Correlation functions of the Kitaev model with a spatially modulated phase in the superconducting order parameter

The Kitaev chain model with a spatially modulated phase in the superconducting order parameter exhibits two types of topological transitions, namely a band topology transition between trivial and topological gapped phases, and a Fermi surface Lifshitz transition from a gapped to a gapless superconducting state. We investigate the correlation functions of the model for arbitrary values of superconducting coupling~$\Delta_0$, chemical potential $\mu$, and phase modulation wavevector $Q$, characterizing the current flowing through the system. In the cases $\mu=0$ or $Q=\pm \pi/2$ the model turns out to exhibit special symmetries, which are proven to induce an even/odd effect in the correlations as a function of the distance $l$ between two lattice sites, as they are non-vanishing or strictly vanishing depending on the parity of $l$, measured in the lattice spacing unit. We identify a clear difference between the band topology and the Lifshitz transition through the $Q$-dependence of the short distance correlation functions, which, in particular, exhibit pronounced cusps with discontinuous derivatives across the Lifshitz transition. We also determine the long distance behavior of correlations, finding that in the gapped phase there can be various types of exponential decays and that in the gapless phase the algebraic decay is characterized by two different spatial periods, depending on the model parameters. Furthermore, we establish a connection between the gapless superconducting phase of the Kitaev chain and the chiral phase of spin models with Dzyaloshinskii-Moriya interaction.