  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.
