Theory of Integer Quantum Hall Effect in Irrational Magnetic Field

The conventional theory of the integer quantum Hall effect (IQHE) fails for irrational magnetic fields owing to the breakdown of magnetic translational symmetry. Here, based on the recently proposed incommensurate energy band (IEB) theory, we present a universal IQHE theory that does not rely on magnetic translation symmetry and is applicable to both rational and irrational magnetic fluxes. Using the square lattice as a paradigmatic example, we first show that the IEB framework provides a superior description of its energy spectrum in a magnetic field, as it explicitly reveals the momentum-space distribution of eigenstates. Key to our IQHE theory is that each gap in the IEB spectrum is intrinsically labeled by an integer pair (m,g), defined by the corresponding Bragg planes. When the Fermi energy lies within such a gap, the occupied electron states $N_{\text{occ}}$ is determined by the k-space volume enclosed by these Bragg planes, leading to the fundamental relation $N_{\text{occ}}/N_0 = m(\phi/\phi_0) + g$. Through St\v{r}eda formula, this leads directly to the quantized Hall conductance $\sigma_{xy} = m e^2/h$ under arbitrary magnetic fields. Our work resolves the long-standing problem of IQHE under irrational flux, and establishes a new paradigm for IQHE.