**2. Model and Formalism**

We consider a two-dimensional (2D) non-interacting pseudospin-1 Dirac–Weyl system modulated by two time-dependent electric potential barriers illustrated in Figure 1. The pseudospin-1 Dirac–Weyl fermions are charged quasiparticles originating from free electrons moving in the three-band structure consisting of gapless tip-to-tip two cones intersected by a flat band, which is shown in Figure 1c. Their dynamics are governed by the dot product of the spin-1 operator and the momentum. Matrices of the spin-1 operator *S*ˆ = *S*ˆ *x*, *S*ˆ *y*, *S*ˆ *z* in the *S*ˆ *z*-representation (the representation that *S*ˆ *z* is diagonalized) can be deduced from spin-lifting/lowering operators *S*ˆ ± = *S*ˆ *x* ± *S*ˆ *y* by *S*ˆ ± |*S*, *Sz* = (*S* ∓ *Sz*) (*S* ± *Sz* + <sup>1</sup>)|*<sup>S</sup>*, *Sz* ± 1 [53]. Simple algebra leads to the results that

$$
\hat{S}\_x = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, \quad \hat{S}\_y = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}, \quad \hat{S}\_z = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}. \tag{1}
$$

By applying AC gate voltages, Hamiltonian of the pseudospin-1 Dirac–Weyl fermions has the form

$$
\hat{H} = -i\hbar v\_{\hat{\mathcal{S}}} \hat{\mathbf{S}} \cdot \nabla + V \left( \mathbf{x}, t \right), \tag{2}
$$

where **Sˆ** is the spin-1 operator defined in Equation (1), *vg* ≈ 10<sup>6</sup> m/s is the group velocity associated with the slope of the Dirac cone. As shown in Figure 1a, the potential function has the form

$$V(\mathbf{x},t) = \begin{cases} V\_0 + V\_1(t), & 0 < \mathbf{x} < L\_{1\prime} \\ V\_0 + V\_2(t), & L\_2 < \mathbf{x} < L\_{3\prime} \\ 0, & \text{others} \end{cases} \tag{3}$$

with *<sup>V</sup>*1(*t*) = *<sup>V</sup>*1*ω*cos(*ω<sup>t</sup>* + *ϕ*) and *<sup>V</sup>*2(*t*) = *<sup>V</sup>*2*ω*cos(*ω<sup>t</sup>*). The Fermi energy of the two reservoirs to the two sides of the double-barrier structure are equalized to eliminate the external bias and secure energy-conserved tunneling. While the frequency of the potential modulation *ω* is small compared to the carrier interaction time (Wigner delay time) with the conductor, the quantum pump can be considered "adiabatic" [1,18,23]. In this case, one can employ an instant scattering matrix approach, which depends only parametrically on the time *t*. The Wigner–Smith delay time can be evaluated by *τ* = Tr( − *ih*¯ **s**† *∂***s** *∂EF* ), with **s** the scattering matrix defined in Equation (5). Calculations below show *τ* ≈ 10−<sup>14</sup> s for all the parameter values. Thus, the adiabatic condition can be well justified when *ω* is in the order of MHz [3].

**Figure 1.** (**a**) schematics of the adiabatic quantum pump. Two time-dependent gate voltages with identical width *d* and equilibrium strength *V*0 are applied to the conductor. Time variation of the two potentials *V*1 and and *V*2 is shown in panel (**b**). *V*1 and *V*2 have a phase difference giving rise to a looped trajectory after one driving period; (**c**) two-dimensional band structure of the pseudospin-1 Dirac–Weyl fermions with a flat band intersected two Dirac cones at the apexes; (**d**) conductivity of the pseudospin-1 Dirac–Weyl fermions measured by [54] *σ* = *e*<sup>2</sup>*kF d πh π*/2 −*<sup>π</sup>*/2 |*t*(*EF*, *θ*)|<sup>2</sup> cos *θdθ* in single-barrier tunneling junction as a function of the Fermi energy for three different values of barrier height *V*0. *kF* = *EF*/¯*hvg* is the Fermi wavevector and *t* is the transmission amplitude defined in Equation (5). It can be seen that higher barrier allowing larger conductivity occurs at the Dirac point *EF* = *V*0 and around *EF* = *V*0/2 (see the text).

For studying the transport properties, the flux normalized scattering modes in different regions can be expressed in terms of the eigenspinors as

$$\Psi = \begin{pmatrix} \Psi\_1 \\ \Psi\_2 \\ \Psi\_3 \end{pmatrix} = \begin{cases} a\_l \Psi\_{\to} + b\_l \Psi\_{\leftarrow \leftarrow} & \mathbf{x} < \mathbf{0}, \\\ a\_1 \Psi\_{1 \to} + b\_1 \Psi\_{1 \leftarrow \leftarrow} & \mathbf{0} < \mathbf{x} < L\_1, \\\ a\_2 \Psi\_{\to} + b\_2 \Psi\_{\leftarrow \leftarrow} & L\_1 < \mathbf{x} < L\_2, \\\ a\_3 \Psi\_{2 \to} + b\_3 \Psi\_{2 \leftarrow \leftarrow} & L\_2 < \mathbf{x} < L\_3, \\\ a\_r \Psi\_{\leftarrow} + b\_r \Psi\_{\rightarrow} & \mathbf{x} > L\_3. \end{cases} \tag{4}$$

where *kx* = /*E*2*F*/(*hv*¯ *g*)2 − *k*2*y* with *EF* the quasiparticle energy at the Fermi level of the reservoirs. Ψ→ = 1 <sup>2</sup>√cos *θ e*<sup>−</sup>*iθ*, <sup>√</sup><sup>2</sup>*s*,*eiθ*<sup>T</sup>*eikx x* for *EF* = 0 (quasiparticles on the two cone bands) and √ 1 2 −*e*<sup>−</sup>*iθ*, 0,*ei<sup>θ</sup>*<sup>T</sup> (we also identify it as Ψ0 for discussions in the next section) for *EF* = 0 (quasiparticles on the flat band). *θ* = arctan(*ky*/*kx*), and *s* = sgn(*EF*). Ψ← can be obtained by replacing *kx* with −*kx* in Ψ→; Ψ*i*→/Ψ*i*← (*i* = 1, 2) can be obtained by replacing *kx* with *qxi* = /(*EF* − *V*0 − *Vi*)2/(*hv*¯ *g*)2 − *k*2*y* and *s* with *si* = sgn(*EF* − *V*0 − *Vi*) in Ψ→/Ψ<sup>←</sup>. The flux normalization factor <sup>2</sup>√cos *θ* is obtained [55] by letting Ψ†(*∂H*<sup>ˆ</sup> /*∂kx*)Ψ = 1. *ψi* (*i* = 1, 2, 3) picks up the *i*-th row of the spinor wave function in all the five regions. Note that quasiparticles on the flat band contribute no flux in the *x*-direction. However, it must be taken into account in the pumping mechanisms while the Fermi energy lies close to the Dirac point. We will go to this point again in the next section.

The boundary conditions are that *ψ*1 + *ψ*3 and *ψ*2 are continuous at the interfaces, respectively [48]. The derivation of the boundary condition is provided in Appendix A. After some algebra, the instant scattering matrix connecting the incident and outgoing modes can be expressed as

$$
\begin{pmatrix} b\_l \\ b\_r \end{pmatrix} = \begin{pmatrix} r & t' \\ t & r' \end{pmatrix} \begin{pmatrix} a\_l \\ a\_r \end{pmatrix} = \mathbf{s}(V\_1, V\_2) \begin{pmatrix} a\_l \\ a\_r \end{pmatrix} \tag{5}
$$

where **s** is parameter-dependent. Detailed derivation of the scattering matrix is provided in Appendix B.

The DC pumped current flowing from the *α* reservoir at zero temperature could be expressed in terms of the Berry phase of the scattering matrix formed within the looped trajectory of the two varying parameters as [2,18,23]

$$I\_{\rm p\alpha} = \frac{\omega e}{2\pi} \int\_{A} \Omega\left(\alpha\right) dV\_{1} dV\_{2\prime} \tag{6}$$

where

$$\Omega\left(\alpha\right) = \sum\_{\beta} \text{Im} \frac{\partial \mathbf{s}\_{a\beta}^{\*}}{\partial V\_{1}} \frac{\partial \mathbf{s}\_{a\beta}}{\partial V\_{2}}.\tag{7}$$

*A* is the enclosed area in the *V*1-*V*2 parameter space. While the driving amplitude is small ( *Viω V*0), the Berry curvature can be considered uniform within *A* and we have

$$I\_{\mu\nu} = \frac{\omega e \sin \varphi V\_{1\omega} V\_{2\omega}}{2\pi} \Omega\left(\alpha\right). \tag{8}$$

Conservation of current flux secures that the pumped currents flowing from the left and right reservoirs are equal: *Ipl* = *Ipr*. The angle-averaged pumped current can be obtained as

$$I\_{\rm paT} = \int\_{-\pi/2}^{\pi/2} I\_{\rm pa} \cos \theta d\theta. \tag{9}$$
