Though our primary motivation is to study properties of the ground states of a Hamiltonian for quantum spin chain, in this exposition we propose a general methodology valid for a translation invariant state. To that end we study the associated Popescu systems [Po,BJKW] representing the translation invariant state and find an useful criteria for the state to be pure. As a simple consequence of this criteria we prove a surprising result which says that there exists no translation invariant lattice symmetric and SU(2)-gauge invariant pure state for half-odd integer spin chain. As a consequence of this result we prove that ground state for half-odd integer spin chain is not unique for any SU(2)-gauge invariant translation and lattice symmetric Hamiltonian. In other words this exhibits that there is a phase transition in the ground state for anti-ferromagnetic isotropic Heisenberg half-odd integer spin chain. 2 It has been known in physics literature that the spin 1
Exact results are derived, via the Temperley-Lieb-Jones algebra, for the ground state energy per site, surface energy, gap and central charge of the anisotropic generalization of the spin-1 biquadratic model.A recently observed simplification in the ground state energy of the U.[SU(2)]-invariant Zamolodchikov-Fateev model at q=e'x 14 is shown to be a direct consequence of a trivial representation of the related operator algebra.Similar points are conjectured to exist in the more general u.[SU(2)]-invariant spin-S chains at q=e 1 " 1 < 2 + 25 '.§ 1. Introduction