Localization and persistent currents in a quasiperiodic disordered helical lattice

Abstract

We investigate localization and persistent currents in a helical tight-binding lattice subject to two independent magnetic fluxes and a quasiperiodic on-site potential. Working with non-interacting, spinless fermions under periodic boundary conditions, we solve the model by exact diagonalization and study localization with both inverse and normalized participation ratios. We identify boundaries separating extended, mixed, and localized regimes by constructing a diagram incorporating potential strength and inter-ring coupling. In the metallic regime, persistent currents flowing around both the toroidal and poloidal directions show oscillations whose amplitude decays as disorder grows and vanishes past the localization threshold; in the localized regime, currents become flux-insensitive. We demonstrate that tuning magnetic fluxes, hopping strengths, or quasiperiodic potential amplitudes provides control over the critical disorder threshold. Our results suggest a versatile platform for disorder-and flux-controlled switching between conductive and insulating states.

Figures (8)

• Figure 1: (a) Schematic of the rotated-rings helical lattice. Each ring contains $N$ equally spaced sites and is rotated by $\Delta\theta=2\pi/L$ relative to its neighbor, forming a helical chain. Intra-ring hopping $J$ (solid black lines) connects nearest-neighbor sites around each ring, while inter-ring hopping $J_R$ (solid red line, only displayed through one site for clarity) links corresponding site positions between adjacent rings. (b) Imposing periodic boundary conditions along the ring loops and the stacking direction produces a toroidal geometry with two cycles. The magnetic flux $\phi_T$ encircles each ring loop (short cycle), while the flux $\phi_P$ threads vertically through the torus hole along the helical axis (long cycle).
• Figure 2: Color map of the composite localization metric $\eta$ plotted versus potential strength $\Delta/J$ and $J_R/J$ for a lattice size (a) $N=L=40$ (b) $N=30$, $L=60$. Red regions ($\eta\lesssim-3.2$) correspond to completely extended or localized states, whereas blueish regions ($-3.2\lesssim\eta$) correspond to mixed states. The mixed (blue) region expands with the increase of $J_R$ and produces energy-dependent localization thresholds (mobility edges).
• Figure 3: Spectrum-averaged IPR (solid red) and NPR (dashed blue) versus quasiperiodic potential strength $\Delta/J$ for $J_R=0.5J$ on a $20\times20$ lattice. The shaded region marks the mixed region. The inset shows the corresponding $\langle {\rm IPR}\rangle$ color map with respect to $J_R/J$ and $\Delta/J$; black contour lines correspond to the lower and upper edges of the mixed region.
• Figure 4: $\langle {\rm IPR} \rangle$ on a $20\times20$ helical torus, shown as a function of the dimensionless flux $\phi_T/\phi_0$ (horizontal axis) and $\phi_P/\phi_0$ (vertical axis). (a) for $\Delta=1.75J$ (b) for $\Delta=6J$.
• Figure 5: Persistent currents on a $20\times20$ lattice ($J_R=0.5J$, half-filling) plotted against the corresponding dimensionless flux. Columns correspond to increasing on-site potential strength $\Delta$. $\Delta=0.5J$ for (a) and (d), $1.0J$ for (b) and (e), $1.5J$ for (c) and (f). The top row (a), (b), (c) shows the poloidal current $I_P/I_0$, and the bottom row (d), (e), (f) shows the toroidal current $I_T/I_0$.