# Table 8 Phase parameters required to simulate the time evolution of the vertical hopping $$h_{3}$$ and $$h_{8}$$, see Eq. (108c)

Vertical Hopping

Operator

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

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

$$U_{(9,11)}^{{y,y}^{\dagger }}U_{(4,6)}^{{y,y}^{\dagger }}$$

$$\tilde{\varphi }^{(3),(6),(8),(11)}_{1}= \pi$$

$$\tilde{\varphi }^{(3),(6),(8),(11)}_{2}=\pi$$

$$U_{10}^{x}U_{5}^{x}$$

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

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

$$\operatorname{exp} (-\frac{i\mathcal{A}}{2}(\sigma _{4}^{x}\sigma _{5}^{y}+\sigma _{9}^{x}\sigma _{10}^{y})\frac{t}{n} )$$

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

$$\tilde{\varphi }^{(9)}_{1}= 3/2\pi$$

$$\tilde{\varphi }^{(4)}_{2}=3/2\pi$$

$$\tilde{\varphi }^{(9)}_{2}=3/2\pi$$

$$U_{5}^{x^{\dagger }}U_{10}^{x^{\dagger }}$$

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

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

$$U_{(4,6)}^{y,y}U_{(9,11)}^{y,y}$$

$$\tilde{\varphi }^{(3),(6),(8),(11)}_{1}={2} \pi$$

$$\tilde{\varphi }^{(3),(6),(8),(11)}_{2}={2}\pi$$

$$U_{(9,11)}^{{x,x}^{\dagger }}U_{(4,6)}^{{x,x}^{\dagger }}$$

$$\tilde{\varphi }^{(3),(6),(8),(11)}_{1}=\pi$$

$$\tilde{\varphi }^{(3),(6),(8),(11)}_{2}=2\pi$$

$$U_{10}^{y^{\dagger }}U_{5}^{y^{\dagger }}$$

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

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

$$\operatorname{exp} (-\frac{i\mathcal{A}}{2}(\sigma _{4}^{y}\sigma _{5}^{x}+\sigma _{9}^{y}\sigma _{10}^{x})\frac{t}{n} )$$

$$\tilde{\varphi }^{(4)}_{1}=3/2\pi$$

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

$$\tilde{\varphi }^{(4)}_{2}=3/2\pi$$

$$\tilde{\varphi }^{(9)}_{2}=3/2\pi$$

$$U_{5}^{y}U_{10}^{y}$$

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

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

$$U_{(4,6)}^{x,x}U_{(9,11)}^{x,x}$$

$$\tilde{\varphi }^{(3),(6),(8),(11)}_{1}=2\pi$$

$$\tilde{\varphi }^{(3),(6),(8),(11)}_{2}=\pi$$