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 $H=\int [a{\varphi }^2/2+\kappa |\nabla \varphi {|}^2/2+\gamma ({\nabla }^2\varphi {)}^2/2+B{\varphi }^2|\nabla \varphi {|}^2]$ and an even positive trial kernel $J(q)$, define $S_0={\sum }_q^{+}1/J(q)$ and $S_2={\sum }_q^{+}{q}^2/J(q)$ using a positive-wavevector sum for a real field. Isserlis theorem gives $⟨H_4{⟩}_0=4B{S}_0{S}_2/V$. The Feynman-Bogoliubov inequality therefore has stationary kernelsThus both the mass and gradient coefficient shift. In the thermodynamic limit, their self-consistency equations areA coarse-graining ultraviolet cutoff $\Lambda$ is necessary for the first integral in three dimensions: its large-$k$ radial integrand tends to a constant. Physical solutions require $J(q)>0$.

For an exact dimensionless Hamiltonian $H$ and trial Hamiltonian $H_0$, the free energies obeywhere $F=-\ln Z$, $F_0=-\ln Z_0$, and the expectation value uses the normalized trial equilibrium state. A physical Hamiltonian is multiplied by $\beta =1/(k_{B}T)$ in these formulas; multiplying the bound by $k_{B}T$ restores physical free energy units.

For classical configurations, the identity $e^{-F}=e^{-F_0}⟨e^{-(H-H_0)}{⟩}_0$ and the Jensen inequality for the convex exponential function give $\ln ⟨e^{-(H-H_0)}{⟩}_0\ge -⟨H-H_0{⟩}_0$. Taking minus the logarithm proves the bound.

The same bound holds for finite-dimensional, noncommuting quantum Hamiltonians, although the preceding exponential-factorization identity cannot generally be used. Define the density matrices $\rho =e^{-H}/Z$ and ${\rho }_0=e^{-H_0}/Z_0$. With natural logarithms, nonnegativity of quantum relative entropy givesRearrangement proves the quantum result without requiring $[H,H_0]=0$. Equality holds when ${\rho }_0=\rho$; in this finite-dimensional setting that means $H-H_0$ is a scalar multiple of the identity operator.^[ref]

A tractable trial family therefore supplies upper bounds that can be minimized over its parameters. This variational minimum need not reproduce exact correlations or phase transition locations.