General theory of flat-band ferromagnetism in the SU(n) Hubbard model*

The SU(n) Hubbard model, which describes multi-component fermions with SU(n) symmetric interaction, has attracted much attention recently because of its realization in cold-atom setups [1]. However, its analytical treatment is, in general, hard, and there are few rigorous results established for n>2. An example of a rigorous result that is already known for the SU(n) Hubbard model is flat-band ferromagnetism. In the SU(2) case, a general theory of flat-band ferromagnetism has been established for a general class of models with the lowest flat band [2, 3]. On the other hand, in the SU(n) case of an arbitrary integer n>2, only a specific class of flat-band ferromagnetism has been discussed, and no general theory has been presented.

Here we extend the general theory of flat-band ferromagnetism for the SU(2) to the SU(n). As a general setup in which a flat band appears, we consider the model with a positive-semidefinite hopping matrix whose eigenvalues are extensively degenerate at zero energy. We show that when the number of particles equals the degeneracy of the zero-energy states, certain irreducibility of the projection matrix onto the zero eigenspace of the hopping matrix is equivalent to the emergence of the SU(n) ferromagnetism. 

[1] S. Taie et al., Nat. Phys. 8, 825 (2012)

[2] A. Mielke, Phys. Rev. Lett. 82, 4312 (1999)

[3] A. Mielke, J. Phys. A: Math. Gen. 32 , 8411 (1999)