Chiral currents in one-dimensional fractional quantum Hall states

We study bosonic and fermionic quantum two-leg ladders with orbital magnetic flux. In such systems, the ratio $\nu$ of particle density to magnetic flux shapes the phase space, as in quantum Hall effects. In fermionic (bosonic) ladders, when $\nu$ equals one over an odd (even) integer, Laughlin fractional quantum Hall (FQH) states are stabilized for sufficiently long-ranged repulsive interactions. As a signature of these fractional states, we find a unique dependence of the chiral currents on particle density and on magnetic flux. This dependence is characterized by the fractional filling factor $\nu$, and forms a stringent test for the realization of FQH states in ladders, using either numerical simulations or future ultracold-atom experiments. The two-leg model is equivalent to a single spinful chain with spin-orbit interactions and a Zeeman magnetic field, and results can thus be directly borrowed from one model to the other.

Figure 1

Top: two-leg ladder model with magnetic flux per plaquette $\mathrm{\Phi }$ leading to chiral currents $j_c$ flowing around the ladder. $\xi$ denotes interaction range. Bottom: schematic fermionic chiral current contours of Eq. (2) marked by thin lines in the $n-\mathrm{\Phi }$ plane, within an integer and a fractional QH phase. The phase transition lines out of the QH states are marked by solid thick lines, and the thick dotted lines correspond to fillings $n=\nu \mathrm{\Phi }/\pi$. The validity regime is discussed in the text.

Figure 2

Left: dispersion relation ${\epsilon }_k^{\pm }$ for representative values of $t_{\perp }/t$. Right: phase diagram for noninteracting fermions; $c=0,1,2$ labels the number of pairs of Fermi points (central charge).

Figure 3

Contours of the chiral current $j_c$ marked by thin lines in the $\mathrm{\Phi }-n$ plane for $t_{\perp }=0.5t$ (top) and $t_{\perp }=0.1t$ (bottom). The phase transition lines are marked by solid thick lines, and the thick dotted line corresponds to $n=\mathrm{\Phi }/\pi$. The inset (top) depicts the chiral current along the horizontal dashed line.