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We introduce and solve a two-band model of spinless fermions with $p_x$-wavepairing on a square lattice. The model reduces to the well-known extendedHarper-Hofstadter model with half-flux quanta per plaquette and weakly coupledKitaev chains in two respective limits. We show that its phase diagram containsa topologically nontrivial weak pairing phase as well as a trivial strongpairing phase as the ratio of the pairing amplitude and hopping is tuned.Introducing periodic driving to the model, we observe a cascade of Floquetphases with well defined quasienergy gaps and featuring chiral Majorana edgemodes at the zero- or $\pi$-gap, or both. Dynamical topological invariants areobtained to characterize each phase and to explain the emergence of edge modesin the anomalous phase where all the quasienergy bands have zero Chern number.Analytical solution is achieved by exploiting a generalized mirror symmetry ofthe model, so that the effective Hamiltonian is decomposed into that ofspin-$1/2$ in magnetic field, and the loop unitary operator becomes spinrotations. We further show the dynamical invariants manifest as the Hopflinking numbers.

Tuning the topology of $p$-wave superconductivity in an analytically solvable two-band model