Figure 1. Alkaline-earth(-like) atomic Fermi gas with effective nuclear spin .

(a) Sketch of a one-dimensional gas of fermionic atoms with an effective nuclear spin selected from the larger set of 2I+1 nuclear spin states. (b) Energy bands of the Hamiltonian for and ; an energy shift is inserted for representation clarity, but they are degenerate. In this case it is possible to define a Fermi momentum k_F for each spin state, so that the system has in total edges. When Ω is turned on, fermions with momentum difference Δk=±2k_SO and spin difference (solid arrows) or (dashed arrow) get coupled through and , respectively: if the condition k_SO=k_F is met, the system develops a full gap, corresponding to ν=1. (c) When , the system can develop a gap for lower fillings ν=1/q via higher-order scattering terms. As an example, the picture highlights three intermediate processes that generate a coupling between two Fermi edges with Δm=1 of a gas: their sequence (top to bottom) originates a third-order process, which couples two Fermi surfaces for q=3 and k_F=k_SO/3. The same processes take place for any couple of edges with Δm=1. (d) For the condition k_SO=k_F is not enough because (upper panel): when k_SO=π/2 the identification of momenta modulo 2π allows for the creation of a gap (lower panel).