Gaussian variational approximation

A Gaussian variational approximation chooses a quadratic trial kernel and minimizes a variational upper bound on the interacting free energy. The optimized kernel often equals the bare kernel plus a self-consistent, wavevector-independent mass shift.

For an exact dimensionless Hamiltonian $H$ and trial Hamiltonian $H_0$, the free energies obeywhere $F_0=-\log Z_0$ and the expectation uses the trial Gibbs distribution $Z_0^{-1}{e}^{-H_0}$.