# Table 5 Phase parameters required to simulate the time evolution of horizontal hopping Hamiltonian, see Eq. (105)

Horizontal Hopping

Operator

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

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

$$U_{3}^{x}U_{7}^{x}U_{11}^{x}$$

$$\tilde{\varphi }^{(3),(7),(11)}_{1}= 2\pi$$

$$\tilde{\varphi }^{(3),(7),(11)}_{2}=\pi$$

$$\operatorname{exp} (-{\frac{i\mathcal{A}}{2}(\sigma _{2}^{x}\sigma _{3}^{y}+\sigma _{6}^{x}\sigma _{7}^{y}+\sigma _{10}^{x}\sigma _{11}^{y})\frac{t}{n}} )$$

$$\tilde{\varphi }^{(2),(6),(10)}_{1}= {1}/{2}\pi$$

$$\tilde{\varphi }^{(2),(6),(10)}_{2}={3}/{2}\pi$$

$$U_{3}^{x^{\dagger }}U_{7}^{x^{\dagger }}U_{11}^{x^{\dagger }}$$

$$\tilde{\varphi }^{(3),(7),(11)}_{1}= \pi$$

$$\tilde{\varphi }^{(3),(7),(11)}_{2}=2\pi$$

$$U_{3}^{y^{\dagger }}U_{7}^{y^{\dagger }}U_{11}^{y^{\dagger }}$$

$$\tilde{\varphi }^{(3),(7),(11)}_{1}= \pi$$

$$\tilde{\varphi }^{(3),(7),(11)}_{2}=\pi$$

$$\operatorname{exp} ({\frac{-i\mathcal{A}}{2}(\sigma _{2}^{y}\sigma _{3}^{x}+\sigma _{6}^{y}\sigma _{7}^{x}+\sigma _{10}^{y}\sigma _{11}^{x})\frac{t}{n}} )$$

$$\tilde{\varphi }^{(2),(6),(10)}_{1}={3}/{2} \pi$$

$$\tilde{\varphi }^{(2),(6),(10)}_{2}={3}/{2}\pi$$

$$U_{3}^{y}U_{7}^{y}U_{11}^{y}$$

$$\tilde{\varphi }^{(3),(7),(11)}_{1}=2\pi$$

$$\tilde{\varphi }^{(3),(7),(11)}_{2}=2\pi$$

$$U_{2}^{x}U_{6}^{x}U_{10}^{x}$$

$$\tilde{\varphi }^{(2),(6),(10)}_{1}=2\pi$$

$$\tilde{\varphi }^{(2),(6),(10)}_{2}=\pi$$

$$\operatorname{exp} (-{\frac{i\mathcal{A}}{2}(\sigma _{1}^{x}\sigma _{2}^{y}+\sigma _{5}^{x}\sigma _{6}^{y}+\sigma _{9}^{x}\sigma _{10}^{y})\frac{t}{n}} )$$

$$\tilde{\varphi }^{(1),(5),(9)}_{1}=3/2\pi$$

$$\tilde{\varphi }^{(1),(5),(9)}_{2}=3/2\pi$$

$$U_{2}^{x^{\dagger }}U_{6}^{x^{\dagger }}U_{10}^{x^{\dagger }}$$

$$\tilde{\varphi }^{(2),(6),(10)}_{1}=\pi$$

$$\tilde{\varphi }^{(2),(6),(10)}_{2}=2\pi$$

$$U_{2}^{y^{\dagger }}U_{6}^{y^{\dagger }}U_{10}^{y^{\dagger }}$$

$$\tilde{\varphi }^{(2),(6),(10)}_{1}=\pi$$

$$\tilde{\varphi }^{(2),(6),(10)}_{2}=\pi$$

$$\operatorname{exp} ({-\frac{i\mathcal{A}}{2}(\sigma _{1}^{y}\sigma _{2}^{x}+\sigma _{5}^{y}\sigma _{6}^{x}+\sigma _{9}^{y}\sigma _{10}^{x})\frac{t}{n}} )$$

$$\tilde{\varphi }^{(1),(5),(9)}_{1}=1/2\pi$$

$$\tilde{\varphi }^{(1),(5),(9)}_{2}=3/2\pi$$

$$U_{2}^{y}U_{6}^{y}U_{10}^{y}$$

$$\tilde{\varphi }^{(2),(6),(10)}_{1}=2\pi$$

$$\tilde{\varphi }^{(2),(6),(10)}_{2}=2\pi$$