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 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.

arXiv comment12 pages, 7 figures

References in corpus (9)

• Electronic Structure Calculations with Dynamical Mean-Field Theory: A Spectral Density Functional Approach
• Cellular Dynamical Mean Field Approach to Strongly Correlated Systems
• Self-energy-functional approach to systems of correlated electrons
• Variational cluster approach to correlated electron systems in low dimensions
• Antiferromagnetic to superconducting phase transition in the hole- and electron-doped Hubbard model at zero temperature
• Effective action approach to strongly correlated fermion systems
• Landau Theory of the Mott Transition in the Fully Frustrated Hubbard Model in Infinite Dimensions
• Charge ordering and phase separation in the infinite dimensional extended Hubbard model
• Extended Variational Cluster Approximation

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