Phase transition and split property in quantum spin chain

In this exposition we investigate further the general methodology proposed in [Mo2] to study properties of the ground states of a translation invariant Hamiltonian for one lattice dimensional quantum spin chain $\cla=\otimes_{\IZ}M_d$, where $M_d$ is the matrix of $d \times d$ complex matrices. We introduce a notion of quantum detailed balance [Mo1] for a translation invariant state on $\cla$ and prove that such a pure state is uniformly mixing [BR,Ma2] if and only 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. \vsp As a consequence of these results we conclude that if the ground states for an integer spin SU(2) invariant ($2s+1=d$) detailed balanced Hamiltonian is unique then the state is split. In particular if the ground state for integer spin anti-ferromagnetic Heisenberg chain is unique, then our main result says that the state is uniformly mixing and lattice space correlation functions of the ground state decay exponentially. Our main result is general enough to have application to other well known models such as Ising model, XY model and quasi-one dimensional quantum spin ladder [DR,Ma2] magnetic materials.