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 Φ=2πm/n, where m,n are coprime 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 Φc(m)≡2πm/n(m), separating two different regimes. The first one, for fluxes Φ<Φc(m), is characterized by complete band overlaps, while the second one, for Φ>Φc(m), features isolated band-touching points in the density of states and Weyl points between the mth and the (m+1)-th bands. In the Hasegawa gauge, the minimum of the (m+1)-th band abruptly moves at the critical flux Φc(m) from kz=0 to kz=π. We then argue that the limit for large m of Φc(m) exists and it is finite: limm→∞Φc(m)≡Φc. Our estimate is Φc/2π=0.1296(1). Based on the values of n(m) determined for integers m≤60, we propose a mathematical conjecture for the form of Φc(m) to be used in the large-m limit. The asymptotic critical flux obtained using this conjecture is Φ(conj)c/2π=7/54.
