Anatomy of a periodically driven p-wave superconductor

The topological properties of periodically driven many-body systems often have no static analogs and defy a simple description based on the effective Hamiltonian. To explore the emergent edge modes in driven p-wave superconductors in two dimensions, we analyze a toy model of Kitaev chains (one-dimensional spinless p-wave superconductors with Majorana edge states) coupled by time-periodic hopping. We show that with proper driving, the coupled Kitaev chains can turn into a fully gapped superconductor which is analogous to the $p_x+ip_y$ state but has two, rather than one, chiral edge modes. A different driving protocol turns it into a gapless superconductor with isolated point nodes and completely flat edge states at quasienergy $\omega=0$ or $\pi/T$, with $T$ the driving period. The time evolution operator $U(k_x,k_y,t)$ of the toy model is computed exactly to yield the phase bands. And the "topological singularities" of the phase bands are exhausted and compared to those of a periodically driven Hofstadter model which features counter-propagating chiral edge modes. These examples demonstrate the unique edge states in driven superconducting systems and suggest driving as a potentially fruitful route to engineer new topological superconductors.