# Elitzur theorem and the Ising model

I was recently studying the Elitzur theorem and its application to the Ising model on Kogut: An introduction to lattice gauge theories and spin systems, chapter $5$C. I was wondering how he obtain $\langle \sigma_3 \rangle=\langle-\sigma_3\exp(-h\sum_{l_m}\delta\sigma_3)\rangle$, since under the gauge transformation $\sigma_3 \mapsto {\sigma'_3}$, I expect to get $\langle \sigma_3 \rangle=\langle-\sigma'_3\exp(-h\sum_{l_m}\delta\sigma_3)\rangle$. It's probably a trivial question, however I can't figure it out. Is there anyone who can help me?

• Note the change of sign in front of $\delta\sigma_3$ in the last line of (5.13). He just changed variables from $\sigma$ to $\sigma'$ and used the fact that summing over all $\sigma$ is the same as summing over all $\sigma'$. – Yvan Velenik Aug 16 '18 at 16:26