Time-reversal-broken Weyl semimetal in the Hofstadter regime

We study the phase diagram for a lattice model of a time-reversal-broken three-dimensional Weyl semimetal (WSM) in an orbital magnetic field B with a flux of p/q per unit cell (0 ≤ p ≤ q - 1), with minimal crystalline symmetry. We find several interesting phases: (i) WSM phases with 2q, 4q, 6q, and 8q Weyl nodes and corresponding surface Fermi arcs, (ii) a layered Chern insulating (LCI) phase, gapped in the bulk, but with gapless surface states, (iii) a phase in which some bulk bands are gapless with Weyl nodes, coexisting with others that are gapped but topologically nontrivial, adiabatically connected to an LCI phase, (iv) a new gapped trivially insulating phase (I⁰) with (non-topological) counter-propagating surface states, which could be gapped out in the absence of crystal symmetries. Importantly, we are able to obtain the phase boundaries analytically for all p, q. Analyzing the gaps for p = 1 and very large q enables us to smoothly take the zero-field limit, even though the phase diagrams look ostensibly very different for q = 1, B = 0, and q → ∞, B → 0.