Controlled parity switch of persistent currents in quantum ladders

We investigate the behavior of persistent currents for a fixed number of noninteracting fermions in a periodic quantum ladder threaded by Aharonov-Bohm and transverse magnetic fluxes $\mathrm{\Phi }$ and $\chi$. We show that the coupling between ladder legs provides a way to effectively change the ground-state fermion-number parity, by varying $\chi$. Specifically, we demonstrate that varying $\chi$ by $2\pi$ (one flux quantum) leads to an apparent fermion-number parity switch. We find that persistent currents exhibit a robust $4\pi$ periodicity as a function of $\chi$, despite the fact that $\chi \to \chi +2\pi$ leads to modifications of order $1/N$ of the energy spectrum, where $N$ is the number of sites in each ladder leg. We show that these parity-switch and $4\pi$ periodicity effects are robust with respect to temperature and disorder, and outline potential physical realizations using cold atomic gases and photonic lattices, for bosonic analogs of the effects.

Figure 1

Top: One-dimensional periodic quantum ladder threaded by Aharonov-Bohm flux $\mathrm{\Phi }$ and transverse flux $\chi$. Spinless fermion “hop” between lattice sites along and across ladder legs. This setup can be realized, e.g., using cold atomic gases or photonic lattices, for bosonic analogs. Bottom: Parity switch of persistent currents. When the parity of $\chi /\left(2\pi \right)$ is switched, persistent currents $I\left(\mathrm{\Phi }\right)$ induced by $\mathrm{\Phi }$ behave as if the fermion-number parity $P$ was switched [$I\left(\mathrm{\Phi }\right)$ is in units of the Fermi velocity $v_F$ over the number $N$ of sites per leg].

Figure 2

Energy spectrum of a two-leg ladder around $k=0$, at integer $\mathrm{\Phi }/\left(2\pi \right)$. The transverse flux $\chi$ has two effects: (i) It shifts the bands corresponding to upper and lower ladder legs (red and blue faint lines) by $\chi /N$ with respect to each other; the interleg coupling $h_{\perp }\left(k\right)$ [Eq. (3)] then open “gaps” at the time-reversal-invariant momenta $k=0$ and $\pi$ (not shown) where bands cross, leading to hybridized bands (mixed red and blue lines). (ii) The flux $\chi$ controls allowed momenta (see text). Crucially, single-particle eigenstates are present at $k=0$ when $\chi /\left(2\pi \right)$ is even [solid dots, open dots show allowed states at odd $\chi /\left(2\pi \right)$]. Therefore, the parity of $\chi /\left(2\pi \right)$ controls the state-number parity below an arbitrary energy level $E$ (gray horizontal line) in the gap.

Figure 3

Left: Single-particle spectrum (lower band in Fig. 2 for large interleg coupling) as a function of the transverse flux $\chi$. The effective time-reversal symmetry $\mathrm{\Theta }$ present at integer $\chi /\left(2\pi \right)$ (see text) imposes a twofold degeneracy of the entire spectrum, except at time-reversal-invariant momenta $k=0,\pi$. Occupied/empty levels correspond to solid/open dots. Right: Single-particle eigenstates shown as a function of quasimomentum $k$, for $\chi =0$ and $2\pi$. When $\chi /\left(2\pi \right)$ alternates between even and odd parities, degenerate states at the Fermi energy $E_F$ alternate between double and single occupation. Half-solid dots indicate states sharing a single particle. The spectrum is $4\pi$ periodic in $\chi$, up to corrections $\sim 1/N$.

Figure 4

Density plots of the persistent current as a function of the AB flux $\mathrm{\Phi }$ and the transverse flux $\chi$, for even/odd fermion-number parity $P$ (left/right). The fermion number is fixed with Fermi energy in the “gap” at $k=0$ (as the level $E$ in Fig. 2). Positive/negative values are shown in blue/red. Plots correspond to a minimal lattice model with unit hopping strength between nearest-neighboring sites along and across ladder legs [39]. The left (right) plot corresponds to 61 (60) particles with system size $N=150$. Changing $P$ causes a relative shift $\chi \to \chi \pm 2\pi$ and small differences $\sim 1/N$ that are not visible here.