Time-reversal-broken Weyl semimetal in the Hofstadter regime

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