**5. Diamond Interferometer**

We consider an electron transport in two-dimensional system on [001] surface with setting *x* and *y* axis along the (100) and (010) crystal directions, respectively. The Hamiltonian for the SOI is

$$\mathcal{H}\_{\mathrm{R}} = \begin{array}{c} \frac{\hbar}{m}k\_{\mathrm{R}}(\not p\_{y}\not r\_{x} - \not p\_{x}\not r\_{y}) \end{array} \tag{37}$$

$$
\hat{\mathcal{H}}\_{\rm D} = \frac{\hbar}{m} k\_{\rm D} (\not p\_x \not r\_x - \not p\_y \not r\_y),
\tag{38}
$$

where *k*R and *k*D are Rashba and Dresselhaus parameters, respectively. *p*ˆ*μ* (*μ* = *x*, *y*) are the momentum and *m* is the electron effective mass.

The interferometer made of four sites is configured as in Figure 2 which is attached to the leads at site *u* = 0 and *u* = 1 as discussed in Reference [3]. Other two sites constituting the interferometer are *u* = *b* and *u* = *c*, connected with bonds of length *L*. We also define the opening angle 2*β* and the relative angle *ν* of the diagonal line to *x* axis. The Hamiltonian reads

$$\mathcal{H}\_{\text{IF}} \equiv \sum\_{\text{u}} \epsilon\_{\text{u}} \left| \psi(\boldsymbol{u}) \right\rangle \left\langle \psi(\boldsymbol{u}) \right| - \sum\_{\text{uv}} \mathcal{U}\_{\text{uv}} \left| \psi(\boldsymbol{v}) \right\rangle \left\langle \psi(\boldsymbol{u}) \right|, \tag{39}$$

for *u*, *v* = 0, *c*, *d*, 1 where  *u* is the site energy and *U*˜ *uv* ≡ *Juv U*ˆ *uv*, *Juv* is a hopping energy and *U*ˆ *uv* is a 2 × 2 unitary matrix representing the effect of SOI and AB phase. Total Hamiltonian is Hˆ = Hˆ IF + Hˆ *L* + Hˆ *R*. In the Appendix E, we explain how this problem is reduced to the Schrödinger equations, Equations (3) and (4).

The coordinates of the four sites are *r*0 = (0, <sup>0</sup>), *rb* = (*L* cos(*ν* + *β*), *L* sin(*ν* + *β*)), *r*1 = (2*L* cos(*β*) cos(*ν*), 2*L* cos(*β*) sin(*ν*)), and *rc* = (*L* cos(*ν* − *β*), *L* sin(*ν* − *β*)). We define *α*R ≡ *k*R*L*, *α*D ≡ *k*D*L* and *ζ* ≡ / *α*2 R + *α*2 D and introduce another angle *θ*, such that *α*R = *ζ* cos *θ*, and *α*D = *ζ* sin *θ*. The unitary matrix for the hopping from site at (0, 0) to site at (*ux*, *uy*) is *U*ˆ (0,0),(*ux*,*uy*) = exp [*iK* · *σ*ˆ ] with *K* ≡ *<sup>α</sup>*R(*uy*, <sup>−</sup>*ux*, 0) + *<sup>α</sup>*D(*ux*, <sup>−</sup>*uy*, 0) [27]. Therefore, for the hopping from site 0 to *b*,

$$\mathbb{K}\_{0b} \cdot \hat{\sigma} \quad = \; \mathbb{J}\sin(\mathfrak{J}\_1)\hat{\sigma}\_{\ge} - \mathbb{J}\cos(\mathfrak{J}\_2)\hat{\sigma}\_{\ge} \equiv \mathbb{J}\hat{\sigma}\_1. \tag{40}$$

with *ξ*1 ≡ *β* + *ν* + *θ* and *ξ*2 ≡ *β* + *ν* − *θ*. Similarly, for the hopping from site *c* to 0,

$$\mathbf{K}\_{\mathbf{c}0} \cdot \boldsymbol{\mathfrak{o}}\prime = \ulcorner \zeta \sin(\mathfrak{J}\_4) \mathfrak{d}\_{\mathbf{x}} + \ulcorner \zeta \cos(\mathfrak{J}\_3) \mathfrak{d}\_{\mathbf{y}} \equiv \ulcorner \mathfrak{d}\_2 \,, \tag{41}$$

with *ξ*3 ≡ *β* − *ν* + *θ* and *ξ*4 ≡ *β* − *ν* − *θ*. We introduce factors *F*1 ≡ 1 + sin(<sup>2</sup>*ν* + 2*β*) sin(2*θ*) and *F*2 ≡ 1 + sin(<sup>2</sup>*ν* − 2*β*) sin(2*θ*) such that *σ*ˆ 2 1 = I*F*<sup>2</sup> 1 and *σ*ˆ 2 2 = I*F*<sup>2</sup> 2 . Then, for *n* = 1, 2,

$$\epsilon^{i\mathbb{Z}\bar{\sigma}\_{n}} \equiv \mathbb{I}\mathfrak{c}\_{n} + i s\_{n} \mathfrak{d}\_{n}.\tag{42}$$

where we defined

$$\mathbf{c}\_{\mathfrak{n}} \equiv \cos(F\_{\mathfrak{n}} \zeta), \qquad \mathbf{s}\_{\mathfrak{n}} \equiv \frac{1}{F\_{\mathfrak{n}}} \sin(F\_{\mathfrak{n}} \zeta). \tag{43}$$

Noting that *U* ˆ 0*b* = *eiζσ*ˆ1 , *U*ˆ *b*1 = *e*<sup>−</sup> *iφ*2 −*iζσ*ˆ2 , *U*ˆ 0*c* = *e*<sup>−</sup>*iζσ*ˆ2 and *U*ˆ *c*1 = *eiφ*2 +*iζσ*ˆ1 ,

$$\begin{aligned} \mathcal{Q}\_b &\equiv \quad \mathcal{Q}\_{0b}\mathcal{Q}\_{b1} \\ &= \quad \mathcal{e}^{-\frac{i\boldsymbol{\theta}}{2}} \left\{ \mathbb{I}c\_1c\_2 - ic\_1s\_2\mathfrak{e}\_2 + ic\_2s\_1\mathfrak{e}\_1 + s\_1s\_2\mathfrak{e}\_1\mathfrak{e}\_2 \right\} \\ &= \quad \mathcal{e}^{-\frac{i\boldsymbol{\theta}}{2}} \left( \mathbb{I}\delta + i\boldsymbol{\tau}\cdot\mathfrak{e} \right), \end{aligned} \tag{44}$$

where

$$\begin{array}{rcl}\delta & \equiv & c\_1 c\_2 - s\_1 s\_2 (\sin(2\nu)\sin(2\theta) + \cos(2\beta)),\\\tau\_x & \equiv & -c\_1 s\_2 \sin\xi\_4 + c\_2 s\_1 \sin\xi\_{1'},\\\tau\_y & \equiv & -c\_1 s\_2 \cos\xi\_3 - c\_2 s\_1 \cos\xi\_{2'},\\\tau\_z & \equiv & s\_1 s\_2 \sin(2\beta)\cos(2\theta). \end{array} \tag{45}$$

Similarly,

$$\begin{split} \mathcal{Q}\_{c} & \equiv & \mathcal{Q}\_{0c} \mathcal{Q}\_{c1} \\ & = & e^{\frac{i\boldsymbol{\theta}}{2}} \left\{ \mathbb{I}c\_{2}c\_{1} - i s\_{2}c\_{1}\mathfrak{d}\_{2} + i c\_{2}s\_{1}\mathfrak{d}\_{1} + s\_{2}s\_{1}\mathfrak{d}\_{2}\mathfrak{d}\_{1} \right\} \\ & = & e^{\frac{i\boldsymbol{\theta}}{2}} \left( \mathbb{I}\delta' + i\pi' \cdot \mathfrak{d}^{\star} \right) , \end{split} \tag{46}$$

with *δ* = *δ* and *τ* = (*<sup>τ</sup>x*, *<sup>τ</sup>y*, <sup>−</sup>*τz*). The angle *ω* is determined by cos *ω* = *δδ* + *τ* · *τ* = *δ*2 + *τ*2*x* + *τ*2*y* − *τ*2*z* = 1 − <sup>2</sup>*τ*2*z* .

**Figure 2.** Schematics of the interferometer made of four sites, 0, *b*, *c*, and 1 separated by a length *L*. The opening angle 2*β* and relative angle *ν* from *x* axis determine the geometric structure.

### **6. QAP in the Diamond Interferometer**

We examine the quantum adiabatic spin-pumping by choosing two SOI strengths *g*1 = *α*R and *g*2 = *α*D as control parameters. First we examine the basic property of the function *C<sup>α</sup>*R,*α*D defined in Equation (36) and then evaluate the scattering amplitudes. Using these results, we calculate the Berry curvatures for two spin directions.
