Aharonov-Bohm caging of an electron in a quantum fractal

TL;DR

The paper investigates Aharonov-Bohm caging of an electron in a Vicsek fractal lattice threaded by a uniform magnetic flux. Using a tight-binding framework, exact diagonalization, Green's function-based density of states and transport calculations, and persistent current analyses up to the third generation, it shows that at half the flux quantum the spectrum collapses to a few eigenvalues and transport is completely blocked, signaling AB caging. The phenomenon remains robust against onsite disorder and exhibits a generation-dependent scaling of the persistent current, indicating potential for flux-controlled localization in fractal networks. This work advances the understanding of quantum transport in fractal geometries and suggests applications in quantum information processing with fractal-based networks.

Abstract

Fractal geometries exhibit complex structures with scale invariance self-similar pattern over various length scales. An artificially designed quantum fractal geometry embedded in a uniform magnetic flux has been explored in this study. It has been found that due to quantum mechanical effect, such quantum fractal display an exotic electronic property which is reflected in its transport characteristics. Owing to this uniform magnetic flux piercing through each closed-loop building block of the fractal structure, an electron traversing through such a fractal geometry will pick up a nontrivial Aharonov-Bohm phase factor, which will influence its transport through the system. It is shown that, one can completely block the transmission of an electron in this fractal geometry by setting the value of the uniform magnetic flux to half flux quantum. This phenomenon of Aharonov-Bohm caging of an electron in this quantum fractal geometry has been supported by the computation of the energy spectrum, two-terminal transport and persistent current in its various generations. This result is very robust against disorder and could be useful in designing efficient quantum algorithms using a quantum fractal network.

Figures (6)

• Figure 1: Schematic diagram of various generations, viz., (a) first generation ($\mathcal{G}_{1}$), (b) second generation ($\mathcal{G}_{2}$), and (c) third generation ($\mathcal{G}_{3}$) of a closed-loop Vicsek fractal lattice geometry with all the diamond-shaped closed-loop building blocks pierced by a nonzero uniform magnetic flux $\Phi$.
• Figure 2: Variation in the energy eigenvalue spectrum ($E$) as function of the Aharonov-Bohm flux ($\Phi$) for (a) first generation (with $\mathcal{N}_{1}=16$ sites), (b) second generation (with $\mathcal{N}_{2}=76$ sites), and (c) third generation (with $\mathcal{N}_{3}=376$ sites) Vicsek fractal lattice structure, respectively. We set the onsite potential $\varepsilon_{n}=0$ for all $n$-sites and the nearest-neighbor hopping integral $t_{nm}=t=1$ for all $\langle n,m\rangle$ bonds.
• Figure 3: The plots for the density of states (DOS) as a function of the energy ($E$) of the electron for the (a)-(c) first generation ($\mathcal{G}_{1}$), (d)-(f) second generation ($\mathcal{G}_{2}$), and (g)-(i) third generation ($\mathcal{G}_{3}$) Vicsek fractal lattice. The left column is for $\Phi=0$, the middle column is for $\Phi=\Phi_{0}/4$, and the right column is for $\Phi=\Phi_{0}/2$.
• Figure 4: The plots for the transmission probability ($T$) as a function of the energy ($E$) of the electron for the (a)-(c) first generation ($\mathcal{G}_{1}$), (d)-(f) second generation ($\mathcal{G}_{2}$), and (g)-(i) third generation ($\mathcal{G}_{3}$) Vicsek fractal lattice. The left column is for $\Phi=0$, the middle column is for $\Phi=\Phi_{0}/4$, and the right column is for $\Phi=\Phi_{0}/2$.
• Figure 5: The plots for the DOS and transmission probability ($T$) as a function of the energy ($E$) of the electron for the second generation Vicsek fractal lattice in presence of random onsite disorder of width $\Delta=0.5$ in the system. We have chosen $\Phi=\Phi_{0}/2$.