Topological Fulde-Ferrell and Larkin-Ovchinnikov states in spin-orbit coupled lattice system

The spin-orbit coupled lattice system under Zeeman fields provides an ideal platform to realize exotic pairing states. Notable examples range from the topological superfluid/superconducting (tSC) state, which is gapped in the bulk but metallic at the edge, to the Fulde-Ferrell (FF) state (having a phase-modulated order parameter with a uniform amplitude) and the Larkin-Ovchinnikov (LO) state (having a spatially varying order parameter amplitude). Here, we show that the topological FF state with Chern number ($\mathcal{C}=-1$) (tFF$_{1}$) and topological LO state with $\mathcal{C}=2$ (tLO$_{2}$) can be stabilized in Rashba spin-orbit coupled lattice systems in the presence of both in-plane and out-of-plane Zeeman fields. Besides the inhomogeneous tSC states, in the presence of a weak in-plane Zeeman field, two topological BCS phases may emerge with $\mathcal{C}=-1$ (tBCS$_{1}$) far from half filling and $\mathcal{C}=2$(tBCS$_{2}$) near half filling. We show intriguing effects such as different spatial profiles of order parameters for FF and LO states, the topological evolution among inhomogeneous tSC states, and different non-trivial Chern numbers for the tFF$_{1}$ and tLO$_{1,2}$ states, which are peculiar to the lattice system. Global phase diagrams for various topological phases are presented for both half-filling and doped cases. The edge states as well as local density of states spectra are calculated for tSC states in a 2D strip.