Table 7 Phase parameters to simulate the time evolution of the vertical hopping $$h_{4}$$, see Eq. (108d)

Vertical Hopping

Operator

$$\tilde{\varphi }^{(j)}_{1}$$

$$\tilde{\varphi }^{(j)}_{2}$$

$$U_{(5,7)}^{{y,y}^{\dagger }}$$

$$\tilde{\varphi }^{(4),(7)}_{1}= \pi$$

$$\tilde{\varphi }^{(4),(7)}_{2}=\pi$$

$$U_{6}^{x}$$

$$\tilde{\varphi }^{(6)}_{1}= 2\pi$$

$$\tilde{\varphi }^{(6)}_{2}=\pi$$

$$\operatorname{exp} (-\frac{i\mathcal{A}}{2}\sigma _{5}^{x}\sigma _{6}^{y}\frac{t}{n} )$$

$$\tilde{\varphi }^{(5)}_{1}= 3/2\pi$$

$$\tilde{\varphi }^{(5)}_{2}=3/2\pi$$

$$U_{6}^{x^{\dagger }}$$

$$\tilde{\varphi }^{(6)}_{1}= \pi$$

$$\tilde{\varphi }^{(6)}_{2}={2}\pi$$

$$U_{(5,7)}^{y,y}$$

$$\tilde{\varphi }^{(4),(7)}_{1}={2} \pi$$

$$\tilde{\varphi }^{(4),(7)}_{2}={2}\pi$$

$$U_{(5,7)}^{{x,x}^{\dagger }}$$

$$\tilde{\varphi }^{(4),(7)}_{1}=\pi$$

$$\tilde{\varphi }^{(4),(7)}_{2}=2\pi$$

$$U_{6}^{y^{\dagger }}$$

$$\tilde{\varphi }^{(6)}_{1}=\pi$$

$$\tilde{\varphi }^{(6)}_{2}=\pi$$

$$\operatorname{exp} (-\frac{i\mathcal{A}}{2}\sigma _{5}^{y}\sigma _{6}^{x}\frac{t}{n} )$$

$$\tilde{\varphi }^{(5)}_{1}=1/2\pi$$

$$\tilde{\varphi }^{(5)}_{2}=3/2\pi$$

$$U_{6}^{y}$$

$$\tilde{\varphi }^{(6)}_{1}=2\pi$$

$$\tilde{\varphi }^{(6)}_{2}=2\pi$$

$$U_{(5,7)}^{x,x}$$

$$\tilde{\varphi }^{(4),(7)}_{1}=2\pi$$

$$\tilde{\varphi }^{(4),(7)}_{2}=\pi$$