Spin and Orbital Angular Momentum Polarization in Thouless Topological Charge Pumping

TL;DR

This work investigates orbital and spin responses in Thouless topological charge pumping within 1D chiral wires driven by a circularly polarized field. Using time-dependent tight-binding simulations, it demonstrates a single-parameter, screw-symmetric pumping where the energy gap remains open and the pumped charge per cycle is quantized, ΔQ = -e, when the Fermi level lies in a field-induced topological gap. The study reveals a robust nonequilibrium orbital polarization via Berry-phase dynamics, which is partially converted into spin polarization by intrinsic spin–orbit coupling, providing a bulk mechanism related to CISS. It further shows a field-driven topological phase transition via band inversion, yielding a nonzero Chern number in the synthetic (k,φ) space and a finite pumped charge, underscoring the potential for orbital–spin textures to accompany anomalous Hall-like physics in higher dimensions.

Abstract

Quantized charge pumping in one-dimensional chiral wires has been widely studied in the context of topological physics in a (1+1)-dimensional synthetic space, yet the role of orbital and spin degrees of freedom in such topological pumps remains largely unexplored. Here, we examine how the topologically quantized charge pump in insulators generates spin polarizations, and assess whether this mechanism may offer distinct insight into the widely known spin-selective transport in chiral wires-commonly referred to as chirality-induced spin selectivity. We performed time-dependent Schrodinger equations of multi-orbital tight-binding Hamiltonians driven by a circularly polarized electric field. Our main findings are twofold. First, the intrinsic screw-like geometry of the system generates a distinctive winding structure governed by a single control parameter, in contrast to conventional adiabatic pumping mechanisms that require at least two independently modulated parameters, thereby providing a clear interpretation of one-dimensional pumping in terms of the topological structure in a (1+1)-dimensional Brillouin zone. Second, while the energy gap remains open throughout the pumping cycle, the Berry-phase driven real-time dynamics of the charge flow induces a nonequilibrium orbital polarization. Through spin-orbit coupling, this orbital response is partially converted into spin polarization whose direction is determined by the current and chirality. On the analogy between the synthetic (1+1)- and 2-dimensional topological insulators, we suggest that non-trivial spin-orbital dynamics may accompany the anomalous quantum charge Hall states of even-dimensional real materials.

Figures (5)

• Figure 1: $\bm{|}$ Orbital-spin polarized topological charge pumping in a chiral single wire.a, Schematic of chiral-induced spin polarization via topological charge pumping. A low-frequency, circularly polarized electric field $\bm{E}$ is applied to the chiral wire to act as the adiabatic drive. b, As the electron is pumped along the chiral geometry, a nonequilibrium orbital polarization $L_z$ is generated, which is partially converted into spin through the spin-orbit interaction. b, Illustration of the adiabatic modulation of a spatially and temporally varying potential along a 1D chiral wire, shown at several instantaneous moments. Arrowed spheres in a, b, and c indicate the spin of the electron. The size of the sphere in (a) represents the magnitude of each spin.
• Figure 2: $\bm{|}$ Electronic structure of chiral wires and their response to a static transverse electric fielda, Atomic structure of the chiral hydrocarbon C$_{12}$H$_{12}$. b, TB band structure of C$_{12}$H$_{12}$ chiral wire at zero field, showing Dirac crossings protected by sublattice symmetry ($D_{\hat{\mathcal{S}}}$) and screw symmetry ($D_{\hat{\mathcal{Q}}}$). c, Band structure of C$_{12}$H$_{12}$ chiral wire under a static transverse electric field $E_{\perp}=0.2~\mathrm{V\AA^{-1}}$, showing the field-induced gaps. Later, we show that the symmetry-lowering field $E_{\perp}$ drives the emergence of topological gaps $E_{\mathrm{gap}}^{\mathrm{Topological}}$, while the material itself is featured by a large trivial gap $E_{\mathrm{gap}}^{\mathrm{Trivial}}$. d, Atomic structure of a trigonal chiral wire. e, TB band structure of trigonal selenium wire including SOC at zero field.$D_{\hat{\mathcal{Q}}}$ highlights the symmetry-protected crossing at the BZ boundary. f, Band structure under a static transverse electric field $E_{\perp}=0.4~\mathrm{V\AA^{-1}}$, illustrating the gap opening in the selenium chiral wire, with $E_{\mathrm{gap}}^{\mathrm{Topological}}$ and $E_{\mathrm{gap}}^{\mathrm{Trivial}}$ indicated.
• Figure 3: $\bm{|}$ Numerical simulation of Thouless charge pumping in chiral wires.a,d, Right-handed chiral structures of (a) C$_{12}$H$_{12}$ and (d) trigonal Se, together with the applied circularly polarized driving field acting on each system. b, The transported charge per pump cycle $\Delta Q\,(-e)$ in the chiral hydrocarbon C$_{12}$H$_{12}$ as a function of the Fermi energy, driven by the rotating electric field $\bm{E}(\varphi(t))$ with amplitude $E_0 = 0.2~\mathrm{V\AA^{-1}}$ and period $T=800~\mathrm{fs}$. c, Real-time profile of the Thouless pumped charge over one period $T$ with the Fermi level located within the $E_{\mathrm{gap}}^{\mathrm{Topological}}$ of $E_F=-1.5~\mathrm{eV}$. e,f, Same as panels c,d, but for the trigonal chiral wire. The Fermi level (f) is set to $E_F=-1~\mathrm{eV}$ within the topological gap and the driving field amplitude is $E_0 = 0.4~\mathrm{V\AA^{-1}}$.
• Figure 4: $\bm{|}$ Orbital-spin polarization in a chiral wire during Thouless charge pumping.a, Real-time profile of the OAM $L_z(t)$ (red) and its time-averaged value $\bar{L}_z(t)$ (blue) in a RH chiral wire undergoing topological pumping. b, Corresponding SAM dynamics $S_z(t)$ (yellow) and its time-averaged value $\bar{S}_z(t)$ (green) for SOC strength $\lambda = 0.3~\mathrm{eV}$. c,d Time-averaged orbital (c) and spin (d) polarizations as functions of SOC strength. e, The pumped charge $\Delta Q$ over one cycle obtained with various SOC strength. f, Time-averaged spin and orbital obtained after 4 cycles (denoted by $|\bar{L}_z(\infty)|$ (solid squre) and $|\bar{S}_z(\infty)|$ (solid ball).
• Figure 5: $\bm{|}$ Topological phase transition in 1D a chiral wire.a, (i) Schematic of a trigonal chiral wire subjected to a perpendicular electric field $E_\perp$. Static band structures of the chiral wire calculated with parameters $V_{pp\sigma}^{12} = 1.5~\mathrm{eV}$, $V_{pp\pi}^{12} = -0.8~\mathrm{eV}$, and $\lambda = 0.15~\mathrm{eV}$, for field strengths (ii) $E_\perp = 0.25~\mathrm{V\AA^{-1}}$ and (iii) $E_\perp = 0.5~\mathrm{V\AA^{-1}}$. The band gaps are indicated by $E_{gap}$. b, Evolution of the band gap as a function of the perpendicular field strength $E_\perp$. Insets show the Berry-curvature distributions in the synthetic $(k,\varphi)$ space for a rotating electric field with amplitudes $E_0 = 0.25~\mathrm{V\AA^{-1}}$ (top) and $E_0 = 0.5~\mathrm{V\AA^{-1}}$ (bottom). c, Transported charge per pumping cycle obtained from real-time simulations along the chiral wire driven by a rotating electric field with amplitudes $E_0 = 0.25~\mathrm{V\AA^{-1}}$ (orange dashed curve) and $E_0 = 0.5~\mathrm{V\AA^{-1}}$ (violet curve). d, The total Chern number in the synthetic $(k,\varphi)$ space as a function of the field amplitude. e, Left: colormap of the site-resolved probability along a chain of $N=36$ atoms as a function of time, obtained by summing over occupied states with end-mode contributions. White triangles mark the centers of the probability distribution. Right: schematic snapshots of the electron probability amplitude along the chain at three representative times, $t_1$, $t_2$, and $t_3$.