths ($\alpha_{\text{R}} \neq\alpha_{\text{D}}$), (top right-hand) with equal spin-orbit coupling strengths ($\alpha_{\text{R}} =\alpha_{\text{D}}$). The symbols $\surd$ in blue and $\bf \times$ in red indicate an allowed and an inhibited Zitterbewegung frequency (spin-transition), respectively. (Bottom) Schematic diagram illustrating the contour plots of the ${\cal E}_-$ and ${\cal E}_+$, both for $\alpha_{\text{R}} =\alpha_{\text{D}}$ at a fix energy $E$; (left) depicting the spin-texture $\langle \bm s\rangle$, and (right-hand) with depiction of the pseudospin texture $\langle \boldsymbol{{\mathcal{S}}}_{\text{R}}+ \boldsymbol{{\mathcal{S}}}_{\text{D}}\rangle$ for each band.
• Figure 2: (Colour online) Energy dispersion diagram of graphene Kek-Y Hamiltonian (left-hand). The symbols $\surd$ in blue and $\bf \times$ in red indicate allowed and forbidden Zitterbewegung frequencies (spin-transitions), respectively. Note that the electron-hole symmetric bands ($E_{++}\leftrightarrow E_{-+}$ and $E_{+-}\leftrightarrow E_{--}$) do not present Zitterbewegung transition amplitudes between them. (Right-hand panels) pseudospin textures for each band at a fix electon/hole energy. (Top) Sublattice pseudospin textures. (Bottom) valley pseudospin textures.
• Figure 3: (Colour online) (Top) Band dispersion for the $\alpha-T_3$ model Hamiltonian with $\alpha=1$. Here, the Zitterbewegung transition amplitudes that are allowed are only these between the electon/hole band to the zero energy flat band $E_0$. Diagrams illustrate the contour plots of the $E_-$ and $E_+$ bands at a fixed electron/hole energy, respectively. The behavior of its pseudospin textures are also shown.