Argument on why spin correlation functions in Ising model decay exponentially with a correlation length? I'm reading Quantum Field Theory in Strongly Correlated Electronic Systems, Nagaosa.
Consider 1D Ising model,
$$H=J_z\sum_i S^z_iS^z_{i+1}.$$
on page 3, it says

The groud stae is 2-fold degenerate because the Hamiltonian is invariant
under the transformation $S^i_z \rightarrow -S^i_z$, performed at all sites $i$.
Calling these two ground states $A$ and $B$ and assuming that the system at the right-hand side is in state $A$, and at the left-hand side in state $B$, then somewhere there
must exist a boundary between region $A$ and region $B$. This boundary is called a kink or soliton. Because at finite temperature this excitation occurs
with a finite density, the spin correlation function $F(r) =\langle S^z_iS^z_{i+r}\rangle$ will decay exponentially with a correlation length $\xi$.

I know how to directly calculate the correlation function, but I wonder how the argument for exponential decay of correlation function is made here and how to understand it.
Any help would be highly appriciated!!
 A: Let me write the Hamiltonian
$$
H = -J \sum_i S_i^z S_{i+1}^z.
$$
This choice will avoid some annoying (and irrelevant) signs.
One way to formulate the statement in the OP precisely is as follows.
Consider the variables $\delta_i=S_i^zS_{i+1}^z$. Since $\delta_i=1$ when the spins at $i$ and $i+1$ agree and $\delta_i=-1$ when the spins at $i$ and $i+1$ disagree, you can identify them with the kinks in your question (that is, there is a kink between $i$ and $i+1$ when $\delta_i=-1$).
Introducing the variables $\delta_i=S_i^zS_{i+1}^z$, the Hamiltonian becomes
$$
H = J^z \sum_i \delta_i.
$$
It follows that the random variables $\delta_i$ are independent and identically distributed. One can easily compute their expectation: since
$$
P(\delta_i = 1) = \frac{e^{\beta J^z}}{e^{\beta J^z} + e^{-\beta J^z}},
$$
one has
$$
\langle \delta_i \rangle = \frac{e^{\beta J^z} - e^{-\beta J^z}}{e^{\beta J^z} + e^{-\beta J^z}} = \tanh(\beta J^z).
$$
Finally, noting that $S_i^zS_{i+r}^z = \delta_i\delta_{i+1}\cdots\delta_{i+r-1}$, we obtain
$$
\langle{S_i^zS_{i+r}^z}\rangle = \langle\delta_i\delta_{i+1}\cdots\delta_{i+r-1}\rangle = \langle \delta_i \rangle^r = (\tanh(\beta J^z))^r.
$$

In words, the fact that kinks proliferate in the system (at each $i$, there is a positive probability that a kink is present, so there will be a positive density of them in the system) prevents the ordering of the spins.
