Quantization and quantum oscillations of the sublattice charge order in Dirac insulators

TL;DR

The paper addresses how sublattice charge order (SCO) behaves in 2D Dirac insulators at charge neutrality under perpendicular magnetic fields when a staggered sublattice potential opens a gap. It combines analytical low-energy Dirac theory with Hofstadter-butterfly lattice calculations to map SCO across weak, intermediate, and strong magnetic fields, revealing a singular $|B|$ response at weak fields, universally quantized SCO plateaus for $oldsymbol{ω_c \\lesssim Δ}$ due to zeroth-Landau-level polarization (a topological Thouless pump), and pronounced quantum magneto-oscillations for $oldsymbol{ω_c \\gtrsim Δ}$; the universal plateaus persist only when the gap is much smaller than the Dirac cutoff $α$. The authors also show that NN hopping and NNN corrections modify SCO only subleadingly, and provide concrete experimental guidance, notably using STM to detect the energy gap via SCO independent of its size. Overall, the work establishes SCO as a sensitive probe of the Dirac gap and highlights topological pumping physics in spinful and spinless Dirac materials under magnetic fields.

Abstract

We report the quantization, quantum oscillations, and singular behavior of sublattice symmetry-breaking sublattice charge order (SCO) in two-dimensional Dirac insulators at charge neutrality under perpendicular magnetic fields $B$. SCO is induced by staggered sublattice potentials, such as those originating from substrates, strains, hydrogenation, and chemical doping. In small non-quantizing magnetic fields that result in less than a flux quantum threading the system, and small sublattice symmetry breaking potentials, SCO exhibits perturbative singular magnetic field dependence, $\sim |B|$, originating from hopping between neighboring sites of the same sublattice. At intermediate magnetic fields, when the cyclotron gap between the zeroth Landau level and the first Landau level, $ω_c$, is smaller than the sublattice potential, $ω_c\lesssim Δ$, SCO shows $\textit{universally}$ quantized plateaus owing to discrete Landau-level degeneracy. As the magnetic flux increases by one flux quantum, one electron (per spin) is transferred from the sublattice with a higher chemical potential to the sublattice with a lower chemical potential. One electron transfer between sublattices per flux quantum results from the sublattice polarization of the zeroth Landau level in gapped Dirac materials, realizing the topological Thouless pump effect. At stronger magnetic fields, $ω_c\gtrsim \ Δ$, corresponding to integer quantum Hall regimes, SCO displays singularities based on the physics of quantum magneto-oscillations. Our findings suggest new ways to experimentally detect the presence of the energy gap in Dirac materials, irrespective of the gap size.

Figures (13)

• Figure 1: Universal quantization of SCO at $\omega_c\lesssim\Delta$. Here $N_s$ is the number of sites in the finite-size system, $L$ is the system size, $\phi_B$ is the magnetic flux threading the unit cell of the honeycomb lattice, and $\phi_0$ is the flux quantum.
• Figure 2: Tight-binding model of graphene with nearest- and next-nearest-neighbor hopping in the presence of a staggered sublattice potential, $\pm\Delta$, and an external magnetic field, ${\bf B}$.
• Figure 3: Energy spectrum of lattice electrons in graphene with sublattice potential, $\Delta=0.1$ (spectral gap measured in the units of NN hopping). Here $\phi_B$ is the flux through the unit cell and $\phi_0$ is the flux quantum.
• Figure 4: $SCO$ per site obtained from the Hofstadter butterfly showing the nonuniversal plateau at weak magnetic fields such that the net flux threading the system is smaller than a flux quantum. The plateau value is related to the cutoff $\alpha$ via Eq. \ref{['sco0']}, resulting in the cutoff value in Eq. \ref{['afa']}.
• Figure 5: The behavior of SCO per site is shown across different regimes of the external magnetic field. Panels (a) and (c) display the SCO over the full range of the magnetic field, from weak to strong fields separated by the deep minimum corresponding to $\omega_c=\Delta$. At strong fields corresponding to $\omega_c>\Delta$, SCO exhibits magneto-oscillatory behavior. In contrast, panels (b) and (d) provide a magnified view of the intermediate-field regime, where the SCO per site exhibits universal quantization.