Weak Self-Consistent Approximation Scheme

U. Brandt, Hajo Leschke

Stephen J. Gustafson, Israel Michael Sigal

Stephen J. Gustafson, Israel Michael Sigal

Jan Gukelberger, Li Huang, Philipp Werner

We study the two-dimensional Hubbard model in the weak-coupling regime and compare the self-energy obtained from various approximate diagrammatic schemes to the result of diagrammatic Monte Carlo simulations, which sum up all weak-coupling diagrams up to a given order. While dynamical mean-field theory provides a good approximation for the local part of the self-energy, including its frequency dependence, the partial summation of bubble and/or ladder diagrams typically yields worse results than second-order perturbation theory. Even widely used self-consistent schemes such as $GW$ or the fluctuation-exchange approximation (FLEX) are found to be unreliable. Combining the dynamical mean-field self-energy with the nonlocal component of $GW$ in $GW + \mathrm{DMFT}$ yields improved results for the local self-energy and nonlocal self-energies of the correct order of magnitude, but here, too, a more reliable scheme is obtained by restricting the nonlocal contribution to the second-order diagram. FLEX + DMFT is found to give accurate results in the low-density regime, but even worse results than FLEX near half-filling.

S. L. Trubatch

Abstract The self‐energy of the one‐mode polaron model is calculated in both the weak and strong coupling regimes by using a variational principle which removes all linear Bose coupling terms and diagonalizes all quadratic Bose terms in the Hamiltonian. It is clearly demonstrated that this approximation scheme is not powerful enough to yield the correct strong coupling behavior.

V. Janiš, J. Schlipf

A general expansion scheme based on the concept of linked-cluster expansion from the theory of classical spin systems is constructed for models of interacting electrons. It is shown that with a suitable variational formulation of mean-field theories at weak (Hartree-Fock) and strong (Hubbard-III) coupling the expansion represents a universal and comprehensive tool for systematic improvements of static mean-field theories. As an example of the general formalism we investigate in detail an analytically tractable series of ring diagrams that correctly capture dynamical fluctuations at weak coupling. We introduce renormalizations of the diagrammatic expansion at various levels and show how the resultant theories are related to other approximations of similar origin. We demonstrate that only fully self-consistent approximations produce global and thermodynamically consistent extensions of static mean-field theories. A fully self-consistent theory for the ring diagrams is reached by summing the so-called noncrossing diagrams.