Spin Gap in a Doped Kondo Chain

We show that the Kondo chain away from half filling has a spin gap upon the introduction of an additional direct Heisenberg coupling between localized spins. This is understood in the weak-Kondo-coupling limit of the Heisenberg-Kondo lattice model by bosonization and in the strong-coupling limit by a mapping to a modified $t-J$ model. Only for certain ranges of filling and Heisenberg coupling does the spin gap phase extend from weak to strong coupling.