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The quantity $\langle c_{i\sigma}^{\dagger}c_{j\sigma}\rangle$ is in general not real and positive. Hence the contribution of the hopping term to the total energy doesn't necessarily become more negative as more particles are added. Note also that to be precise, the left-hand side of Eq. 1 should be $H-\mu N$, where $N$ is the number operator. Many people ...