\documentclass[12pt]{article}
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\begin{document}
\begin{equation*}
H=-\sum_{i\sigma} J^\sigma_i (a_{i\sigma}^\dagger
a_{i+1 \sigma}+ a_{i\sigma}
a_{i+1 \sigma}^\dagger)+\frac{1}{2} \sum _{i
\sigma \sigma'} U_{\sigma \sigma'}
a^{\dagger}_{i\sigma}a^{\dagger}_{i\sigma'}a_{i\sigma'}a_{i\sigma}
\end{equation*}
\end{document}