\begin{block}{Results: intermediate energies, integer spin}
\begin{itemize}
\item $\max\left(\mathcal{J}^2 T,\mathcal{J}V\right)\ll |\mathcal{D}_{zz}|\ll\max\left(T,V\right)$
\end{itemize}
\vspace{15pt}
\begin{equation}
\notag \frac{d\rho_S^a}{dt}=\sum_b w_{a\leftarrow b}\rho_S^b-\rho_S^a\sum_n w_{b\leftarrow a},\quad \rho_S^a = \langle \psi_a | \rho_S|\psi_a \rangle
\end{equation}
\begin{itemize}
    \item transition rates
\end{itemize}
\begin{equation}
\notag
w_{a\leftarrow b} \approx \gamma_V^{jk}(0){\mathcal{J}}_{ij}{\mathcal{J}}_{lk}S_i^{ab}S_l^{ba}
\end{equation}
\begin{columns}
    \begin{column}{.25\textwidth}
        \begin{figure}
        \includegraphics[scale=0.4]{pic8}
        \label{fig:8}
        \end{figure}
    \end{column}
    \begin{column}{.65\textwidth}
        \begin{itemize}
            \item effective "Gibbs" distribution
        \end{itemize}
        \begin{align}\notag
            \rho_S^a &\sim \exp\left(\beta a\right),\quad a=S,S-1,...,-S+1,-S
        \\ \notag
            \beta &= 2 \mathrm{arccoth}\left( \frac{{\mathcal{J}}_{xz}^2+{\mathcal{J}}_{zy}^2+\sum_{i,j=x,y}{\mathcal{J}}_{ij}^2\frac{V}{2T}\coth\frac{V}{2T}}{\epsilon_{zij} {\mathcal{J}}_{ix}{\mathcal{J}}_{jy} \frac{V}{T}}\right)
        \end{align}
    \end{column}
\end{columns}