Fig. 1. Sketch of the hopping term of the Hamiltonian.

Fig. 2. (a) Singularities in derivative of ground state energy indicate phase transitions. Data is from , ; (b) phase diagram for non-interacting case; (c1)−(c3) the current pattern used to distinguish the 3 phases. Direction and thickness of an arrow indicate the direction and strength of the current on the plotted bond. The strengths are normalized by the strongest local current. The sizes of the points indicate the density strengths and are normalized by the largest density. The current patterns are from ED calculations for when , .

Fig. 3. (a) Nearest overlaps from ED calculations. Data is from ; (b) when , as decays, the overlap with state decays from 1 smoothly; (c) generally all local currents in the system reverse sign in . The staggered hopping makes the process the first order phase transition. Data is from .

Fig. 4. (a) Phase diagram for HCB when , , ; (b) VN entropy and the corresponding fitted central charge for three phases in the phase diagram above. are from horizontal current, vortex and vertical current phase respectively. The points are VN entropy data in PBC finite system. Solid lines and central charge are fitted from formula(7). The staggered hopping has made the VN entropy also staggered. Data is from , ; (c) horizontal current phase to vertical current phase transition for . The average horizontal current and average vertical current will swap their strong and weak relations in the transition point. In order to reduce the boundary effect in average currents, we used the part in the middle of the ladder; (d) horizontal current to vortex and vortex to vertical current phase transition. The nearest overlap shows that in horizontal and vertical current phases the ground state changes smoothly, and numerically oscillates in the vortex phase. Data is from .

Fig. 5. (a) Phase diagram for filling when ; (b) the homogenous phase and vortex phase are distinguished from the inhomogeneity of densities and currents. Data is from , ; (c) the Mott line for , . The average vertical current surpasses the average horizontal current smoothly, during the process the current patterns have perfect periodicity of 2 and the density is homogenous.