Rigorous determination of spontaneously dimerized ground states in 1D Heisenberg antiferromagnets

Starting from the exact result of Majumdar and Ghosh (1969) for the dimerized ground state of a Heisenberg S=¹/₂ antiferromagnet with competing next-nearest-neighbour interaction J ₂+1/2J₁ on a linear chain, it is shown that this state is the ground state of the same kind of Hamiltonian, if couplings are extended up to 2n-nearest-neighbours, and the relation J₁=2n, J₂=2n-1, ..., J_2n=1 holds. Furthermore, it is proved rigorously that this state is the ground state when n=2 and J₁-2J₂+J ₃=0, J₃=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<or=J₄<or=¹/ ₄J₁. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising model is discussed.