Unified Integer and Fractional Quantum Hall Effects from Boundary-Induced Edge-State Quantization

Abstract

Despite the success of Landau-level theory and edge-state transport formalisms, a direct microscopic link between bulk quantization and the observed hierarchy of quantum Hall plateaus has not been established. In particular, no unified microscopic mechanism accounting simultaneously for integer and fractional sequences has been derived within standard quantum mechanics. Here we show that boundary-induced quantization of edge states provides this missing bridge. Starting from the Landau problem in laterally confined two-dimensional electron systems, we demonstrate that the imposition of Dirichlet, Neumann, and mixed (Robin) boundary conditions discretizes both the guiding-center coordinate and the longitudinal momentum of chiral edge states. The resulting boundary-dependent spectra generate families of edge channels with well-defined multiplicities that couple to electronic transport. When incorporated into an edge-state transport description, this boundary quantization reproduces the integer Hall sequence and simultaneously yields a structured hierarchy of fractional filling factors without invoking separate microscopic mechanisms. We further show that a weak Hall-induced parity-breaking contribution reorganizes the low-energy edge spectrum while leaving the bulk Landau levels intact. This controlled symmetry breaking enhances edge-state multiplicities at small Landau indices and stabilizes the fractional plateaus observed at strong magnetic fields. The quantized Hall response thus emerges from the interplay between Landau quantization and boundary-induced guiding-center discretization, which together determine the spectrum and occupation of chiral edge channels. These results establish boundary-induced quantization as the microscopic origin of quantum Hall transport and provide a unified description of both integer and fractional regimes within conventional quantum mechanics.

Figures (12)

• Figure 1: Quantized guiding centers obtained from Dirichlet and Neumann boundary conditions, together with the corresponding Landau wave functions evaluated at those centers. Panels (a) and (b) show the guiding-center positions $y_{0,nj}$ obtained from Eqs. (\ref{['BC_Dirichlet']}) and (\ref{['BC_Neumann']}), respectively. In the right-hand column, the Landau functions for $n=4$ are plotted at the guiding-center positions indicated in the left column. The boundary conditions place the guiding centers near the sample edges and produce similar edge localization. The Robin spectrum, shown in Fig. \ref{['Robinyon']}, consists of the Dirichlet spectrum supplemented by two additional guiding-center positions located near the inflection points of the Landau wave packet envelope. The magnetic field is $B=15$ T and the stripe width is $L_y=0.2~\mu$m.
• Figure 2: Kinetic and total energies of edge channels as functions of magnetic field. Left panel: longitudinal kinetic energies $K_{nj}=\hbar^2 k_{x,nj}^2/2m^*$ for the first five Landau levels, computed from the exact quantized momenta in Eq. (\ref{['defkxnj']}). Approximately half of the kinetic branches remain below the Fermi energy. Right panel: total energies $E_{nj}=K_{nj}+U_{nj}$. Including the magnetic confinement potential $U_{nj}$ reorders the spectrum: branches associated with large $|\xi_{nj}|$ are shifted upward, while those with small or vanishing $\xi_{nj}$ experience smaller energy shifts. As a result, fewer channels remain below $E_F$, whereas the $\xi_{nj}=0$ branches remain purely kinetic and evolve together with the $n=1$ Landau level. The persistence (pinning) of selected channels after the bulk Landau level crosses $E_F$ constitutes a microscopic mechanism responsible for residual chiral edge channels and, consequently, for the emergence of fractional Hall plateaus. Parameters are indicated in the plots; details on the effective stripe width $L_y$ are provided in the Supplementary Information.
• Figure 3: Effective eigenfunction index $n'$ and corresponding number of edge states, obtained under the Neumann boundary condition, as functions of the Landau index $n$ and the parity-breaking parameter $b$. The solid curves show the analytical expression of $n'(n,b)$ in (\ref{['Landaunp']}). Horizontal dashed lines indicate integer values of $n'$. Small circles denote the number of zeros of the Neumann boundary condition, $\left.d\eta/dy\right|_{y=\pm L_y/2}=0$, which correspond to local extrema of the Hermite--Gaussian envelope where the guiding centers are pinned by the boundary. For finite parity breaking (e.g. $b\simeq0.03$), the resulting integer sequences exhibit a non-monotonic dependence on $n$, signaling an enhancement of low-energy edge-state multiplicity at small Landau indices.
• Figure 4: Schematic representation of the Landauer–Büttiker transport picture applied to the parity-quantized Landau spectrum. Electrons are injected from the left reservoir (chemical potential $\mu_1$) into the quantized edge channels whose total energies $E_{nj}$ match the Fermi level $E_F$; resonant tunneling across the contact barrier selectively populates these discrete states, while electrons injected into non-resonant states relax inelastically toward them. Inside the sample, edge electrons propagate chirally along equipotential contours with constant longitudinal momentum $k_{x,nj}$ and strongly suppressed backscattering. At the drain contact (chemical potential $\mu_2$), part of the incoming flux is transmitted into the reservoir (forming the transmitted current $I_t$) and part is reflected back into lower edge channels, contributing to the reflected edge flux. The transmission coefficient $T_j$ for individual channel $j$ results from the sum over reservoir feeding states, and the net transmission summed over the $\nu_p$ populated channels equals the $T_{1p}$ used in the text (see Eqs. \ref{['T1']}--\ref{['FC']}), providing the flux-conservation link between microscopic channel transmissions and the macroscopic currents.
• Figure 5: Experimental and theoretical Hall resistance (in units of $h/e^2$). (a) Theoretical prediction compared with the experimental Hall-resistance curve measured in a two-dimensional electron system at a GaInAs/InP interface, reported in Ref. Koch1993 (reproduced with permission from Elsevier). (b) Comparison between theory and experiment for a two-dimensional electron system in a GaAs/(AlGa)As heterostructure, reported in Ref. WeisVonKlitzing (reproduced with permission from The Royal Society, UK). Theoretical curves were obtained from Eq. (\ref{['Hallreslfloor']}) using the boundary- determined effective filling factor for the Fermi energies indicated in each panel, without additional fitting parameters. Both integer and fractional plateaus arise from the same boundary-quantized edge spectrum.