Fig. 2: The lattice Schwinger and the model the Creutz-ladder model.

a Schwinger model in 1 + 1 dimension. The lattice comprises vertices containing matter particles (red and green) connected by links carrying the associated gauge field (blue for positive and yellow for negative values). The Fermionic particles \({\hat{\Psi }}_{i}\) that can hope from one site to the other, with a hopping strength mediated by the gauge field \({\hat{U}}_{i,i+1}\), with an energy contribution in the Hamiltonian given by \({\hat{\Psi }}_{i}{\hat{U}}_{i,i+1}{\hat{\Psi }}_{i+1}^{{{{\dagger}}} }\). The energy density of the electromagnetic field is given by the square of the Electric field. b Two-leg ladder associated with the Creutz-ladder model. Without vertical tunneling (t_v), the two-site paths depicted in yellow and blue can interfere constructively or destructively depending on the choice of horizontal and diagonal tunneling terms (t_h and t_d, respectively).