Edge states of a Bi$_2$Se$_3$ nanosheet in a perpendicular magnetic field

Conventional wisdom dictates that the conducting edge states of two-dimensional topological insulators of the Bi$_2$Se$_3$ family are protected by time-reversal symmetry. However, theoretical bulk calculations and a recent experiment show that the edge states persist in the presence of large external magnetic fields. To address this apparent contradiction, we have developed an analytical description for the edge-state wave function of a semi-infinite sample in a perpendicular magnetic field. Our description relies on the usual bulk Landau levels, together with additional states arising due to the presence of the hard wall, which are unnormalizable in the infinite system. The analytical wave functions agree extremely well with numerical calculations and can be used to directly analyze the behavior of the edge states in a magnetic field.

💡 Research Summary

The paper presents a comprehensive analytical treatment of edge states in a Bi₂Se₃ nanosheet subjected to a perpendicular magnetic field. Starting from the well‑established three‑dimensional k·p Hamiltonian for bulk Bi₂Se₃, the authors project onto a thin film geometry, replace the out‑of‑plane momentum by a differential operator, and obtain a four‑band effective Hamiltonian that decouples into two blocks identical in form to the Bernevig–Hughes–Zhang (BHZ) model. Introducing a magnetic field B along the z‑axis via the vector potential A = (0, Bx, 0) leads to the familiar Landau‑level problem. By defining creation and annihilation operators a† and a, the Hamiltonian is rewritten as a pair of coupled harmonic oscillators. The bulk eigenstates are expressed as linear combinations of Hermite functions φₙ(ξ) with coefficients (α, β) that depend on the mass term Δ, Zeeman term μ_eff B, Fermi velocity v_F, and the parameters D and B. The resulting Landau‑level energies are given by Eq. (9)–(10) and include both positive and negative branches for each Landau index n ≥ 1, together with a special n = 0 solution.

To capture the physics of a semi‑infinite sheet (x ≥ 0) with a hard‑wall (Dirichlet) boundary at x = 0, the authors modify the bulk Landau wavefunctions by multiplying them with a factor (1 − e^{−Λ(E)x}). The decay constant Λ(E) is derived analytically (Eq. 13) and ensures the wavefunction vanishes at the edge while leaving the bulk form essentially unchanged far from the boundary. These “Dirichlet Landau states” (D‑LS) constitute the basis for the edge‑state solution. The authors classify D‑LS into three families:

1. Dirichlet bulk states (D‑bulk) – ordinary bulk Landau states dressed with the edge factor; they dominate when the edge dispersion crosses a bulk Landau level.
2. Dirichlet non‑normalizable Landau states (D‑NLS) – states that are not square‑integrable in the infinite plane but become physical once the hard wall is imposed; they describe portions of the spectrum where the edge mode lies in the continuum.
3. Dirichlet complex Landau states (D‑CLS) – built from Landau solutions with complex wave numbers, capturing simultaneous propagation and exponential decay along the edge.

Each family uses the same coefficient vectors v_±^{τ_z n} obtained from the bulk solution, with appropriate choice of the ± branch determined by the sign of τ_z and the direction of k_y. Normalization constants N_±^{τ_z n} guarantee unit probability. The edge dispersion E(k_y) is obtained by projecting the Hamiltonian onto this finite basis and solving the resulting secular equation. Crucially, the authors show that whenever E(k_y) coincides with a bulk Landau energy, the corresponding D‑LS provides an extremely accurate approximation to the true edge wavefunction. This observation leads to a self‑consistency condition that determines the specific k_y values (denoted k_{y,τ_z n}) at which the edge mode hybridizes with a given Landau level, without requiring a full numerical solution of the differential equation.

Numerical calculations for B = 1 T and τ_z = +1 illustrate the method. The edge band (plotted in red) intersects the bulk Landau levels (dashed lines) at discrete k_y points. At each intersection, the analytically constructed D‑bulk wavefunction reproduces the numerically obtained edge state with excellent fidelity, confirming the validity of the approach. The authors also include the Zeeman term with an effective g‑factor of 2; even with this term the edge channel persists up to the largest fields considered, demonstrating that the protection of the edge mode does not rely solely on time‑reversal symmetry but rather on the interplay between the hard wall and the Landau quantization.

In the appendices the derivation of Λ(E) is detailed, the zero‑field edge solution is revisited, and an explicit algorithm for extracting k_{y,τ_z n} from the self‑consistency relation is presented. This algorithm requires only the model parameters and the Landau energies, offering a practical tool for predicting edge‑state behavior in experimental settings.

In summary, the paper delivers a rigorous analytical framework that combines bulk Landau quantization with hard‑wall boundary conditions to describe edge states in Bi₂Se₃ nanosheets under perpendicular magnetic fields. The method reconciles the apparent paradox of edge‑state survival despite broken time‑reversal symmetry, matches numerical simulations with high precision, and provides clear physical insight into the robustness of topological edge channels in magnetic environments. This work lays a solid theoretical foundation for future studies of magnetic‑field‑tuned topological devices and may guide experimental investigations of quantum spin‑Hall physics in thin‑film topological insulators.