Magnetic Bloch bands and Weiss oscillations in Dirac mass superlattices

TL;DR

The paper investigates 2D Dirac fermions under a perpendicular magnetic field subjected to a 1D periodic mass modulation, revealing persistent Jackiw–Rebbi modes whose velocity is renormalized by the field. It develops two complementary tools—the transfer-matrix method for finite kink–antikink arrays and a gauge-invariant magnetic Bloch-state projection for infinite lattices—to compute dispersive magnetic minibands and analyze transport. The authors predict modified quantum Hall plateaus and Weiss-like magnetoconductivity oscillations with strongly reduced amplitude and a $\\pi/2$ phase shift relative to electrostatic superlattices. The framework provides a versatile route to miniband engineering in graphene and topological insulator surfaces, with potential experimental realization and broad applicability to Dirac materials under periodic mass textures.

Abstract

We study two-dimensional Dirac fermions in a one-dimensional mass superlattice under a perpendicular magnetic field. Using exact solutions for isolated and finite arrays of domain walls, we demonstrate the persistence of Jackiw-Rebbi modes with a field-dependent renormalized velocity. For the periodic case, we adopt a gauge-invariant projection method onto magnetic Bloch states, valid for arbitrary fields and mass profiles, which yields dispersive Landau levels, and confirm its accuracy by comparison with finite arrays spectra. From the miniband spectra we predict modified quantum Hall plateaus and Weiss-like magnetoconductivity oscillations, characterized by a strongly reduced amplitude and a $π/2$ phase shift compared to electrostatic superlattices.

Figures (11)

• Figure 1: Evolution of the energy spectrum for the mass profile \ref{['kink']} with a single mass kink, obtained by numerical solution of Eq. \ref{['QCkinkMF']} with $M=\bar{\varepsilon}_c$, as the magnetic field increases: $B=0.5, 1, 2, 5$ T, see panels (a), (b), (c), and (d), respectively. Energy is expressed in units of $\bar{\varepsilon}_c$ and $k_y$ in units of $\bar{\ell}_B^{-1}$, see Eq. \ref{['scalesdef2']}. For comparison, we also plot the linear dispersion of the interface chiral mode at $B=0$ (red dashed line).
• Figure 2: Renormalized velocity $v_r$ (in units of $v_{\text{F}}$) of the chiral 1D Jackiw-Rebbi mode propagating along a single sharp mass kink, see Eq. \ref{['renvelocity']}, vs inverse magnetic field $1/B$ (with $B$ given in Tesla) for different mass amplitudes: $M=1.5\bar{\varepsilon}_c$ (black), $M=\bar{\varepsilon}_c$ (blue), $M=0.5\bar{\varepsilon}_c$ (red), with $\bar{\varepsilon}_c$ in Eq. \ref{['scalesdef2']}. The black dashed curves illustrate the asymptotic $B^{-\frac{1}{2}}$ scaling at large field. The black dotted line shows the Jackiw-Rebbi mode velocity at $B=0$.
• Figure 3: Mass profile $m(x)$ for a finite array of $N$ kinks and antikinks, with width $L=(N-\frac{1}{2})d$.
• Figure 4: Energy spectra for finite-length kink-antikink arrays with the mass profile in Eq. \ref{['finitearrayprofile']} for $M=\bar{\varepsilon}_c$. We set $B=1$ T and use $\bar{\varepsilon}_c$ and $\bar{\ell}_B^{-1}$ in Eq. \ref{['scalesdef2']} as units for energy and $k_y$, respectively. The mass profiles $m(x)$ for the left and right columns are indicated schematically on top of the figure. Results are shown for $N=1$ (indigo) and $N=7$ (orange curves) kink-antikink pairs and different values of the inter-kink spacing $d$. Panels (a) and (b) are for $d=5\bar{\ell}_B$. Panels (c) and (d) are for $d=2\bar{\ell}_B$. Panels (e) and (f) are for $d=\bar{\ell}_B$.
• Figure 5: Probability density profile for the state in the "zero-energy" band at $k_y =1.25\bar{\ell}_B ^{-1}$ for $B=1$ T, $N = 7$ and different $d$ values, see Figs. \ref{['fig4']}(b,d,e). The corresponding energies are $E =0$, $0.081784\bar{\varepsilon}_c$, and $0.000221\bar{\varepsilon}_c$, for $d/\ell_B = 5,2,1$, respectively.