  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 of the
self-energy. 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.
