Critical phase boundaries of static and periodically kicked long-range Kitaev chain

We study the static and dynamical properties of a long-range Kitaev chain, i.e., a $p$-wave superconducting chain in which the superconducting pairing decays algebraically as $1/l^{alpha}$, where $l$ is the distance between the two sites and $alpha$ is a positive constant. Considering very large system sizes, we show that when $alpha >1$, the system is topologically equivalent to the short-range Kitaev chain with massless Majorana modes at the ends of the system; on the contrary, for $alpha <1$, there exist symmetry protected massive Dirac end modes. We further study the dynamical phase boundary of the model when periodic $delta$-function kicks are applied to the chemical potential; we specially focus on the case $alpha >1$ and analyze the corresponding Floquet quasienergies. Interestingly, we find that new topologically protected massless end modes are generated at the quasienergy $pi/T$ (where $T$ is the time period of driving) in addition to the end modes at zero energies which exist in the static case. By varying the frequency of kicking, we can produce topological phase transitions between different dynamical phases. Finally, we propose some bulk topological invariants which correctly predict the number of massless end modes at quasienergies equal to 0 and $pi/T$ for a periodically kicked system with $alpha > 1$.