Figure 1

(Color online) The staggered flux phase in the two-leg ladder penetrated by a magnetic field. The arrows inside the plaquettes indicate the direction of the applied uniform field (directed inside the plane of the figure). The arrows along the links indicate directions of the staggered currents. The central plaquette is the domain wall between the two degenerate ground states. This excitation carries the charge $e∕2$.Reuse & Permissions

Figure 2

(Color online) The spectrum in the region $\tau >1$, $\tau \cos \pi f>{\sin }^2\pi f$.Reuse & Permissions

Figure 3

(Color online) The spectrum in the region ${\left(\sin \pi f\right)}^2∕\tau >\cos \pi f>\tau$. The horizontal line depicts the chemical potential such that there are only two Fermi points.Reuse & Permissions