Dynamical Variational Principles for Strongly Correlated Electron Systems

The self-energy-functional approach (SFA) is discussed in the context of different variational principles for strongly correlated electron systems. Formal analogies between static and dynamical variational approaches, different types of approximation strategies and the relations to density-functional and dynamical mean-field theory are emphasized. The discussion elucidates the strengths of the SFA in the construction of new non-perturbative approximations but also the limitations of the approach and thereby opens up future perspectives. Variational approaches have a long and successful tradition in the theory of condensed-matter systems as they offer a smart, controlled and systematic way to treat the problem of electron-electron interaction. A well-known variational approach is Hartree-Fock (HF) theory. It is based on the RayleighRitz principle and provides a practicable and consistent mean-field description of an interacting electron system. As quantum fluctuations are neglected completely, HF theory must be classified as a static mean-field approximation. This may be contrasted with dynamical mean-field theory (DMFT) [1,2] which includes temporal fluctuations in the mean-field picture. The DMFT, however, cannot be derived from the Ritz principle. It must be constructed from some dynamical variational principle which involves a dynamical (i.e. timeor frequency-dependent) quantity as the basic object. Dynamical variational principles have already been suggested in the sixties [3,4] but, compared to the Ritz principle, were employed with rather limited success only. This brings up the following questions: What are the similarities and the differences between different variational principles and approximation strategies? How can the DMFT be considered as an approximation within a variational concept? Can dynamical variational principles be used for constructing practicable and non-perturbative approximations different from the DMFT? An attempt to answer these questions straightforwardly leads to the self-energyfunctional approach (SFA) [5] suggested recently. The purpose of this paper is to discuss different variational approaches and to place the SFA into this context with the objective to explore possible future developments. 1 Variational Principles and Approximation Strategies Consider a many-electron system in the volume V , at temperature T and with chemical potential μ. It is characterized by a Hamiltonian Ht,U =