**1. Introduction**

After quantized particle transport driven by adiabatic cyclic potential variation was proposed by D. J. Thouless in 1983 [1], such a concept has attracted unceasing interest among researchers concerning its theoretical meaning and potential applications in various fields such as a precision current standard and neural networks [2–4]. Mechanism of the quantum pump can be interpreted consistently by the Berry phase of the scattering matrix in the parameter space within the modulation cycle [2] and the classic turnstile picture [5,6]. Usually, the pumped current is unidirectional when the phase difference between the two driving parameters is fixed. In the turnstile picture, the opening order of the two gates is defined by the driving phase. The first-opened gate let in the particle and the second-opened gate let it out forming a direct current (DC) current after a cycle is completed. However, reversed DC current direction has been discovered in various systems even when the driving phase is fixed such as in monolayer graphene [6] and carbon nanotube-superconductor hybrid systems [7]. This is because that conventionally a "gate" is defined by a potential barrier and higher barriers allow smaller transmission probabilities. However, as a result of the Klein tunneling effect, the potential barrier becomes transparent regardless of its height at certain parameter settings. When higher barrier allows even stronger transmission, the opening and closing of a "gate" in the quantum pump is reversed and so the driven current is reversed with the driving phase difference unchanged. The same phenomenon is also discovered in the superconductive carbon nanotube when Andreev reflection again violates the higher-barrier-lower-transmission convention and reversed the pumped current under the same driving forces. This turnstile interpretation of the reversed pumped current coincides with the Berry phase of the scattering matrix in the parameter space within the modulation cycle. However, a clear comparison between the two mechanisms is lacking, which is one of the motivations of this work.

About the significance of the comparison between the Berry phase picture and the classic turnstile mechanism of the adiabatic quantum pumping, we would like to make some further background remarks.

The classic turnstile mechanism and the Berry-phase-of-scattering-matrix picture of adiabatic quantum pumping are proposed based on different physical origin. The former is from classic mechanics and the latter is from quantum mechanics. General agreemen<sup>t</sup> between them is certainly a surprising result because they have at least the following differences on the conceptual level.

(1) In quantum mechanics, the leftward and rightward transmission probabilities of both the symmetric and asymmetric double-barrier structure (The former means the height and width of the two barriers are exactly the same. The latter means the height and width of the two barriers are different.) are exactly the same if a typical two-lead device is considered. The difference between the leftward and rightward transmission is in the phase factor of the transmission amplitudes. Such a phase difference gives rise to a nontrivial Berry phase formed by cyclic modulation of the two barriers with *V*1 = *<sup>V</sup>*1*ω*cos(*ω<sup>t</sup>* + *ϕ*) and *V*2 = *<sup>V</sup>*2*ω*cos(*ω<sup>t</sup>*) when time-reversal symmetry is not conserved such as excluding *ϕ* = 0 or *π*. In the turnstile picture, the two barriers are treated separately like two gates. The opening and closing of the two gates is determined by the transmission probability of the corresponding barrier potential. When higher barrier generates smaller transmission probability, the opening and closing of the gate is defined conventionally: lifting the barrier means closing the gate and lowering the barrier means opening the gate. Because the charge carrier density in a typical semiconductor can be up to 10<sup>10</sup> ∼ 1015/cm3 at room temperature, only a small change in the barrier height and hence in the transmission probability can justify the definition of the "opening" and "closing" of the gate for charge carriers. While the same gate-modulation is applied, charge carriers are driven unidirectionally to one of the reservoirs like the turnstile in daily life. No phase factor is involved in the picture at all.

(2) In the quantum interpretation of parametric pump, time-reversal symmetry is a vital factor. The most prominent case is when the driving phase difference *ϕ* = *π*. In this case, time-reversal symmetry is conserved as the two parameters vary periodically. Because of this, a DC current is forbidden even when the classic turnstile gives rise to the largest mass flow when the phase lag is the largest. At this point, the classic and quantum models become incomparable, which is out of our present discussion.

Therefore, we feel a confirmation of the agreemen<sup>t</sup> between the two mechanisms is a significant step forward to understand the underlying physics of adiabatic quantum pumping. In preparation of this work, we have proved it in various parameter settings in different systems such as two-dimensional electron gas and graphene besides the present pseudospin-1 Dirac–Weyl system by calculating term by term Equation (10). Although we could not provide a general proof, up to now, no numerical evaluation violates such a conclusion.

After the idea of the adiabatic quantum pump (also called Thouless pump and parametric pump) is proposed, such a mechanism has been investigated in various transport devices such as a single spin in diamond [8], quantum-dot structures [9], Rashba nanowires [10], Mach–Zehnder interferometers [11], the magnetic nanowire with double domain walls [12], magnetic-barrier-modulated two dimensional electron gas [13], mesoscopic rings with Aharonov–Casher and Aharonov–Bohm effect [14], magnetic tunnel junctions [15], and monolayer graphene [6,16,17]. Correspondingly, theoretical techniques have been put forward for the treatment of the quantum pumps such as the scattering matrix formalism [18], non-equilibrium Green's function [19–22], and the quantum master equation approach [9]. In this work, we use the scattering matrix approach for alternating current (AC) transport, which defines the Berry phase formed within the looped trajectory of the two varying parameters [2,18,23].

Recently, after realization of the monolayer graphene, which is characterized as a pseudospin-1/2 Dirac–Weyl fermionic material, a family of general pseudospin-*s* (*s* = 1/2, 1, 3/2, ··· ) Dirac–Weyl fermionic materials has been proposed by sharing similar band structure with one or several pairs of Dirac cones. Pseudospin-1 materials with a band structure of two Dirac cones and a flat band through where the cones intersect have attracted intense interest in the physical society currently. Numerical or experimental studies have proposed various host materials of such band structure such as conventional crystal with special space group symmetries [24,25], in the electronic, photonic, and phononic Lieb

lattice [26–33], kagome lattice [31,34], dice or *T*3 lattice [31,35–45], and *K*4 crystal [46]. Along with these progress in material building, various transport properties of the pseudospin-1 Dirac–Weyl fermions have been investigated such as super Klein tunneling effect [47–49], magneto-optics [50], Hall quantization [51], and Hofstadter butterfly [52] in a magnetic field. While the adiabatic quantum pumping process serves as an important platform to detect various properties of novel quantum states, it is worthwhile to apply the idea on newly-emerged pseudospin-1 Dirac–Weyl materials. To understand how their particular transport properties modify the adiabatically-driven pumped current is the other motivation of this work.

The plan of the present work is as follows. In Section 2, the model is introduced and the key formulas for the scattering matrix, Berry phase, and pumped current are given. In Section 3, we present numerical results of the pumped current and discussions of the underlying mechanisms. In Section 4, a rigorous proof of the consistency between the quantum Berry phase picture and the classic turnstile mechanism for adiabatic quantum pumping is provided. A brief summary is given in Section 5. Detailed derivation of the boundary condition and the scattering matrix are provided in Appendices A and B, respectively.
