We introduce an equation for density matrices that ensures a monotonic decrease of the free energy and reaches a fixed point at the Gibbs thermal. We build a variational approach for many-body systems that can be applied to a broad class of states, including all bosonic and fermionic Gaussian, as well as their generalizations obtained by unitary transformations, such as polaron transformations in electron-phonon systems. We apply it to the Holstein model on $20\times 20$ and $50\times 50$ square lattices, and predict phase separation between the superconducting and charge-density wave phases in the strong interaction regime.

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• Electron-phonon coupling
• Pairing mechanisms
• Phase separation
• Quantum phase transitions
• Thermal properties
• 2-dimensional systems
• Lattice models in condensed matter
• Variational approach