  We study the Harper-Hofstadter Hamiltonian and its corresponding
non-perturbative butterfly spectrum. The problem is algebraically solvable
whenever the magnetic flux is a rational multiple of $2\pi$. For such values of
the magnetic flux, the theory allows a formulation with two Bloch or
$\theta$-angles. We treat the problem by the path integral formulation, and
show that the spectrum receives instanton corrections. Instantons as well as
their one loop fluctuation determinants are found explicitly and the finding is
matched with the numerical band width of the butterfly spectrum. We extend the
analysis to all 2-instanton sectors with different $\theta$-angle dependence to
leading order and show consistency with numerics. We further argue that the
instanton--anti-instanton contributions are ambiguous and cancel the ambiguity
of the perturbation series, as they should. We hint at the possibility of exact
2-instanton solutions responsible for such contributions via Picard-Lefschetz
theory. We also present a powerful way to compute the perturbative fluctuations
around the 1-instanton saddle as well as the instanton--anti-instanton
ambiguity by using the topological string formulation.

High Energy Physics - Theory