Lieb-Schultz-Mattis constraints for the insulating phases of the one-dimensional SU($N$) Kondo lattice model

The nature of the insulating phases of the $\mathrm{SU}(N$)-generalization of the one-dimensional Kondo lattice model is investigated by means of nonperturbative approaches. By extending the Lieb-Schultz-Mattis (LSM) argument to multicomponent fermion systems with translation and global $\mathrm{SU}(N$) symmetries, we derive two indices which depend on the filling and the "$\mathrm{SU}(N$)-spin" (representation) of the local moments. These indices strongly constrain possible insulating phases; for instance, when the local moments transform in the $N$-dimensional (defining) representation of $\mathrm{SU}(N$), a featureless Kondo insulator is possible only at filling $f=1-1/N$. To obtain further insight into the insulating phases suggested by the LSM argument, we derive low-energy effective theories by adding an antiferromagnetic Heisenberg exchange interaction among the local moments [the $\mathrm{SU}(N$) Kondo-Heisenberg model]. A conjectured global phase diagram of the $\mathrm{SU}(N$) Kondo lattice model as a function of the filling and the Kondo coupling is then obtained by a combination of different analytical approaches.