We study the Hofstadter model on a hexagonal lattice with irrational magnetic flux in this work. The Hofstadter model of the square lattice with irrational flux has been solved mathematically by Avila [Invent. Math. 210, 283 (2017);Acta Math. 215, 1 (2015)] and his collaborators in his Fields medal work. However, this theory is usually not applicable to lattices with internal degrees of freedom, such as spin or sublattices. In this work, we show that for the hexagonal lattice with only nearest-neighbor hoppings, the system can still be characterized by a $2\times 2$ transfer matrix and solved exactly by Avila's global theory, although this lattice has two sublattices. We obtained the exact localization phase diagram of the hexagonal lattice with irrational flux by this theory, which reveals three pure phases, i.e., the extended, localized, and critical states but no mobility edge due to the chiral symmetry. We used the renormalization group (RG) theory to verify these results, which can determine part of the phase diagram. We then computed the fractal dimension of the remaining part numerically. The results from both the RG theory and numerical analysis confirmed the phase diagram we get from Avila's global theory. Our results may be tested in various hexagonal moiré lattices and artificial superlattices achieved in recent experiments.

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• Anderson localization
• Honeycomb lattice
• Transfer matrix method