**1. Introduction**

Coherent transport in mesoscopic systems is of fundamental interest since it allows realization of various phenomena observed in quantum optics in a solid-state system. Furthermore, the electron spin degree of freedom adds an intriguing knob for the manipulation and observation of the transport phenomena. Spin-orbit interaction (SOI) effect [1] is one of the key ingredients in narrow-gap semiconductor devices, whose strength can be controlled by external gates [2], in principle, without changing the electron density. Introducing the effect of SOI to the electron interferometer structure is quite attractive since it enables perfect spin filtering effect [3–5]. Moreover, transient behavior in such an interferometer has been investigated [6].

In addition to passive functional devices such as filters, the active functions, for example, spin-pumping or spin manipulation effect by dynamically modulating the gate voltages [7–9], magnetic field [10–13], or magnetization of the ferromagnets [14–17], has been investigated. In particular, quantum adiabatic pumping (QAP) phenomena [18,19], which stems from geometrical properties of the dynamics, is an active field of research [20–25]. In the non-interacting limit, QAP is related to the scattering matrix of the coherent transport. We have investigated the QAP effect by adiabatically modulating the Aharonov-Bohm (AB) phase [26] of the interferometer as well as the local potential in the interferometer. However, it seems no studies have been made of the adiabatic spin-pumping with purely geometric means such as Aharonov-Casher phase or AB phase. The fundamental question here is whether QAP is possible by only modulating the electron geometric phase.

In this work, we studied spin-QAP in Rashba-Dresselhaus-Aharonov-Bohm interferometer introduced in [3] using Brouwer's formula [19] and derived an explicit formula of the Berry curvature for each spin component. Using the obtained result, we clarified the condition of finite spin-pumping. In Section 2, we introduce a simple two-terminal setup and the expressions of the scattering amplitudes. Section 3 explains the details of the eigenstates of the scattering problem. Then, with these states, the formula of the QAP is derived in Section 4. It is shown that the modulation of the AB phase cannot induce QAP. Section 5 explains the properties of the diamond-shape interferometer, and is applied to study QAP assuming Rashba SOI and Dresselhaus SOI strengths as control parameters in Section 6. Finally, discussions follow in Section 7 and Appendices are included for the detailed derivations of the formula used in the main text.

We consider a standard setup of scattering problem of spin 1/2 electrons as shown in Figure 1. A coherent scattering region (interferometer) is attached at the site *u* = 0 with the one-dimensional left lead made of sites *u* = −1, −2, ... and is attached at the site *u* = 1 with the one-dimensional right lead made of sites *u* = 2, 3, .... The assumption of one-dimensional leads is not essential as far as the interferometer is coupled to the leads via single mode scattering channels. However, the one-dimensional tight-binding formalism benefits from its simplicity. Although the analysis is standard, the obtained rigorous scattering amplitudes and corresponding scattering eigenstates are essential to clarify the condition and to quantify the quantum adiabatic spin-pumping, as will be shown in the later sections. We introduce the spinor ket vector at site *u*,

$$|\psi(u)\rangle \equiv \begin{pmatrix} \varepsilon\_{u\uparrow} \\ \varepsilon\_{u\downarrow} \end{pmatrix},\tag{1}$$

where the two amplitudes *cuσ* for spin *σ* <sup>=</sup>↑, ↓ satisfy normalization condition *cu*↑ 2 + *cu*↓ 2 = 1. The total Hamiltonian in the tight-binding approximation is given in general

$$\hat{\mathcal{H}}\_{\text{TB}} \equiv \sum\_{\boldsymbol{u}} \epsilon\_{\boldsymbol{u}} \left| \psi(\boldsymbol{u}) \right\rangle \left\langle \psi(\boldsymbol{u}) \right| + \sum\_{\boldsymbol{u}\boldsymbol{v}} \hat{\mathcal{W}}\_{\text{uv}} \left| \psi(\boldsymbol{v}) \right\rangle \left\langle \psi(\boldsymbol{u}) \right|, \tag{2}$$

where the site index *u* and *v* run the entire system. The real parameter  *u* is spin-degenerate site energy and *W*ˆ *uv* is a 2 × 2 hopping matrix satisfying *W*ˆ †*uv* = *W*ˆ *vu*. We assume that the hopping matrix *W*ˆ *uv* is only non-diagonal in the scattering region between *u* = 0 and *u* = 1. We neglect the electron-electron interaction.

**Figure 1.** Schematics of the model of a scattering (shaded) region connected with two semi-infinite one-dimensional leads.
