Unified microscopic theory of integer and fractional quantum Hall effects

We demonstrate that the integer and fractional quantum Hall effects originate from a common microscopic mechanism: boundary-induced quantization of edge states in confined two-dimensional electron systems. Although integer and fractional Hall plateaus are traditionally described using distinct theoretical frameworks, their universal coexistence in finite samples has lacked a unified microscopic explanation. Here we show that this missing link is provided by the quantization of edge states by physically consistent boundary conditions. Starting from the Landau problem in finite geometries, we show that Dirichlet, Neumann, and mixed (Robin) boundary conditions discretize the guiding-center coordinate and the longitudinal momentum of edge states, generating families of chiral edge channels with boundary-dependent multiplicities. When incorporated into the Landauer–Büttiker transport formalism, these multiplicities naturally produce the integer Hall plateaus and a structured hierarchy of fractional filling factors. We further show that retaining a weak Hall-induced parity-breaking contribution reorganizes the low-energy edge spectrum without modifying the bulk Landau levels, enabling the strong-field fractional plateaus observed experimentally. Boundary quantization, refined by weak parity breaking, produces a fine structure of edge energies in which selected chiral branches remain pinned to the Fermi level, providing a unified microscopic foundation for quantized Hall transport across both integer and fractional regimes.