Lemma: The exact four-state generator

In the ordered state space

the heat-bath generator and invariant row vector are

For $z(\sigma )={\sigma }_1{\sigma }_2$,

Proof. A flip at either site has rate

It is $b$ in aligned states and $c$ in anti-aligned states, which gives $Q_a$. The Gibbs weights are $e^{\kappa }$ in aligned states and $e^{-\kappa }$ otherwise. Dividing by $4\cosh \kappa$ gives ${\pi }_a$.

The spin actions follow from (eq:spin-action) because $\tanh (\kappa {\sigma }_j)=a{\sigma }_j$. At an update of site one, the conditional mean of the new product is $a$, and the same holds at site two. Therefore $Lz=2(a-z)$.