# Table 9 Phase parameters required to simulate the time evolution of the vertical hopping $$h_{2}$$ and $$h_{7}$$, see Eq. (108b)

Vertical Hopping

Operator

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

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

$$U_{(8,10)}^{{y,y}^{\dagger }}U_{(3,5)}^{{y,y}^{\dagger }}$$

$$\tilde{\varphi }^{(2),(5),(7),(10)}_{1}= \pi$$

$$\tilde{\varphi }^{(2),(5),(7),(10)}_{2}=\pi$$

$$U_{9}^{x}U_{4}^{x}$$

$$\tilde{\varphi }^{(4),(9)}_{1}=2 \pi$$

$$\tilde{\varphi }^{(4),(9)}_{2}=\pi$$

$$\operatorname{exp} (-\frac{i\mathcal{A}}{2}(\sigma _{3}^{x}\sigma _{4}^{y}+\sigma _{8}^{x}\sigma _{9}^{y})\frac{t}{n} )$$

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

$$\tilde{\varphi }^{(8)}_{1}= 1/2\pi$$

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

$$\tilde{\varphi }^{(8)}_{2}=3/2\pi$$

$$U_{4}^{x ^{\dagger }}U_{9}^{x^{\dagger }}$$

$$\tilde{\varphi }^{(4),(9)}_{1}= \pi$$

$$\tilde{\varphi }^{(4),(9)}_{2}=2\pi$$

$$U_{(3,5)}^{y,y}U_{(8,10)}^{y,y}$$

$$\tilde{\varphi }^{(2),(5),(7),(10)}_{1}={2} \pi$$

$$\tilde{\varphi }^{(2),(5),(7),(10)}_{2}={2}\pi$$

$$U_{(8,10)}^{{x,x}^{\dagger }}U_{(3,5)}^{{x,x}^{\dagger }}$$

$$\tilde{\varphi }^{(2),(5),(7),(10)}_{1}=\pi$$

$$\tilde{\varphi }^{(2),(5),(7),(10)}_{2}=2\pi$$

$$U_{9}^{y^{\dagger }}U_{4}^{y^{\dagger }}$$

$$\tilde{\varphi }^{(4),(9)}_{1}=\pi$$

$$\tilde{\varphi }^{(4),(9)}_{2}=\pi$$

$$\operatorname{exp} (-\frac{i\mathcal{A}}{2}(\sigma _{3}^{y}\sigma _{4}^{x}+\sigma _{8}^{y}\sigma _{9}^{x})\frac{t}{n} )$$

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

$$\tilde{\varphi }^{(8)}_{1}=3/2\pi$$

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

$$\tilde{\varphi }^{(8)}_{2}=3/2\pi$$

$$U_{4}^{y}U_{9}^{y}$$

$$\tilde{\varphi }^{(4),(9)}_{1}=2\pi$$

$$\tilde{\varphi }^{(4),(9)}_{2}=2\pi$$

$$U_{(3,5)}^{x,x}U_{(8,10)}^{x,x}$$

$$\tilde{\varphi }^{(2),(5),(7),(10)}_{1}=2\pi$$

$$\tilde{\varphi }^{(2),(5),(7),(10)}_{2}=\pi$$