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Recently three dimensional topological quantum materials with gapless energyspectra have attracted considerable interests in many branches of physics.Besides the celebrated example, Dirac and Weyl points which possess gaplesspoint structures in the underlying energy dispersion, the topologicallyprotected gapless spectrum can also occur along a ring, named Dirac and Weylnodal rings. Ultra-cold atomic gases provide an ideal platform for exploringnew topological materials with designed symmetries. However, whether Dirac andWeyl rings can exist in the single-particle spectrum of cold atoms remainselusive. Here we propose a realistic model for realizing Dirac and Weyl ringsin the single-particle band dispersion of a cold atom optical lattice. Ourscheme is based on previously experimentally already implemented Raman couplingsetup for realizing spin-orbit coupling. Without the Zeeman field, the modelpreserves both pseudo-time-reversal and inversion symmetries, allowing Diracrings. The Dirac rings split into Weyl rings with a Zeeman field that breaksthe pseudo-time-reversal symmetry. We examine the superfluidity of attractiveFermi gases in this model and also find Dirac and Weyl rings in thequasiparticle spectrum.

Dirac and Weyl Rings in Three Dimensional Cold Atom Optical Lattices