Phase transition and split property in quantum spin chain

In this exposition we investigate further the methodology developed in [Mo2] to study properties of the ground states of a translation invariant Hamiltonian for one lattice dimensional quantum spin chain A = ⊗ZZMd, where Md is the space of d × d complex matrices. To that end we study the associated Cuntz elements [Mo2] representing a translation invariant pure state and find an useful criteria for the state to be in quantum detailed balance [Mo1]. This criteria is further explored to prove that such a pure state is split or uniformly mixing [BR,Ma2] if the lattice space correlation functions decay exponentially. Furthermore we also prove that a pure lattice symmetric, translation and SU(2) gauge invariant state give rise to a canonical Popescu systems acting on a finite dimensional Hilbert space and thus the lattice space correlation functions of the pure state decay exponentially. The above result has a ready generalization where the symmetry group SU(2) can be replaced by a class of simply connected compact Lie-group.