Self-Energy-Functional Theory

Joachim Alexander Maruhn, P.‐G. Reinhard, E. Suraud

Felix Hofmann, Martin Eckstein, Enrico Arrigoni +1 more

The self-energy functional theory (SFT) is generalized to describe the real-time dynamics of correlated lattice-fermion models far from thermal equilibrium. This is achieved by starting from a reformulation of the original equilibrium theory in terms of double-time Green's functions on the Keldysh-Matsubara contour. With the help of a generalized Luttinger-Ward functional, we construct a functional $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\Omega}}[\ensuremath{\Sigma}]$ which is stationary at the physical (nonequilibrium) self-energy $\ensuremath{\Sigma}$ and which yields the grand potential of the initial thermal state $\ensuremath{\Omega}$ at the physical point. Nonperturbative approximations can be defined by specifying a reference system that serves to generate trial self-energies. These self-energies are varied by varying the reference system's one-particle parameters on the Keldysh-Matsubara contour. In the case of thermal equilibrium, this approach reduces to the conventional SFT. Contrary to the equilibrium theory, however, ``unphysical'' variations, i.e., variations that are different on the upper and the lower branches of the Keldysh contour, must be considered to fix the time dependence of the optimal physical parameters via the variational principle. Functional derivatives in the nonequilibrium SFT Euler equation are carried out analytically to derive conditional equations for the variational parameters that are accessible to a numerical evaluation via a time-propagation scheme. Approximations constructed by means of the nonequilibrium SFT are shown to be inherently causal, internally consistent, and to respect macroscopic conservation laws resulting from gauge symmetries of the Hamiltonian. This comprises the nonequilibrium dynamical mean-field theory but also dynamical-impurity and variational-cluster approximations that are specified by reference systems with a finite number of degrees of freedom. In this way, nonperturbative and consistent approximations can be set up, the numerical evaluation of which is accessible to an exact-diagonalization approach.

G. F. Bertsch, Kazuhiro Yabana

Christopher J. N. Coveney, David P. Tew

We present the diagrammatic theory of the irreducible self-energy and Bethe-Salpeter kernel that naturally arises within the Green’s function formalism for a general N -body non-Hermitian interaction. In this work, we focus specifically on the coupled-cluster (CC) self-energy generated by the similarity transformation of the electronic structure Hamiltonian. The main result of this work is a Green’s function formalism for non-Hermitian many-body quantum systems. We develop the biorthogonal quantum theory to construct dynamical correlation functions where the time dependence of operators is governed by a non-Hermitian Hamiltonian. We extend the Gell-Mann and Low theorem to include non-Hermitian interactions and to generate perturbative expansions of many-body Green’s functions. We introduce the single-particle coupled-cluster Green’s function and derive the perturbative diagrammatic expansion for the non-Hermitian coupled-cluster self-energy in terms of the “noninteracting” reference Green’s function, Σ ̃ [ G 0 ] . From the exact equation of motion of the single-particle coupled-cluster Green’s function, we derive the self-consistent renormalized coupled-cluster self-energy, Σ ̃ [ G ̃ ] , and demonstrate its relationship to the perturbative expansion of the self-energy, Σ ̃ [ G 0 ] . Subsequently, we show that the usual electronic self-energy can be recovered from the coupled-cluster self-energy by neglecting the effects of the similarity transformation. We show how the coupled-cluster ground-state energy can be obtained from the coupled-cluster self-energy and provide an overview of the relationship between approximations for the coupled-cluster self-energy, ionization potential/electron affinity equation-of-motion coupled-cluster theory, and the G 0 W 0 approximation. As a result, we introduce the CC- G 0 W 0 self-energy by leveraging the connections between Green’s function and coupled-cluster theory. Finally, we derive the diagrammatic expansion of the coupled-cluster Bethe-Salpeter kernel.

Michael Potthoff, Matthias Balzer

Based on a functional-integral formalism, a generalization of the self-energy-functional theory (SFT) is proposed which is applicable to systems of interacting electrons with disorder. Similar to the pure case without disorder, a variational principle is set up which gives the physical (disorder) self-energy as a stationary point of the (averaged) grand potential. Although the resulting self-energy functional turns out to be more complicated, the formal structure of the theory can be retained since the unknown part of the functional is universal. This allows one to construct nonperturbative and thermodynamically consistent approximations via searching for a stationary point on a restricted domain of the functional. The theory and the possible approximations are worked out for models with local interactions and local disorder. This results in a derivation of different mean-field approaches and various cluster extensions, including well-known concepts such as the statistical dynamical mean-field theory, the molecular coherent-potential approximation, and the dynamical cluster approximation. Due to the common formal framework provided by the SFT, one achieves a general systematization of dynamical approaches, i.e., approaches based on the spectrum of one-particle excitations. Different mean-field and different cluster schemes naturally appear in this framework and complement the existing ones. Their prospects for future applications are discussed.