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Topological superfluids are recently discovered quantum matters that hosttopologically protected gapless edge states known as Majorana fermions - exoticquantum particles that act as their own anti-particles and obey non-Abelianstatistics. Their realizations are believed to lie at the heart of futuretechnologies such as fault-tolerant quantum computation. To date, the mostefficient scheme to create topological superfluids and Majorana fermions isbased on the Sau-Lutchyn-Tewari-Das Sarma model with a Rashba-type spin-orbitcoupling on the }\textbf{\textit{x-y}}\textbf{ plane and a large out-of-plane(perpendicular) Zeeman field along the }\textbf{\textit{z}}\textbf{-direction.Here we propose an alternative setup, where the topological superfluid phase isdriven by applying an in-plane Zeeman field. This scheme offers a number of newfeatures, notably Cooper pairings at finite centre-of-mass momentum (i.e.,Fulde-Ferrell pairing) and gapless excitations in the bulk. As a result, anovel gapless topological quantum matter with inhomogeneous pairing orderparameter appears. It features unidirected Majorana surface states atboundaries, which propagate in the same direction and connect two Weyl nodes inthe bulk. We demonstrate the emergence of such an exotic topological matter andthe associated Majorana fermions in spin-orbit coupled atomic Fermi gases anddetermine its parameter space. The implementation of our scheme insemiconductor/superconductor heterostructures is briefly discussed.

Gapless topological Fulde-Ferrell superfluidity induced by in-plane Zeeman field