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We theoretically study a one-dimensional quasi-periodic Fermi system withtopological $p$-wave superfluidity, which can be deduced from a topologicallynon-trivial tight-binding model on the square lattice in a uniform magneticfield and subject to a non-Abelian gauge field. The system may be regarded anon-Abelian generalization of the well-known Aubry-Andr\'e-Harper model. Weinvestigate its phase diagram as functions of the strength of thequasi-disorder and the amplitude of the $p$-wave order parameter, through anumber of numerical investigations, including a multifractal analysis. Thereare four distinct phases separated by three critical lines, i.e., two phaseswith all extended wave-functions (I and IV), a topologically trivial phase (II)with all localized wave-functions and a critical phase (III) with allmultifractal wave-functions. The phase I is related to the phase IV by duality.It also seems to be related to the phase II by duality. Our proposed phasediagram may be observable in current cold-atom experiments, in view ofsimulating non-Abelian gauge fields and topological insulators/superfluids withultracold atoms.

Phase diagram of a non-Abelian Aubry-Andr\'e-Harper model with $p$-wave superfluidity