From SU(2) to SU(N)

From two-component Fermi gas, we already know that one could get SU(2) Fermi gas by simply picking two hyperfine state.

You might wonder if one could get SU(N) by picking up \(N\) hyperfine state, the general answer for alkaline atoms is no. When two atoms collide, the scattering length depends on whether the outer most electrons pairs up in a singlet or triplet state. Thus, different hyperfine states are not identical to each other, indicating the lose of SU(N) symmetry.

For alkaline-earth-like atoms, two outer most electrons pairs up in a total \(S=0\) state. The total hyperfine spin \(F\) number is entirely determined by the nuclear spin $I$ since the ground state angular momentum \(L=0\). This renders every component identical to each other.

In experiment, one could even control the number of hyperfine spins by selectively projecting out the undesired spin.

Experimental Realization

Prof. Gyu-Boong Jo’s group at HKUST realized the SU(N) Fermi gas in low temperature They are able to measure the contact after time-of-flight expansion as a function of both the temperature and the number of component.


C. N. Yang predicted bosonization of SU(N) fermions at zero temperature. That is, thermodynamics properties of the SU(N) Fermi gas would approach that of the single component Bose gas as \(N\) approaches infinity.

Virial expansion

We also independently verify the bosonization using the virial expansion of the thermodynamic potential for the SU(N) Fermi gas. Because we are still in the weakly interacting regime, the virial coefficients for the SU(N) Fermi gas could be deduced from the SU(2) Fermi gas.

The contact is proportional to the partial derivative of the thermodynamic potential with respect to inverse scattering length.

We thus have the contact as a function of \(N\) and temperature in pure theory as well. Comparing to the experiment, we find the \(N\) and temperature dependency agrees well for temperature around Fermi temperature.


Song, Bo, Yan, Yangqian, He, Chengdong, Ren, Zejian, Zhou, Qi, and Jo, Gyu-Boong, Evidence for bosonization in a three-dimensional gas of SU (N) fermions, Phys. Rev. X 10, 041053 (2020)