Critical magnetic flux for Weyl points in the three-dimensional Hofstadter model

We investigate the band structure of the three-dimensional Hofstadter model on cubic lattices, with an isotropic magnetic field oriented along the diagonal of the cube with flux $\Phi=2 \pi \cdot m /n$, where $m,n$ are co-prime integers. Using reduced exact diagonalization in momentum space, we show that, at fixed $m$, there exists an integer $n(m)$ associated with a specific value of the magnetic flux, that we denote by $\Phi_c(m) \equiv 2 \pi \cdot m/n(m)$, separating two different regimes. The first one, for fluxes $\Phi<\Phi_c(m)$, is characterized by complete band overlaps, while the second one, for $\Phi>\Phi_c(m)$, features isolated band touching points in the density of states and Weyl points between the $m$- and the $(m+1)$-th bands. In the Hasegawa gauge, the minimum of the $(m+1)$-th band abruptly moves at the critical flux $\Phi_c(m)$ from $k_z=0$ to $k_z=\pi$. We then argue that the limit for large $m$ of $\Phi_c(m)$ exists and it is finite: $\lim_{m\to \infty} \Phi_c(m) \equiv \Phi_c$. Our estimate is $\Phi_c/2\pi=0.1296(1)$. Based on the values of $n(m)$ determined for integers $m\leq60$, we propose a mathematical conjecture for the form of $\Phi_c(m)$ to be used in the large-$m$ limit. The asymptotic critical flux obtained using this conjecture is $\Phi_c^{{\rm (conj)}}/2\pi=7/54$.