Spin and Orbital Angular Momentum Polarization in Thouless Topological Charge Pumping

Abstract

Quantized charge pumping in one-dimensional chiral wires has been widely studied in the context of topological physics in (1 + 1)-dimensional synthetic space, yet the role of orbital and spin degrees of freedom remains largely unexplored. Here, we show that topological charge pumping in insulating chiral systems intrinsically generates orbital and spin polarization, providing a new perspective on spin-selective transport in chiral materials, often associated with chirality-induced spin selectivity. Using time-dependent Schrödinger dynamics of multiorbital tight-binding models driven by circularly polarized light, we identify two key results. First, the screw-like geometry enables a single-parameter topological charge pumping. Second, while the energy gap remains open throughout the pumping cycle, Berry phase driven dynamics induces nonequilibrium orbital polarization. Through spin–orbit coupling, this orbital response is partially converted into spin polarization. By analogy between synthetic (1 + 1)- and two-dimensional topological insulators, we suggest that nontrivial spin–orbital dynamics may accompany anomalous quantum charge Hall states.

The quantum pump, as a dynamical manifestation of topological phases, has attracted significant interest for its fundamental physics and perspective applications of topological charge transport. − The underlying theoretical foundations can be traced back to Laughlin, whose explanation of the quantum Hall effect in terms of a cylindrical geometry with a varying magnetic flux directly inspired the concept of topological pumping. Similarly, as demonstrated by Thouless, the quantized charge transport in a cyclic one-dimensional (1D) Hilbert space can be mathematically mapped to the two-dimensional (2D) quantized Hall conductance. Beyond this mathematical analogy between 1D and 2D, efforts have been made to realize the Thouless pump by driving a one-dimensional Hamiltonian with a time-periodic potential, thereby forming an effective (1 + 1)-dimensional system that uses time as a synthetic second dimension. , In this context, various experimental platforms have been explored, including photonic, magneto-mechanical, electro-mechanical, acoustic, and ultracold atom systems. − In pure condensed matter settings, however, the one-dimensional charge pump has not yet been well realized experimentally and has largely remained at the level of model Hamiltonians, such as the Rice-Mele model. On the road toward the real-material realization of topological Thouless charge pumping, valuable clues can be found in previous studies of one- dimensional chiral materials, such as DNA and α-helical proteins, which have been shown to host topological states when subjected to a low-frequency rotating electric field. ,

While these developments have so far primarily focused on topology-driven charge transport, the role of orbital degree of freedom and the possibility of orbital-induced phenomena remains comparatively unexplored. Traditionally, orbital angular momentum (OAM) in crystalline solids has been regarded as largely quenched by crystal-field effects. However, recent studies have challenged this view by demonstrating that OAM can be nontrivially effective, leading to a variety of orbital-related responses in solids. − In particular, the locking between linear momentum and angular momentum (spin or orbital) has been shown to play a fundamental role in several transport phenomena, including the Edelstein effect, − the spin Hall and orbital Hall effects, and the valley orbital Hall effect. , Notably, recent studies have also explored orbital-spin conversion mechanisms mediated by topological phase transitions. For chiral one-dimensional materials, the orbital-momentum texture is also known to be critically significant, and the particular geometric chacter is thought to provide fertile ground for various chirallity-driven orbital phenomena. , Among the chirality-driven effects that have recently attracted significant attention, a particularly intriguing example is chirality-induced spin selectivity (CISS). − In this context, the OAM polarization has increasingly become attractive, being recognized as a fundamental ingredient for various orbital-based phenomena.

In this work, using dynamical time-dependent tight-binding (TB) simulations together with density functional theory calculations, we demonstrate that quantized Thouless charge pumping can be realized in real chiral materials. We show that the intrinsic chiral feature of the wire impose a distinct winding structure of the Bloch states from the conventional two-parameter Rice-Mele pump, thereby reducing the realization of topological Thouless pumping to a single control parameter. We trace the real-time dynamics and examine the role of orbital and spin degrees of freedom in such topological pumps. We show that the resulting topological charge transport is accompanied by a finite nonequilibrium orbital polarization, arising from the momentum-dependent orbital texture of the chiral band structure. Through the local SOC, this orbital response is partially converted into spin angular momentum (SAM) polarization, which can be considered as a topology-based CISS effect in gapped chiral systems. Our findings establish chiral wires as realistic platforms for orbital- and spin-polarized Thouless charge pumping, and motivate a broader search for Berry-phase-driven spin–orbital textures in chiral quantum materials and equivalent anomalous Hall systems.

Results

Adiabatic modulation of the energy profile along a 1D chiral geometry provides a natural route for achieving directional transport of particles (see schematic Figure ). In the quantum dynamics, the corresponding sliding potential minimum not only transports a trapped particle but also allows its wave function to acquire a geometric phase. As shown by Thouless, the accumulated phase over a full cycle leads to quantized transport, with the pumped charge determined by topological singularities enclosed in the parameter space. Based on this idea, many proposed topological pumps utilize time-dependent superlattices to simulate the Rice-Mele model, ,,, which requires at least two sets of independently tunable parameters. Such modulation schemes are typically incompatible with solid-state systems, where the lattice structure is rigid and dynamic control demands complex experimental architectures.

1.

|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 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. c, 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.

We demonstrate that topological Thouless pumping can be realized in a real material with chiral structure using a single periodic parameter, enabled by a fundamentally distinct topological winding structure compared to the Rice-Mele model. Moreover, we reveal that the robust topological charge pumping is accompanied by dynamical orbital polarization, which arises from the coupling between the chiral geometry and the external polarized field (Figure b,c). We then show that the intrinsic SOC of the chiral system further acts as a transducer, converting this orbital flow into spin-polarized transport, thereby realizing a nonmagnetic mode of spin transport in nontrivial insulator chiral wires (Figure a–c).

Electronic Properties of 1D Chiral Wires

Chiral structures are known to host nontrivial momentum-dependent orbital and spin textures in their electronic band structures. , To capture the essential features of the electronic structure and the orbital-spin textures of a chiral wire, we employ a three-orbital TB model in the basis {↑, ↓}⊗{A ₁, A ₂, A ₃, ... A _N }⊗{p _x , p _y , p _z }, where N denotes the number of atoms in the unit cell. We introduce the reference screw-symmetric TB Hamiltonian ${Ĥ}_0$ , including the local atomic SOC as (see Supplementary Note 1 for the details):

$${Ĥ}_0=\sum _{ij,\mu \nu ,\alpha }{t}_{i\mu ,j\nu }{c}_{i\mu \alpha }^{†}{c}_{j\nu \alpha }+{\lambda }_{\mathrm{S}\mathrm{O}\mathrm{C}}\sum _{i,\mu \nu ,\alpha \beta }{c}_{i\mu \alpha }^{†}({\mathit{\sigma ̂}}^{\alpha \beta }·{\mathit{L̂}}^{i,\mu \nu })c_{i\nu \beta }$$ 1

Here, t _iμ,jν denotes the spin-independent hopping amplitude between the basis orbitals, and ${\mathit{\sigma ̂}}^{\alpha \beta }$ are the vector of Pauli matrices acting in the spin space. Also, ${\mathit{L̂}}^{i,\mu \nu }$ represents the OAM matrix elements (in ℏ units) between orbitals μ and ν on the same site i, and λ_SOC is the SOC strength. Using eq S7, the TB Hamiltonian in the momentum space can be written in the compact form

$${Ĥ}_0(k)={Î}_{2\times 2}^{(\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n})}\otimes \mathrm{T̂}+{\lambda }_{\mathrm{S}\mathrm{O}\mathrm{C}}\mathit{\sigma ̂}·\mathbf{L̂}$$ 2

where $\mathrm{T̂}(k)$ represents the spin-independent hopping matrix. For a 1D wire with an N-fold screw symmetry, the corresponding screw operator is defined as

$${\mathcal{Q̂}}_N={\mathrm{T̂}}_z(a/N){\mathrm{R̂}}_z(2\pi /N)$$ 3

which combines a translation by a/N along the chiral axis with a rotation of θ = 2π/N about the z axis. In Supplementary Note 2 and 3, we show how the Bloch eigenstates of eq , denoted by |ψ _n,k ⟩, transform under the relevant symmetry operations. As a result, the orbital and spin textures satisfy the constraint ${\mathrm{\Lambda }}_i(n,k)=⟨{\psi }_{n,k}|{\mathrm{\Lambda ̂}}_i|{\psi }_{n,k}⟩=0$ for i = x, y at generic k, where Λ = L, S. By contrast, the out-of-plane component Λ _z (n, k) is not symmetry constrained. Although the lack of additional lattice symmetries allows finite orbital and spin components along the screw axis, time-reversal symmetry enforces the antisymmetric momentum dependence L _z (n, k) = −L _z (n, – k) and S _z (n, k) = −S _z (n, – k).

In Supplementary Note 4 (Figure S1 and Figure S2), we present and confirm that the electronic band structure and the spin–orbital textures of this three-orbital TB model are farily consistent with those of ab initio DFT calculations.

Electronic properties of 1D chiral wires under an electric field

Trigonal-stacked chiral materials such as Te and Se have been widely synthesized, − and a broad variety of chiral organic molecules have also been extensively investigated. − , However, the realization of topological Thouless charge pumping in such chiral structures and in particular, the resulting orbital and spin responses during topological transport remains largely unexplored. In the present study, to factorize out the effect of SOC, we consider chiral hydrocarbon C₁₂H₁₂ (Figure a) as a model chiral system which enables us to investigate purely geometric contributions during topological charge pumping. This 1D chiral wire inherits an extended sublattice symmetry from its carbon-nanotube precursor and additionally hosts a 6-fold screw symmetry, which together enforce characteristic band degeneracies in its electronic structure. As displayed in Figure b, the orbital resolved band structure of this spinless chiral system shows the expected symmetry-protected crossings. In the TB formulation (see Supplementary Note 5), one may adopt the basis {A ₁, ..., A ₆, B ₁, ..., B ₆}⊗{p _x , p _y , p _z }, (Figure a) which makes the two underlying symmetries discussed above particularly transparent. In this basis, the Hamiltonian can be written in the off-diagonal form $Ĥ={\sigma }_{+}\otimes {Ĥ}^{AB}+{\sigma }_{-}\otimes {Ĥ}^{BA}$ where σ_± are the raising and lowering Pauli matrices acting on the extended sublattice space. Using the sublattice operator $\mathcal{Ŝ}={\sigma }_z$ and the identity σ _z σ_±σ _z = – σ_±, one directly obtains the sublattice-symmetry relation $\mathcal{Ŝ}Ĥ{\mathcal{Ŝ}}^{-1}=-Ĥ$ . As shown, the crossing at the Γ point is protected by this sublattice symmetry (as denoted by $D_{\mathcal{Ŝ}}$ in Figure b). In addition, the nonsymmorphic 6-fold screw symmetry acting within each extended sublattice enforces Dirac-type $(D_{\mathcal{Q̂}})$ crossings at the Brillouin-zone boundary ± Z.

2.

Electronic structure of chiral wires and their response to a static transverse electric field. a, Atomic structure of the chiral hydrocarbon C₁₂H₁₂. b, TB band structure of C₁₂H₁₂ chiral wire at zero field, showing Dirac crossings protected by sublattice symmetry ( $D_{\mathcal{Ŝ}}$ ) and screw symmetry ( $D_{\mathcal{Q̂}}$ ). c, Band structure of C₁₂H₁₂ chiral wire under a static transverse electric field $E_{\perp }=0.2\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ , showing the field-induced gaps. Later, we show that the symmetry-lowering field E _⊥ drives the emergence of topological gaps ${\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}$ , while the material itself is featured by a large trivial gap ${\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{l}}$ . d, Atomic structure of a trigonal chiral wire. e, TB band structure of trigonal selenium wire including SOC at zero field. $D_{\mathcal{Q̂}}$ highlights the symmetry-protected crossing at the BZ boundary. f, Band structure under a static transverse electric field $E_{\perp }=0.4\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ , illustrating the gap opening in the selenium chiral wire, with ${\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}$ and ${\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{l}}$ indicated.

When a static electric field E _⊥ is applied (Figure a), its length-gauge coupling is effectively captured by the onsite dipole term $-e{\mathbf{E}}_{\perp }·\mathbf{r̂}$ , where $\mathbf{r̂}$ denotes the intracell position operator. Consequently, the A/B extended sublattices experience different onsite energy shifts, explicitly breaking the sublattice symmetry. Moreover, because $\mathbf{r̂}$ transforms as a polar vector in real space, any in-plane rotation of the field generates a potential that does not commute with the screw operator, ${\mathcal{Q̂}}_N^{-1}{\mathbf{E}}_{\perp }·\mathbf{r̂}{\mathcal{Q̂}}_N\ne {\mathbf{E}}_{\perp }·\mathbf{r̂}$ . Thus, in the modified Hamiltonian $Ĥ$ , the rotational part of the screw symmetry about the chiral axis is violated, i.e. ${\mathcal{Q̂}}_N^{-1}Ĥ{\mathcal{Q̂}}_N\ne Ĥ$ , while the translational component along the axis remains preserved. As a result, both protecting symmetries are lifted and the Dirac points evolve into finite gaps, as shown in Figure c.

Our symmetry argument remains valid in the presence of SOC. For the trigonal selenium chiral wire (Figure d), the nonsymmorphic 3-fold screw symmetry produces Dirac-type crossings at both Γ and the zone boundary, as seen in the TB band structure with OAM resolution (Figure e). Including SOC splits each band into Kramers doublets, while time-reversal symmetry preserves the crossings between these pairs at the time-reversal-invariant momenta. Applying a static electric field breaks the rotational part of the screw operation, reducing the space group from P3₁21 (or P3₂21) to P1 and the point group from D ₃ to C ₁. This symmetry lowering lifts the screw-protected degeneracies and opens finite gaps at the former crossing points, as shown in Figure f.

Thouless Charge Pumping in 1D Chiral Wires

Applying a circularly polarized electric field $\mathbf{E}(t)=E_0\cos \phi (t)\mathbf{x̂}+E_0\sin \phi (t)\mathbf{ŷ}$ , with φ­(t) = ωt + φ₀ (ω the driving frequency and φ₀ the initial phase), modifies the reference Hamiltonian ${Ĥ}_0$ (eq ) and introduces a parametric dependence on φ in the TB Hamiltonian:

$$Ĥ[\phi ]={Ĥ}_0-e\sum _{i,\mu ,\alpha }(\mathbf{E}(\phi )·{\mathbf{r}}_i)c_{i\mu \alpha }^{†}{c}_{i\mu \alpha }$$ 4

Because the screw symmetry is never restored at any point during the driving cycle, these field-induced gaps remain open during the full period T = 2π/ω. These persistent gaps allow topological features to emerge and enable modulating bands to contribute to Thouless charge pumping.

To demonstrate the topological nature of these gaps, we drive the two systems, shown in Figure , with a right-handed circularly polarized (RCP) electric field (Figure a,d) and probe their real-time charge dynamics. Details of the current evaluation and the calculation of the pumped charge ΔQ over a full driving cycle are provided in the Supplementary Note 6. Figure b presents the transported charge ΔQ (−e), over one pumping cycle in C₁₂H₁₂ for various Fermi levels. When the Fermi level lies within the trivial gap ( ${\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{l}}$ in Figure c), the pumped charge remains zero. However, it is very remarkable that when the Fermi level is located inside the field induced gap ( ${\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}$ in Figure c), each cycle of the drive produces exactly one quantum of pumped charge. The detailed real-time profile of the pumped charge over a single cycle is shown in Figure c.

3.

|Numerical simulation of Thouless charge pumping in chiral wires. a,d, Right-handed chiral structures of (a) C₁₂H₁₂ and (d) trigonal Se, together with the applied circularly polarized driving field acting on each system. b, The transported charge per pump cycle ΔQ (−e) in the chiral hydrocarbon C₁₂H₁₂ as a function of the Fermi energy, driven by the rotating electric field E (φ­(t)) with amplitude $E_0=0.2\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ and period T = 800 fs. c, Real-time profile of the Thouless pumped charge over one period T with the Fermi level located within the ${\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}$ of ε _F = – 1.5 eV. e,f, Same as panels c,d, but for the trigonal chiral wire. The Fermi level (f) is set to ε _F = – 1 eV within the topological gap and the driving field amplitude is $E_0=0.4\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ .

The charge pumping along the SOC trigonal chiral wire exhibits analogous dependence on the Fermi level, as presented in Figure e. One-cycle real-time profile of the charge for the case of nontrivial pumping is shown in Figure f. In both systems, the driving period T = 800 fs (corresponding to ℏω ≃ 5.2 meV) remains much smaller than the instantaneous band gaps during the entire cycle, ensuring adiabatic evolution by suppressing interband transitions.

Together with the crystal momentum k, the phase φ defines a synthetic (1 + 1)-D parameter space $\mathcal{R}=(k,\phi )\in {\mathbb{T}}^2$ . The resulting Berry curvature two-form on this even-dimensional torus constitutes the integer Chern number corresponding to the quantized Thouless pump. Because the Chern number defined on (k, φ) is equivalent to the 1D winding number of the Bloch band in the chiral wire, the pumping mechanism here is intrinsically different from that of the Rice-Mele model. Whereas Rice-Mele requires two independent modulations to open a gap and execute a nontrivial cycle, the geometric winding of the chiral system already provides the necessary phase structure. Consequently, only a single control parameter φ is needed to maintain the gap and realize a quantized Thouless pump.

Orbital and Spin Angular Momentum Dynamics in a Thouless Pump

Time-reversal symmetry enforces the antisymmetric orbital-momentum locking L _z (n, k) = – L _z (n, – k) in chiral structure. Orbital polarization effect (OPE) refers to the generation of OAM polarization in these materials when an applied bias field or driven current violates this time-reversal protected constraint. , Here, we extend the concept of OPE beyond metallic charge transport to the regime of topological Thouless charge pumping, specifically investigating how OAM accompanies and responds to the quantized topological charge pump. We begin by examining the Heisenberg equation of motion for spin and orbitals. As described in Supplementary Note 1 (eq S11), the effective Hamiltonian in the synthetic (1 + 1)-dimensional can be derived from eq in the following form:

$$Ĥ(k,\phi )={Î}_{2\times 2}^{(\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n})}\otimes \mathrm{T̂}(k)+{\lambda }_{\mathrm{S}\mathrm{O}\mathrm{C}}\mathit{\sigma ̂}·\mathbf{L̂}-e\mathbf{E}(\phi )·\mathbf{r̂}$$ 5

Detailed derivation of the Heisenberg equation of motions are summarized in Supplementary Note 7, which can be summarized as

$$\frac{\mathrm{d}\mathbf{L̂}}{\mathrm{d}t}=\frac{i}{\hslash }[Ĥ(k,\phi ),\mathrm{L̂}]=e\mathbf{r̂}\times \mathbf{E}(\phi )+{\lambda }_{\mathrm{S}\mathrm{O}\mathrm{C}}\mathbf{L̂}\times \mathit{\sigma ̂}+\frac{i}{\hslash }[{Î}_{2\times 2}^{(\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n})}\otimes \mathrm{T̂}(k),\mathbf{L̂}]$$ 6

$$\frac{\mathrm{d}\mathbf{Ŝ}}{\mathrm{d}t}=\frac{i}{\hslash }[Ĥ(k,\phi ),Ŝ]={\lambda }_{\mathrm{S}\mathrm{O}\mathrm{C}}\mathit{\sigma ̂}\times \mathbf{L̂}$$ 7

Important lessons can be derived from these equations. The longitudinal component of orbital L _z (t) is driven by the OPE torque $\mathbf{r̂}\times \mathbf{E}(\phi )$ with the in-plane rotating driving field even when the SOC is absent (λ_SOC = 0). The rate of spin generation is linearly proportional to the SOC-induced torque $\mathit{\sigma ̂}\times \mathbf{L̂}$ , which is adversely affected by the rate of orbital generation $(\mathbf{L̂}\times \mathit{\sigma ̂}=-\mathit{\sigma ̂}\times \mathbf{L̂})$ . By explicitly calculating the time-dependent Schrödinger equation, we traced the real-time evolution of the OAM and SAM during topological charge pumping in a trigonal chiral wire, as shown in Figure (see Supplementary Note 6 for details). Figure a displays the OAM dynamics in the system without SOC with the Fermi level sitting in the topological gap (see Figure a). As seen, the instantaneous polarization L _z (t) (red) shows an oscillatory profile, and its time-averaged value (blue), ${\mathrm{L̅}}_z=\frac{1}{t-t_0}{\int }_{t_0}^t{L}_z(t^{\prime })\mathrm{d}{t}^{\prime }$ , gradually develops and stabilizes over successive pumping cycles. This behavior provides direct evidence of OPE induced by the Thouless charge pumping in chiral wires. When SOC is included, part of the OAM generated by the OPE torque is converted into spin. As depicted in Figure b, the SAM polarization S _z (t) (yellow) exhibits a similar oscillatory profile as the orbital dynamics, and its time-averaged value (green), ${\mathrm{S̅}}_z=\frac{1}{t-t_0}{\int }_{t_0}^t{S}_z(t^{\prime })\mathrm{d}{t}^{\prime }$ , reveals the emergence of a net spin polarization over several pumping cycles. To elucidate the effect of SOC in the interconversion behavior between OAM and SAM, we simulated the time-evolution with various strengths of SOC: the results of the time-averaged spin and orbital are presented in Figures c and d. As expected from the term SOC-induced torque ( $\mathbf{L̂}\times \mathit{\sigma ̂}=-\mathit{\sigma ̂}\times \mathbf{L̂}$ in eq ) ${\mathrm{S̅}}_z$ monotonically increases with the strength of SOC, whereas the ${\mathrm{L̅}}_z$ decreases. The time-averaged values after 4 cycles (3200 fs) are denoted by $|{\mathrm{L̅}}_z(\infty )|$ (solid square) and $|{\mathrm{S̅}}_z(\infty )|$ (solid ball) and presented in Figure f with various SOC strengths.

4.

Orbital-spin polarization in a chiral Se trigonal wire during Thouless charge pumping. a, Real-time profile of the OAM L _z (t) (red) and its time-averaged value ${\mathrm{L̅}}_z(t)$ (blue) in a right handed Se trigonal wire undergoing topological pumping. b, Corresponding SAM dynamics S _z (t) (yellow) and its time-averaged value ${\mathrm{S̅}}_z(t)$ (green) for SOC strength λ = 0.3 eV. c, d, Time-averaged orbital (c) and spin (d) polarizations as functions of SOC strength. e, The pumped charge ΔQ over one cycle obtained with various SOC strength. f, Time-averaged spin and orbital obtained after 4 cycles (denoted by $|{\mathrm{L̅}}_z(\infty )|$ (solid squre) and $|{\mathrm{S̅}}_z(\infty )|$ (solid ball).

Remarkably, the amount of pumped charge over one cycle remains as one quantum, irrespective of the SOC strength (Figure e), which obviously proves the topological nature of the charge pumping. These results demonstrate that the orbital dynamics in the chiral Thouless pump are governed predominantly by the OPE torque $e\mathbf{r̂}\times \mathbf{E}(\phi )$ . Together with the intrinsic orbital-momentum locking term, this reveals a purely geometric consequence of the structural chirality. By contrast, the spin response arises solely through SOC, which converts the OAM generated by the OPE mechanism into SAM. Consequently, our findings demonstrate that a topologically quantized charge pump in insulating chiral wires can generate spin polarization through an orbital-to-spin conversion pathway. This result implies that spin–orbit transport or potentially even CISS can emerge from a purely topological origin. Moreover, although quantized Thouless pumping requires an insulating gap, cyclic driving in helical metallic systems can still generate finite, albeit nonquantized, Berry-phase-driven charge, orbital, and spin responses, without topological protection.

Thouless topological phase transition in a chiral wire

Above in Figure and Figure , we demonstrated that the perpendicular bias field (E _⊥) opens a topological gap $({\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{o}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}})$ and induces the Thouless charge pumping, whose topological nontriviality depends on the selection of the Fermi level. We now address the question of whether applying a perpendicular electric field to a chiral wire can induce band inversion and thereby tune the topological character of its originally trivial gap ( ${\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{l}}$ in Figures c and f)? To mimic the typical reduced band gap in a bundle of 1D trigonal wire, we adopt a refined parameter set with hopping amplitudes $V_{pp\sigma }^{12}=1.5$ eV and $V_{pp\pi }^{12}=-0.8$ eV (see Supplementary Note 1). This choice preserves the essential features of the electronic structure in a trigonal wire while producing a narrower gap. Figure a presents the static band structure of the trigonal chiral wire under a perpendicular electric field (see panel (i), for two representative field strengths, $E_{\perp }=0.25\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ (panel (ii) and $E_{\perp }=0.5\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ (panel (iii). The corresponding field-dependent evolution of the gap is summarized in Figure b. Notably, the originally trivial band gap $({\epsilon }_{\mathrm{G}\mathrm{a}\mathrm{p}}^{\mathrm{T}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{a}\mathrm{l}})$ closes at $E_{\perp }=0.33\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ , and it reopens upon further increasing the field. This behavior indicates a field-driven band inversion in the electronic spectrum. We now show that this reopened gap becomes topological. Figure c displays the real-time dynamics of the transported charge per pumping cycle. At $E_0=0.25\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ , the total pumped charge (orange curve) vanishes, arising from the topologically trivial nature of the band gap. In contrast, at field amplitude $E_0=0.5\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ , the contributions from the Kramers doublets no longer cancel, yielding a finite net pumped charge (violet curve) of ΔQ = – 6­(-e). To further confirm the topological character, we calculate the Berry curvature Ω­(k, φ) (see Supplementary Note 8) of the static band structure defined in the (1 + 1)-dimensional Brillouin zone (0 ≤ k ≤ 2π/a, 0 ≤ φ ≤ 2π). As shown in the inset of Figure b, the Berry curvatures corresponding to the trivial and topological responses in Figure c exhibit qualitatively distinct behaviors. In the trivial phase, the Berry curvature develops sharp positive peaks (red) accompanied by a diffuse negative background (blue) whose contributions give zero integration over the full Brillouin zone. By contrast, under the stronger drive, the Berry-curvature profile is reshaped, indicative of an inverted band ordering and resulting in a finite quantized value. Figure d summarizes the total Chern number as a function of field strength, clearly revealing the associated Thouless topological phase transition.

5.

Topological phase transition in 1D a chiral wire. a, (i) Schematic of a trigonal chiral wire subjected to a perpendicular electric field E _⊥. Static band structures of the chiral wire calculated with parameters $V_{pp\sigma }^{12}=1.5\mathrm{e}\mathrm{V}$ , $V_{pp\pi }^{12}=-0.8\mathrm{e}\mathrm{V}$ , and λ = 0.15 eV, for field strengths (ii) $E_{\perp }=0.25\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ and (iii) $E_{\perp }=0.5\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ . The band gaps are indicated by ε _Gap . b, Evolution of the band gap as a function of the perpendicular field strength E _⊥. Insets show the Berry-curvature distributions in the synthetic (k, φ) space for a rotating electric field with amplitudes $E_0=0.25\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ (top) and $E_0=0.5\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ (bottom). c, Transported charge per pumping cycle obtained from real-time simulations along the chiral wire driven by a rotating electric field with amplitudes $E_0=0.25\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ (orange dashed curve) and $E_0=0.5\mathrm{V}{\mathrm{\AA }}^{-\mathrm{1}}$ (violet curve). d, The total Chern number in the synthetic (k, φ) 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.

To further elucidate Thouless charge pumping, we compute the site-resolved probability amplitudes in a finite chiral chain. Let |Ψ _n (t)⟩ denote the instantaneous (time-evolved) wave function of the n-th state in a finite system comprising N atoms. The site resolved probability for the n-th state at the site i is calculated as

$$\begin{array}{ll}{p}_n^i(t)& =\sum _{\mu \sigma }⟨{\mathrm{\Psi }}_n(t)|i,\mu ,\sigma ⟩⟨i,\mu ,\sigma |{\mathrm{\Psi }}_n(t)⟩\\ & =\sum _{\mu }({|{\varphi }_{n,\mu \uparrow }^i(t)|}^2+{|{\varphi }_{n,\mu \downarrow }^i(t)|}^2)\end{array}$$ 8

where | i, μ, σ ⟩ denotes the local orbital μ ∈ {p _x , p _y , p _z } with spin σ centered at the site i. Figure e (left) shows the colormap of the site-resolved probability along a chain of N = 36 atoms as a function of time, obtained by summing over the occupied states that contain end-mode contributions. We define this summed probability as $P_i(t)=\sum _{n\in em}{f}_n^{em}{p}_n^i(t)$ , where $f_n^{em}$ denotes the occupation factor of the end-mode states. From this site-resolved probability, we obtain the time-dependent center of the wave packet (white triangles in Figure e) along the chain as $z_c(t)=\frac{1}{N_{\mathcal{B}}}\sum _{i=1}^N{z}^i{P}_i(t)$ , where z ⁱ is the position of the site i along the chain, and $N_{\mathcal{B}}$ is the number of occupied states with end-mode components. The figure corresponds to the parameter set of the topological phase. The evolution of the site-resolved probability and the corresponding wave packet center clearly reveal the transport characteristic of the pumping process. In the right panel of the figure, the schematic snapshots at t ₁, t ₂, and t ₃ highlight representative configurations of the probability profile during the cycle. Therefore, for chiral molecules such as helicenes, where external fields can induce level inversion, similar pumping mechanism can be formulated in real space via the evolution of electronic polarization (e.g., Wannier centers), providing a direct counterpart to the Berry-phase picture and extending our mechanism beyond periodic systems.

In summary, we demonstrated that a one-dimensional chiral system provides a real-material realization of Thouless pumping, in which quantized topological charge transport is accompanied by the transfer of orbital and spin angular momentum. The key ingredient is the Dirac-cone-type degeneracy protected by the material’s screw-rotation symmetry, together with its controlled breaking by an external perpendicular electric field. Using the time-evolving Schrödinger equation of a multiorbital TB Hamiltonian, we obtained real-time profiles of the charge current, orbital angular momentum, and spin. The torque mechanism, originating from the synergistic combination of geometrical chirality and field polarization, predominantly drives the orbital response, which is subsequently converted into spin angular momentum through spin–orbit coupling. We discussed that this orbital-spin interconversion provides a bulk topological pathway toward the chirality-induced spin–orbit transport originated from the Berry-phase physics of topological charge transport of the insulator. Our results indicates that a new topology-based CISS system can be devised, and this system offers a salient example in which bulk topological numbers may be identified through directly measurable physical observables-such as orbital or spin angular momentumwithout relying on bulk-edge correspondence.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.6c01053.

• Details of the TB models, symmetry considerations, DFT calculations, real-time simulations, field-induced orbital torque, and Chern number calculation in synthetic space (PDF)

The manuscript was written through the contributions of all authors. All authors have given approval to the final version of the manuscript.

This research appeared as a preprint: Sisakht, E. T.; Jeong, U.; Oh, J.; Liu, Y.; Yan, B.; Park, N. Spin and Orbital Angular Momentum Polarization in Thouless Topological Charge Pumping. arXiv preprint, arXiv:2512.20879, 2025.

The authors declare no competing financial interest.