Zigzag Spin Model at F-AF Transition Point

Let us consider the s = spin chain with nearest- and next-nearest interactions given by the Hamiltonian 

We note that the Hamiltonian (2) has a symmetry, its spectrum coincides with the spectrum of H(x) obtained by the following transformation 

This transformation permutes the factors at the first and the second terms in the Hamiltonian (2). Thus, due to the symmetry it is sufficient to consider the range 

we will show that the ground-state energy of (2) is zero. Let us represent the Hamiltonian (2) as a sum of Hamiltonians hn of cells containing three sites 

We will present a singlet wave function which is an exact zero-energy eigenvector to each hn and, therefore, it is the exact ground-state wave function of (2). This function has a form 

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