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 $\Phi =2\pi \cdot m/n$, 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 $m\le 60$, 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 ${\Phi }_c^{(\mathrm{conj})}/2\pi =7/54$.

13 pages + 7 pages appendix/references, 9 figures

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