Emergent topological properties in spatially modulated sub-wavelength barrier lattices

TL;DR

The paper studies a one-dimensional continuum with spatially modulated Dirac-delta barriers that mimic Hofstadter physics. By treating the modulation parameters as synthetic dimensions, it derives Chern numbers and demonstrates Thouless pumping and Wannier-center shifts, establishing a bulk–pump correspondence in a continuum system. The Hofstadter butterfly appears as the spectrum fragments into $q$ sub-bands for rational $\beta=\frac{p}{q}$, and a Diophantine relation links two pumping schemes, connecting the modulated Kronig-Penney model to the Harper-Hofstadter problem. Finally, it proposes a realistic experimental route using a three-level dark-state lattice in ultracold atoms, enabling Hofstadter-type topology in a controllable, continuum platform.

Abstract

We investigate topological phenomena in a spatially modulated Dirac-$δ$ lattice, where the scattering potential varies periodically in space. Changing the potential modulation frequency leads to Hofstadter's butterfly-like energy spectrum and enables the emergence of topological transport regimes characterized by non-trivial Chern numbers. We show how the considered modulated system is connected to the Hofstadter model via the Harper equation. By adiabatically varying spatial modulation parameters, we demonstrate controllable quantum transport and verify the topological nature of these effects through Wannier center displacement and bulk invariant calculations. We also propose an experimentally feasible realization of such a system using optically controlled three-level atoms. Our findings showcase spatially engineered Kronig-Penney-type systems as versatile platforms for investigating and exploiting different topological quantum transport regimes.

Figures (7)

• Figure 1: (a) Equidistantly spaced sub-wavelength barriers with scattering amplitudes $h^{\alpha \beta \gamma \Delta}$ for modulation parameters $\beta=\frac{1}{5}$, $\Delta = 0$, $\gamma=0$, $h_0=10$ and $\alpha=0.5$. The black dashed line indicates the modulation envelope. (b) Evolution of the barrier heights $h^{\alpha \beta \gamma \Delta}$ as $\gamma$ is changed while $\Delta = 0$. (c) Evolution of the barrier heights and positions as $\Delta$ is changed while $\gamma = 0$.
• Figure 2: Emergence of the butterfly-like energy spectrum in the lowest projected energy band $E(\beta)$ as the modulation amplitude $\alpha$ is increased. Panels (a)-(e) correspond to $\alpha = 0.1, 0.25, 0.5, 0.75$ and $1$. The energies of the upper bound $E_1^{\mathrm{sup}}$, standing-wave solution $E_{K_1}^{\mathrm{mid}}$ and the lower bound $E_1^{\mathrm{inf}}$ are indicated by cyan, red and magenta dashed lines respectively and their dependence on $\alpha$ is shown in (f). The unmodulated height is fixed at $h_0=10$.
• Figure 3: (a) The total Chern number $C^{(\gamma)}$ represented as an in-gap color, indicating the sum of Chern numbers of the sub-bands below Fermi energy $E_\mathrm{F}$. (b) Same color scale is used to mark $t_{n_\mathrm{F}}$ obtained from Diophantine's equation Eq. (\ref{['eq:Diophantine']}). The model parameters are $\alpha=0.5$ and $h_0=10$.
• Figure 4: (a) The total Chern number $C^{(\Delta)}$ as an in-gap color, indicating the sum of Chern numbers $\sum_{n=1}^{n_{\mathrm{F}}}C^{(\Delta)}_n$ of the sub-bands below Fermi energy $E_\mathrm{F}$. (b) Coloring given by $s_{n_\mathrm{F}}$ obtained from Diophantine's equation Eq. (\ref{['eq:Diophantine']}). The parameters used are $\alpha=0.5$ and $h_0=10$.
• Figure 5: Spatial transfer of localized Wannier function density $|w_0(x)|^2$ under adiabatic change of $\gamma$ and $\Delta$ when the lowest energy sub-band is filled for modulation $\beta = \frac{1}{3}$ in (a,b) and $\beta=\frac{2}{3}$ in (c,d). Two elementary cells are shown with barrier positions and heights indicated by $h^{\alpha\beta\gamma\Delta}$. The dashed white curves are position expectation values of Wannier functions of neighboring elementary cells.