viewed as the coherent-state amplitude of the slave bosons in the Kotliar-Ruckenstein theory, whose saddle-point solution corresponds to the GA.

To obtain the Gutzwiller density matrix $\rho_G$, one needs to self-consistently solve the quasi-particle density matrix $\rho^{\rm qp}$ and the SB amplitudes $\Phi_i$. For a given set of atomic displacements$\{\mathbf u_i \}$, which defines the tight-binding $t_{ij}$ through Eq. (\[eq:t\_ij\]), the Gutzwiller solution is obtained by minimizing the electron free energy $$\begin{aligned}
    \label{eq:free-energy}
    & & \mathcal{F} = {\rm Tr}(\rho^{\rm qp} \mathcal{H}^{\rm qp}) + U \sum_i {\rm Tr}(\Phi^\dagger_i n_{i, \uparrow} n_{i, \downarrow} \Phi_i) \nonumber \\ 
    & &  \quad +  \, T \, {\rm Tr}(\rho^{\rm qp} \ln \rho^{\rm qp}) + T \sum_i {\rm Tr}\left[ \Phi^\dagger_i \Phi^{\;}_i \ln\left(\Phi^\dagger_i \Phi^{\;}_i / \Pi_i\right) \right] \nonumber \\
     & & \quad + \, \sum_{i, \alpha} \mu_{i,\alpha} \left[ {\rm Tr}(\Phi^\dagger_i \, \hat{n}_{i, \alpha} \Phi^{\;}_i)  - {\rm Tr}(\rho^{\rm qp} \hat{n}_{i, \alpha})\right].\end{aligned}$$ The first line above denotes the total energy within the Gutzwiller approximation. The two terms in the second line correspond to quasi-particle entropy and the relative entropy between correlated distribution $\Phi^\dagger_i \Phi_i$ and the corresponding uncorrelated $\Pi_i$ [@wang10; @sandri13]. The atom-dependent Lagrangian multipliers $\mu_{i, \alpha}$ are introduced to enforce the Gutzwiller constraints in the third line, and $\hat{n}_{i\alpha} 