# Sign conventions in Hubbard model

The Hubbard model is often expressed as $$H=-J\sum\limits_{<i,j>} \sum_\sigma c_{i,\sigma}^{\dagger}c_{j,\sigma} +h.c.+U\sum\limits_{i} c_{i,\uparrow}^{\dagger} c_{i,\downarrow}^{\dagger} c_{i,\downarrow} c_{i,\uparrow} -\mu\sum\limits_i n_i~~~~~~~~~(1)$$ My question is now about the signs. As it appears, the factors $J$, $U$ and $\mu$ are supposed to be positive. Therefore it appears like the hopping term decreases the energy of the system and also more particles (larger $\sum_i n_i$) seem to decrease the energy of the system. The $U$ term increases the energy, if two particles with different spin occupy the same site.

Why should more particles decrease the energy ($\mu>0$)? As I understand it, the chemical potential is the energy needed, to increase the particle number by 1. Here it seems like the chemical potential acts like it lowers the energy, if particles are added.

## 1 Answer

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 call it the "Hamiltonian" and even denote it as $H$. But whenever the term $-\mu\sum_{i} n_{i}$ explicitly appears, the "Hamiltonian" really means $H-\mu N$. The chemical potential, just like the temperature, is a thermodynamic variable defined independent of the Hamiltonian (in the precise sense).

• Despite the nomenclature, when I solve this numerically, the average particle number $\langle N\rangle=\langle \psi_0 | N | \psi_0 \rangle$ in the ground state will shift to higher values, if $\mu$ is increased ($\mu>0$)! So: lowers a higher particle number the energy? – Merlin1896 Oct 29 '15 at 7:49
• @Merlin1896 As I stated above, the left-hand side of your Eq. 1 should really be $H-\mu N$, and its expectation value $\langle H-\mu N\rangle$ is not the energy. It is the energy subtracted by $\mu \langle N\rangle$. That $\langle N\rangle$ is an increasing function of $\mu$ is a separate issue. It is a condition following from thermodynamic stability. – higgsss Oct 29 '15 at 8:43