Hubbard Model

$$H=-t\sum _{i,j,\sigma }({\hat{c}}_{i\sigma }^{†}{\hat{c}}_{j\sigma }+{\hat{c}}_{j\sigma }^{†}{\hat{c}}_{i\sigma })+U\sum _i{\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }-\mu \sum _i\widehat{n_i}$$