Orbital magnetization in Sierpinski fractals

Abstract

Orbital magnetization (OM) in Sierpinski carpet (SC) and triangle (ST) fractal is theoretically investigated by using Haldane model as a prototypical example. The OM calculation is performed following two distinct approaches; employing the definition and local markers formalism. Both methods coincides for all systems analyzed. For the SC, higher fractal generations create a dense set of edge states, resulting in a staircase profile, leading to oscillations in the magnetization as a function of the chemical potential. In contrast, the ST self-similarity produces distinct fractal-induced spectral gaps, which manifest as constant plateaus in the magnetization. The STs exhibit a pronounced sensitivity to edge terminations. Our results reveal how quantum confinement in fractal structures affects the electronic orbital angular momentum, pointing to possible pathways for exploring novel orbitronics in systems with complex geometries.

Figures (4)

• Figure 1: (a) Topological phase diagram of the 2D Haldane model. The red mark indicates the parameters used in this work. (b) Four generations of the Sierpinski carpet (SC), with a magnified view of the underlying honeycomb lattice. (c) Energy spectra and density of states for SC G(0-4). The inset shows in-gap states from the topological phase. Yellow shading indicates the Haldane bulk gap for parameters $V_{ab}/t_2=3.0$, $\phi=0.7\pi$ [red mark in (a)].
• Figure 2: (a) Orbital magnetization vs. chemical potential in the Haldane model for Sierpinski carpet generations G(0$-$3) (a.1$-$4) and also for (b) Sierpinski triangle generations G(1$-$4) (b.1$-$4). Solid lines represent the results obtained using the definition of orbital magnetization [Eq. (\ref{['Mdef']})], while circles represent the results obtained from the local markers formulation. The shaded yellow region corresponds to 2D bulk gap in the Haldane model with $V_{ab}/t_2=3.0$, $\phi=0.7\pi$. Blue areas in panels (b.1$-$4) mark fractal-induced gaps in ZZ ST. The insets in panels (a.1$-$4) show magnified views of orbital magnetization in the 2D bulk gap region for SC.
• Figure 3: (a) Illustration of five generations of the Sierpinski Triangle with zigzag edges [ZZ ST G(0$-$4)] considered in our calculations. (b) Energy spectra with the corresponding density of states for ZZ ST G(0$-$4). The inset magnifies the region of in-gap states, which appear in the system subjected to open boundary conditions when parameters are adjusted to reach the topological phase of the model. The yellow shaded region represents the band gap of the 2D bulk Haldane model, obtained with the same coupling constants ($V_{ab}/t_2=3.0$ and $\phi=0.7\pi$) used in our calculations within the fractal geometry (red mark in Fig. \ref{['FIG1']} (a)). The blue shaded strips correspond to the fractal induced energy gaps opened in the energy spectra of ZZ ST.
• Figure 4: (a) Sierpinski triangle, G(6), armchair edges [AC ST G(6)]. (b) Energy spectrum and density of states for the Haldane model ($V_{ab}/t_2=3.0$, $\phi=0.7\pi$). Yellow shading: bulk band gap; blue: fractal-induced gaps. (c) Orbital magnetization versus chemical potential in circles and lines corresponding with definition (Eq. \ref{['Mdef']}), and local markers formulation, respectively, with blue areas indicating energy gaps.