Free ends and energy

For the specific state $E_1$, we have chosen all spins up. The energy of this configuration becomes then $$\begin{equation*} E_1=E_{\uparrow\uparrow}=-J. \end{equation*}$$ The other configurations give $$\begin{equation*} E_2=E_{\uparrow\downarrow}=+J, \end{equation*}$$ $$\begin{equation*} E_3=E_{\downarrow\uparrow}=+J, \end{equation*}$$ and $$\begin{equation*} E_4=E_{\downarrow\downarrow}=-J. \end{equation*}$$

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