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I am currently reading the chapter 10 of Quantum Phase Transitions by Subir Sachdev (1st ed.). Chapter 10 consists of introductory remarks on the Bose-Hubbard Model. Equation (10.8) [Eq. (9.8) in 2nd ed.] gives an expression for the mean-field value of the ground state energy, which is the following:

$$ \frac{E_0}{N} = \frac{E_{\text{MF}}(\Psi_B)}{N} - zw\langle\hat{b}^\dagger \rangle\langle \hat{b}\rangle + \langle \hat{b}\rangle \Psi^*_B + \langle \hat{b}^\dagger\rangle \Psi_B $$

Here $E_{\text{MF}}(\Psi_B)$ is the ground state energy of the mean-field Hamiltonian as a function of $\Psi_B$, a field that represents the influence of neighboring sites. $N$ is the number of sites on the lattice, $z$ is the coordination number of the lattice, $w$ is the hopping amplitude, and the expectation values are evaluated with respect to the ground state of the mean field Hamiltonian.

The text says that by taking the derivative of the above equation with respect to $\Psi_B$, the optimal value of $\Psi_B$ is $zw\langle\hat{b} \rangle$. In order to prove this I took the derivative of the right hand side and set that to zero. This gave me

$$ \frac{1}{N}\frac{\partial }{\partial \Psi_B}E_{\text{MF}}(\Psi_B) +\langle \hat{b}^\dagger\rangle = 0 $$

This does not give me the claimed optimal value. Where did I make my mistake? How can I prove that the optimal value of $\Psi_B$ is $zw\langle\hat{b} \rangle$?

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To derive the correct relation, first one needs to remember that $\langle b\rangle$ is not some independent parameter, it is the expectation value of $b$ in the mean-field ground state which also depends on $\Psi_B$. Then we also need to evaluate $\partial E_\text{MF}/\partial \Psi_B$, which can be obtained using the Hellman-Feynman theorem.

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