Theory of Integer Quantum Hall Effect in Irrational Magnetic Field

TL;DR

This work presents a universal theory for the integer quantum Hall effect under irrational magnetic flux by leveraging the incommensurate energy band (IEB) framework, which does not rely on magnetic translation symmetry. It shows that each energy gap in the IEB is labeled by a pair $(m,g)$ tied to Bragg planes, and that the filled-state count obeys $N_{\text{occ}}/N_0 = m(\phi/\phi_0) + g$, leading, via the Středa formula, to a quantized Hall conductance $\sigma_{xy} = m e^2/h$ for arbitrary flux. The theory unifies the rational and irrational flux cases, reinterpreting quantization as a geometric constraint from Bragg planes rather than a Chern-number framework. It also provides a momentum-space perspective on the Hofstadter spectrum through IEBs, connecting gap labeling to the Wannier diagram and offering practical computational routes via the similar AAH/quasiperiodic mappings. This establishes a new paradigm for IQHE with broad applicability to quasiperiodic lattice systems.

Abstract

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 $N_{\text{occ}}/N_0 = m(φ/φ_0) + g$. Through Středa formula, this leads directly to the quantized Hall conductance $σ_{xy} = m e^2/h$ under arbitrary magnetic fields. Our work resolves the long-standing problem of IQHE under irrational flux, and establishes a new paradigm for IQHE.

Figures (5)

• Figure 1: IEBs and Hofstadter spectrum of a square lattice with $t_2 = 0.75t_1$. (a) is the IEB of $\alpha = 3/7$ (red lines, rational case) along $k_y=k_x$ direction in BZ. (b) is the Hofstadter spectrum in selected energy region. (c) is the IEB of $\alpha = (\sqrt{5}-1)/2$ (blue lines, irrational case). In (a) and (c), the replica bands corresponding to $|K_1\rangle$ and $|K_{-1}\rangle$ are highlighted with orange and azure lines, respectively. The vertical dashed lines represent the Bragg planes. Here, the cutoff $n_c = 12$ is used, and $t_1=t$ as energy unit.
• Figure 2: Enlarged IEBs of $\alpha = (\sqrt{5}-1)/2$ in different energy regions. Shadow region is the corresponding Hofstadter spectrum line. Vertical dashed lines represent the Bragg planes associated with the gaps. Gap labels $(m,g)$ are given as well. All the parameters are the same as Fig. \ref{['Hofstadter1']}.
• Figure 3: (a) is the Hofstadter spectrum with $t_2 = t_1$. Several largest enclosed domains are colored according to their gap labels $(m,g)$. (b) is the corresponding Wannier diagram.
• Figure S1: IEBs and Hofstadter spectrum of a square lattice with $t_2 = t_1$. (a) is the IEB of $\alpha = 3/7$ (red lines, rational case) along $k_y=k_x$ direction in BZ. (b) is the Hofstadter spectrum in selected energy region. (c) is the IEB of $\alpha = (\sqrt{5}-1)/2$ (blue lines, irrational case). In (a) and (c), the replica bands corresponding to $|K_1\rangle$ and $|K_{-1}\rangle$ are highlighted with orange and azure lines, respectively. The vertical dashed lines represent the Bragg planes. Here, the cutoff $n_c = 44$ is used, and $t_1=t$ as energy unit.
• Figure S2: IEBs and Hofstadter spectrum of a square lattice with $t_2 = 0$. (a) is the IEB of $\alpha = 3/7$ (red lines, rational case) along $k_y=k_x$ direction in BZ. (b) is the Hofstadter spectrum in selected energy region. (c) is the IEB of $\alpha = (\sqrt{5}-1)/2$ (blue lines, irrational case). In (a) and (c), the replica bands corresponding to $|K_1\rangle$ and $|K_{-1}\rangle$ are highlighted with orange and azure lines, respectively. The vertical dashed lines represent the Bragg planes. Here, the cutoff $n_c = 44$ is used, and $t_1=t$ as energy unit.