Convexity of the self-energy functional in the variational cluster approximation

Abstract

In the variational cluster approximation (VCA) (or variational cluster perturbation theory), widely used to study the Hubbard model, a fundamental problem that renders variational solutions difficult in practice is its known lack of convexity at stationary points, i.e., the physical solutions can be saddle points rather than extrema of the self-energy functional. Here, we suggest two different approaches to construct a convex functional $\ensuremath{\Omega}[\ensuremath{\Sigma}]$. In the first approach, one can show analytically that in the approximation where the irreducible particle-hole vertex depends only on center of mass coordinates, the functional is convex away from phase transitions in the corresponding channel. Numerical tests on a tractable version of that functional show that convexity can be a nuisance when looking for instabilities both in the pairing and particle-hole channels. Therefore, an alternative phenomenological functional is proposed. Convexity is explicitly enforced only with respect to a restricted set of variables, such as the cluster chemical potential that is known to be otherwise problematic. Numerical tests show that our functional is convex at the physical solutions of VCA and allows second-order phase transitions in the pairing channel as well. This opens the way to the use of more efficient algorithms to find solutions of the VCA equations.