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                \begin{document}$$\text {d}_{i} f({\vec {x}}) = f({\vec {x}}) - f({\vec {x}} - {\vec {\ell }}_{i})$$\end{document}$ is the lattice derivative along the $\documentclass[12pt]{minimal}
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                \begin{document}$${\vec {\ell }}_{i}$$\end{document}$ direction, and:$$\documentclass[12pt]{minimal}
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                \begin{document}$$\begin{aligned} \tilde{\text {J}}_{1,{\vec {x}}} = -j_{{\vec {x}}, {\vec {x}}+ {\vec {\ell }}_{1}} - j_{{\vec {x}}, {\vec {x}}+ {\vec {\ell }}_{1} - {\vec {\ell }}_{2}}\;,\qquad \tilde{\text {J}}_{2,{\vec {x}}} = -j_{{\vec {x}}, {\vec {x}} + {\vec {\ell }}_{2}} - j_{{\vec {x}}, {\vec {x}} - {\vec {\ell }}_{1} + {\vec {\ell }}_{2}}\;. \end{aligned}$$\end{document}$$The operators $\documentclass[12pt]{minimal}
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                \begin{document}