Spontaneously dimerized ground states in 1D frustrated Heisenberg antiferromagnets

We give an explicit proof that the completely dimerized ('Kekulè or nearest-neighbor resonant valence bond) state is the degenerate ground state of a linear chain S =1/2 Heisenberg antiferromagnet, with competing couplings extending up to 2n nearest neighbors, if the relation J₁ = 2n, J ₂ = 2n - 1,.,J_2n = 1, holds between the various exchange integrals. The previously known result for J₂ =1/2 J₁ is then just a special case corresponding to n=1. We study in detail n=2 (fourth-nearest neighbors) and show that this state is the ground state of the Hamiltonian when J₁ - 2J₂ + J₃ = 0, J ₃ = 2J₄, in a finite interval of values of the parameter J₄. A rigorous lower limit for the extension of this interval is found to be 0 ≤ J₄ ≤1/4 J₁. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising models is made.