**5. Conclusions**

In summary, adiabatic quantum pumping in a periodically modulated pseudospin-1 Dirac–Weyl system is studied. By using two AC electric gate-potentials as the driving parameters, a direction-reversed pumped current is found by the Berry phase of the scattering matrix at certain parameter regimes as a result of super Klein tunneling and particle-hole symmetry close to the Dirac point of the band structure. Such a phenomenon originates from the abnormal transmission behavior of the Dirac–Weyl quasiparticles that sometimes they transmit more through a higher electric potential barrier. As a result, definition of the "opening" and "closing" of a gate is reversed in the classic turnstile picture and hence direction of the pumped DC current is reversed. We also provide rigorous proof of the consistency between the quantum Berry phase picture and the classic turnstile mechanism.

**Author Contributions:** Conceptualization, R.Z.; Methodology, R.Z.; Validation, R.Z.; Formal Analysis, X.C.; Investigation, X.C.; Resources, X.C.; Data Curation, X.C.; Writing—Original Draft Preparation, R.Z.; Writing—Review & Editing, R.Z.

**Funding:** The work is supported by the National Natural Science Foundation of China (No. 11004063) and the Fundamental Research Funds for the Central Universities, SCUT (No. 2017ZD099).

**Acknowledgments:** R.Z. is grateful for enlightening discussions with Pak Ming Hui.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Appendix A. Derivation of the Boundary Condition of the Spinor Wavefunction**

In Equation (4), we have defined the spinor wavefunction Ψ of the double-barrier pseudospin-1 Dirac–Weyl system. Within the instant scattering matrix approach, we solve the static Schrödinger equation

$$
\hat{H}\Psi = E\Psi,\tag{A1}
$$

where *H*ˆ is defined in Equation (2) and the time *t* is taken as a constant. Substituting the spin-1 operator defined in Equation (1), this equation becomes

$$\begin{aligned} & -i\hbar v\_{\mathcal{S}} \begin{bmatrix} 0 & \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) & 0\\ \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) & 0 & \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) \\\ 0 & \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) & 0 \end{bmatrix} \begin{pmatrix} \psi\_{1} \\ \psi\_{2} \\ \psi\_{3} \end{pmatrix} \\\ & + V \begin{pmatrix} \psi\_{1} \\ \psi\_{2} \\ \psi\_{3} \end{pmatrix} = E \begin{pmatrix} \psi\_{1} \\ \psi\_{2} \\ \psi\_{3} \end{pmatrix}, \end{aligned} \tag{A2}$$

which is

$$-i\hbar v\_{\mathcal{S}} \begin{bmatrix} \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) \psi\_2 \\\ \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) \psi\_1 + \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right) \psi\_3 \\\ \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right) \psi\_2 \end{bmatrix} = \left[ E - V\left(x, t\right) \right] \begin{pmatrix} \psi\_1 \\\ \psi\_2 \\\ \psi\_3 \end{pmatrix} . \tag{A3}$$

This spinor equation equalizes to three scalar equations

$$-i\hbar v\_{\mathcal{S}} \frac{1}{\sqrt{2}} \left(\frac{\partial}{\partial \mathbf{x}} - i\frac{\partial}{\partial y}\right) \Psi\_2 = \left[E - V\left(\mathbf{x}, t\right)\right] \Psi\_{1\prime} \tag{A4}$$

$$-i\hbar v\_{\mathcal{S}} \left[ \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial \mathbf{x}} + i \frac{\partial}{\partial y} \right) \psi\_1 + \frac{1}{\sqrt{2}} \left( \frac{\partial}{\partial \mathbf{x}} - i \frac{\partial}{\partial y} \right) \psi\_3 \right] = \left[ E - V \left( \mathbf{x}, t \right) \right] \psi\_2. \tag{A5}$$

and

$$-i\hbar v\_{\mathcal{S}} \frac{1}{\sqrt{2}} \left(\frac{\partial}{\partial \mathbf{x}} + i \frac{\partial}{\partial y}\right) \Psi\_2 = \left[E - V\left(\mathbf{x}, t\right)\right] \Psi\_3. \tag{A6}$$

The boundary condition at *x*0 can be obtained from

$$\int\_{x\_0-\varepsilon}^{x\_0+\varepsilon} \hat{H} \Psi d\mathbf{x} = \int\_{x\_0-\varepsilon}^{x\_0+\varepsilon} E \Psi d\mathbf{x}.\tag{A7}$$

By substituting Equations (A3) and (A4) into Equation (A7), we have

$$\begin{cases} \frac{-i\hbar v\_{\text{f}}}{\sqrt{2}} \left\{ \left[ \psi\_{2} \left( \mathbf{x}\_{0} + \varepsilon, y \right) - \psi\_{2} \left( \mathbf{x}\_{0} - \varepsilon, y \right) \right] - i \frac{\partial}{\partial y} \psi\_{2} \left( \mathbf{x}\_{0}, y \right) 2\varepsilon \right\} \\ = \left[ E - V \left( \mathbf{x}\_{0}, t \right) \right] \psi\_{1} \left( \mathbf{x}\_{0}, y \right) 2\varepsilon. \end{cases} \tag{A8}$$

Because the energy *E*, the electric potential *<sup>V</sup>*(*<sup>x</sup>*, *t*), and the spinor wavefunction <sup>Ψ</sup>(*<sup>x</sup>*, *y*) is finite, in the limit of *ε* → 0, we have the boundary condition

$$\left[\psi\_2\left(\mathbf{x}\_0 + \varepsilon, \boldsymbol{y}\right) - \psi\_2\left(\mathbf{x}\_0 - \varepsilon, \boldsymbol{y}\right)\right] = 0,\tag{A9}$$

which is that *ψ*2 is continuous at *x* = *x*0. Similarly, by substituting Equations. (A3) and (A5) into Equation (A7), we have

$$\begin{cases} \frac{1}{\sqrt{2}} \left\{ \left[ \psi\_1 \left( \mathbf{x}\_0 + \varepsilon\_\prime \mathbf{y} \right) - \psi\_1 \left( \mathbf{x}\_0 - \varepsilon\_\prime \mathbf{y} \right) \right] + i \frac{\partial}{\partial \mathbf{y}} \psi\_1 \left( \mathbf{x}\_0, \mathbf{y} \right) 2\varepsilon \right\} \\ + \left[ \psi\_3 \left( \mathbf{x}\_0 + \varepsilon\_\prime \mathbf{y} \right) - \psi\_3 \left( \mathbf{x}\_0 - \varepsilon\_\prime \mathbf{y} \right) \right] - i \frac{\partial}{\partial \mathbf{y}} \psi\_3 \left( \mathbf{x}\_0, \mathbf{y} \right) 2\varepsilon \right] \\ = \left[ E - V \left( \mathbf{x}\_0, t \right) \right] \psi\_2 \left( \mathbf{x}\_0, \mathbf{y} \right) 2\varepsilon. \end{cases} \tag{A10}$$

In the limit of *ε* → 0, we have the boundary condition

$$\left[\psi\_1\left(\mathbf{x}\_0+\varepsilon,\mathbf{y}\right)+\psi\_3\left(\mathbf{x}\_0+\varepsilon,\mathbf{y}\right)\right]-\left[\psi\_1\left(\mathbf{x}\_0-\varepsilon,\mathbf{y}\right)+\psi\_3\left(\mathbf{x}\_0-\varepsilon,\mathbf{y}\right)\right]=0,\tag{A11}$$

which is that *ψ*1 + *ψ*3 is continuous at *x* = *x*0. Reproducing the procedure in Equations (A3), (A6), and (A7), we can reobtain the boundary condition of *ψ*2 in Equation (A9).

Therefore, in the case of pseudospin-1 Dirac–Weyl fermions, the boundary condition is that the second component of the spinor wavefunction *ψ*2 is continuous and the first component plus the third component of the spinor wavefunction *ψ*1 + *ψ*3 is continuous. No derivative of the wavefunction is involved in the continuity condition.

### **Appendix B. Detailed Algebra for Obtaining the Scattering Matrix**

We use the transfer-matrix method to obtain the instant scattering matrix **s** defined in Equation (5). Using the obtained spinor wavefunction (4) and the continuity relation (A9) and (A11), we can have the following matrix equations:

$$\mathbf{M}\_1 \left( \begin{array}{c} a\_l \\ b\_l \end{array} \right) = \mathbf{M}\_2 \left( \begin{array}{c} a\_1 \\ b\_1 \end{array} \right),\tag{A12}$$

$$\mathbf{M}\_3 \left( \begin{array}{c} a\_1 \\ b\_1 \end{array} \right) = \mathbf{M}\_4 \left( \begin{array}{c} a\_2 \\ b\_2 \end{array} \right),\tag{A13}$$

$$\mathbf{M}\_5 \left( \begin{array}{c} a\_2 \\ b\_2 \end{array} \right) = \mathbf{M}\_6 \left( \begin{array}{c} a\_3 \\ b\_3 \end{array} \right), \tag{A14}$$

$$\mathbf{M}\_{\mathcal{T}} \begin{pmatrix} a\_3 \\ b\_3 \end{pmatrix} = \mathbf{M}\_8 \begin{pmatrix} a\_r \\ b\_r \end{pmatrix},\tag{A15}$$

with

$$\mathbf{M}\_1 = \begin{pmatrix} \frac{k\_x - ik\_y}{k\_x + ik\_y} + 1 & \frac{-k\_x - ik\_y}{-k\_x + ik\_y} + 1\\ \sqrt{2} \frac{k\_x - ik\_y}{\mathbb{E}\_F / \hbar v\_g} & \sqrt{2} \frac{-k\_x - ik\_y}{\mathbb{E}\_F / \hbar v\_g} \end{pmatrix} \tag{A16}$$

$$\mathbf{M}\_2 = \begin{pmatrix} \frac{q\_{x1} - ik\_y}{q\_{x1} + ik\_y} + 1 & \frac{-q\_{x1} - ik\_y}{-q\_{x1} + ik\_y} + 1\\ \sqrt{2} \frac{q\_{x1} - ik\_y}{(E\_F - V\_0 - V\_1) / \hbar v\_{\mathcal{S}}} & \sqrt{2} \frac{-q\_{x1} - ik\_y}{(E\_F - V\_0 - V\_1) / \hbar v\_{\mathcal{S}}} \end{pmatrix} \tag{A17}$$

$$\mathbf{M}\_3 = \begin{pmatrix} \frac{(q\_{x1} - ik\_y \\ q\_{x1} + ik\_y \\ \sqrt{2} \frac{q\_{x1} - ik\_y}{(E\_F - V\_0 - V\_1)/\hbar v\_\mathcal{J}} e^{iq\_{x1}L\_1}}{\sqrt{2} \frac{q\_{x1} - ik\_y}{(E\_F - V\_0 - V\_1)/\hbar v\_\mathcal{J}} e^{iq\_{x1}L\_1}} & \sqrt{2} \frac{-q\_{x1} - ik\_y}{(E\_F - V\_0 - V\_1)/\hbar v\_\mathcal{J}} e^{-iq\_{x1}L\_1} \end{pmatrix} \tag{A18}$$

$$\mathbf{M}\_{4} = \begin{pmatrix} \begin{pmatrix} \frac{k\_{x} - ik\_{y}}{k\_{x} + ik\_{y}} + 1 \end{pmatrix} e^{ik\_{x}L\_{1}} & \begin{pmatrix} \frac{-k\_{x} - ik\_{y}}{-k\_{x} + ik\_{y}} + 1 \end{pmatrix} e^{-ik\_{x}L\_{1}} \\ \sqrt{2} \frac{k\_{x} - ik\_{y}}{E\_{F}/\hbar v\_{\mathcal{S}}} e^{ik\_{x}L\_{1}} & \sqrt{2} \frac{-k\_{x} - ik\_{y}}{E\_{F}/\hbar v\_{\mathcal{S}}} e^{-ik\_{x}L\_{1}} \end{pmatrix} \end{pmatrix} \tag{A19}$$

$$\mathbf{M}\_{5} = \begin{pmatrix} \begin{pmatrix} \frac{k\_{x} - ik\_{y}}{k\_{x} + ik\_{y}} + 1 \end{pmatrix} e^{i\mathbf{k}\_{x}L\_{2}} & \begin{pmatrix} \frac{-k\_{x} - ik\_{y}}{-k\_{x} + ik\_{y}} + 1 \end{pmatrix} e^{-i\mathbf{k}\_{x}L\_{2}}\\ \sqrt{2} \frac{k\_{x} - ik\_{y}}{\mathrm{E}\_{\mathrm{F}} / \hbar v\_{\mathrm{g}}} e^{i\mathbf{k}\_{x}L\_{2}} & \sqrt{2} \frac{-k\_{x} - ik\_{y}}{\mathrm{E}\_{\mathrm{F}} / \hbar v\_{\mathrm{g}}} e^{-i\mathbf{k}\_{x}L\_{2}} \end{pmatrix},\tag{A20}$$

$$\mathbf{M}\_{6} = \begin{pmatrix} \frac{(q\_{x2} - ik\_{y} \\ \frac{q\_{x2} + ik\_{y}}{q\_{x2} + ik\_{y}} + 1)}{\sqrt{2} \frac{q\_{x2} - ik\_{y}}{\left(\frac{q\_{x2} - ik\_{y}}{\left(\frac{E\_{F}}{E\_{F}} - \frac{V\_{0}}{V\_{1}}\right) / \hbar v\_{\mathcal{S}}}} e^{iq\_{\mathcal{S}}L\_{2}} & \sqrt{2} \frac{-q\_{x2} - ik\_{y}}{\left(\frac{-q\_{x2} - ik\_{y}}{\left(\frac{E\_{F}}{E\_{F}} - \frac{V\_{0}}{V\_{1}}\right) / \hbar v\_{\mathcal{S}}} e^{-iq\_{\mathcal{S}}L\_{2}}} \end{pmatrix},\tag{A21}$$

$$\mathbf{M}\_{7} = \begin{pmatrix} \left(\frac{q\_{12} - ik\_{\rm g}}{q\_{12} + ik\_{\rm g}} + 1\right) e^{iq\_{12}L\_{3}} & \left(\frac{-q\_{12} - ik\_{\rm g}}{-q\_{12} + ik\_{\rm g}} + 1\right) e^{-iq\_{12}L\_{3}} \\ \sqrt{2} \frac{q\_{12} - ik\_{\rm g}}{\left(E - V\_{0} - V\_{1}\right) / \hbar v\_{\rm g}} e^{iq\_{12}L\_{3}} & \sqrt{2} \frac{-q\_{12} - ik\_{\rm g}}{\left(E - V\_{0} - V\_{1}\right) / \hbar v\_{\rm g}} e^{-iq\_{12}L\_{3}} \end{pmatrix} \tag{A22}$$

$$\mathbf{M}\_8 = \begin{pmatrix} \frac{-k\_x - ik\_y}{-k\_x + ik\_y} + 1 & \frac{k\_x - ik\_y}{k\_x + ik\_y} + 1\\ \sqrt{2} \frac{-k\_x - ik\_y}{\frac{E\_F / \hbar v\_\mathcal{R}}{E\_F / \hbar v\_\mathcal{R}}} & \sqrt{2} \frac{k\_x - ik\_y}{\frac{E\_F / \hbar v\_\mathcal{R}}{E\_F / \hbar v\_\mathcal{R}}} \end{pmatrix}. \tag{A23}$$

It should be noted that it is more convenient to use the *kx*/*ky* version of the eigenspinors than the exp(*<sup>i</sup>θ*) version in the numerical treatment because when *qxi* becomes imaginary, *θ* becomes ill-defined. In addition, the sign function is avoided in the eigenspinors accordingly. For the wavefunction in the *x* > *L*3 region, we used the translated plane wave exp[±*ikx*(*x* − *<sup>L</sup>*3)]. From Equations (A12) to (A23), we can have 

$$
\mathbf{M} \begin{pmatrix} a\_l \\ b\_l \end{pmatrix} = \mathbf{M} \begin{pmatrix} a\_r \\ b\_r \end{pmatrix},\tag{A24}
$$

with

$$\mathbf{M} = \mathbf{M}\_1^{-1} \mathbf{M}\_2 \mathbf{M}\_3^{-1} \mathbf{M}\_4 \mathbf{M}\_5^{-1} \mathbf{M}\_6 \mathbf{M}\_7^{-1} \mathbf{M}\_8. \tag{A25}$$

Then, the scattering matrix **s** defined in Equation (5) can be obtained by

$$\mathbf{s} = \begin{pmatrix} 0 & -M\_{12} \\ 1 & -M\_{22} \end{pmatrix}^{-1} \begin{pmatrix} -1 & M\_{11} \\ 0 & M\_{21} \end{pmatrix} . \tag{A26}$$

Numerical results of the transmission coefficients *T* = |*t*|<sup>2</sup> are given in Figure A1, which reproduced the results reported in Ref. [57]. From Figure A1, we could find the three characteristic transport properties of the pseudospin-1 Dirac–Weyl fermions: Super Klein tunneling, Klein tunneling, and transmission minimum at the Dirac point. Super Klein tunneling means perfect transmission regardless of the incident angle when the incident energy levels with one half of the electric potential barrier, which is shown in the black solid line of Figure A1a. This is a unique property demonstrated in the pseudospin-1 Dirac–Weyl system. Klein tunneling means perfect transmission at normal incidence regardless of the quasiparticle energy, which is shown in the black solid line of Figure A1b and is also visible in Figure A1a when the horizontal coordinate equates 0. This is a property shared between the pseudospin-1 Dirac–Weyl system and the monolayer graphene, the latter of which also belongs to the pseudospin-1/2 Dirac–Weyl system. Transmission minimum at the Dirac point is shown in the dashed red and dotted blue curves of Figure A1b. This is also a property shared between the pseudospin-1 Dirac–Weyl system and the monolayer graphene. The transmission of normal incidence at the Dirac point is not well-defined, which is overlooked in the numerical treatment.

**Figure A1.** Static transmission probabilities *T* = |*t*|<sup>2</sup> as a function of the incident angle (**a**) and the Fermi energy (**b**), respectively [57]. Parameters *V*0, *d*, and *L*2− *L*1are the same as those in Figure 2.
