There has been growing interest in investigating properties of elementary particles predicted by the standard model. Examples of such studies include exploring their low-energy analogs in a condensed-matter system, where they arise as collective states or quasiparticles. Here we show that a toolbox for systematically engineering the emergent elementary fermions, i.e., Dirac, Weyl, and Majorana fermions, can be built in a single atomic system composed of a spinless magnetic dipolar Fermi gas in a three-dimensional optical lattice. The designed direction-dependent dipole-dipole interaction leads to both the basic building block, i.e., in-plane $p+ip$ superfluid pairing instability, and the manipulating tool, i.e., out-of-plane Peierls instability. It is shown that the Peierls instability provides a natural way of tuning the topological nature of $p+ip$ superfluids and can transform the fermion's nature between distinct emergent particles. Our scheme should contribute to the search for elementary particles through manipulating the topology.

• Received 18 May 2021
• Revised 3 August 2021
• Accepted 5 August 2021

DOI:https://doi.org/10.1103/PhysRevA.104.033312