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

Abstract

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 $\mathrm{\Phi }=2\pi m/n$, where $m,n$ are coprime integers. Using reduced exact diagonalization in momentum space, we show that, at fixed $m$, there exists an integer $n\left(m\right)$ associated with a specific value of the magnetic flux, that we denote by ${\mathrm{\Phi }}_c\left(m\right)\equiv 2\pi m/n\left(m\right)$, separating two different regimes. The first one, for fluxes $\mathrm{\Phi }<{\mathrm{\Phi }}_c\left(m\right)$, is characterized by complete band overlaps, while the second one, for $\mathrm{\Phi }>{\mathrm{\Phi }}_c\left(m\right)$, features isolated band-touching points in the density of states and Weyl points between the $m\mathrm{th}$ and the $(m+1)$-th bands. In the Hasegawa gauge, the minimum of the $(m+1)$-th band abruptly moves at the critical flux ${\mathrm{\Phi }}_c\left(m\right)$ from $k_z=0$ to $k_z=\pi$. We then argue that the limit for large $m$ of ${\mathrm{\Phi }}_c\left(m\right)$ exists and it is finite: ${lim}_{m\to \infty }{\mathrm{\Phi }}_c\left(m\right)\equiv {\mathrm{\Phi }}_c$. Our estimate is ${\mathrm{\Phi }}_c/2\pi =0.1296\left(1\right)$. Based on the values of $n\left(m\right)$ determined for integers $m\le 60$, we propose a mathematical conjecture for the form of ${\mathrm{\Phi }}_c\left(m\right)$ to be used in the large-$m$ limit. The asymptotic critical flux obtained using this conjecture is ${\mathrm{\Phi }}_c^{\left(\mathrm{conj}\right)}/2\pi =7/54$.