The systematics of different approximations within the self‐energy‐functional theory (SFT) is discussed for fermionic lattice models with local interactions. In the context of the SFT, an approximation is essentially given by specifying a reference system with the same interaction but a modified non‐interacting part of the Hamiltonian which leads to a partial decoupling of degrees of freedom. The reference system defines a space of trial self‐energies on which an optimization of the grand potential as a functional of the self‐energy Ω[Σ] is performed. As a stationary point is not a minimum in general and does not provide a bound for the exact grand potential, however, it is a priori unclear how to judge on the relative quality of two different approximations. By analyzing the Euler equation of the SFT variational principle, it is shown that a stationary point of the functional on a subspace given by a reference system composed of decoupled subsystems is also a stationary point in case of the coupled reference system. On this basis a strategy is suggested which generates a sequence of systematically improving approximations. The discussion is actually relevant for any variational approach that is not based on wave functions and the Rayleigh‐Ritz principle.
The systematics of different approximations within the self‐energy‐functional theory (SFT) is discussed for fermionic lattice models with local interactions. In the context of the SFT, an approximation is essentially given by specifying a reference system with the same interaction but a modified non‐interacting part of the Hamiltonian which leads to a partial decoupling of degrees of freedom. The reference system defines a space of trial self‐energies on which an optimization of the grand potential as a functional of the self‐energy Ω[Σ] is performed. As a stationary point is not a minimum in general and does not provide a bound for the exact grand potential, however, it is a priori unclear how to judge on the relative quality of two different approximations. By analyzing the Euler equation of the SFT variational principle, it is shown that a stationary point of the functional on a subspace given by a reference system composed of decoupled subsystems is also a stationary point in case of the coupled reference system. On this basis a strategy is suggested which generates a sequence of systematically improving approximations. The discussion is actually relevant for any variational approach that is not based on wave functions and the Rayleigh‐Ritz principle.
Abstract. The systematics of different approximations within the self-energy-functional theory (SFT) is discussed for fermionic lattice models with local interactions. In the context of the SFT, an approximation is essentially given by specifying a reference system with the same interaction but a modified non-interacting part of the Hamiltonian which leads to a partial decoupling of degrees of freedom. The reference system defines a space of trial self-energies on which an optimization of the grand potential as a functional of the self-energy Ω[Σ] is performed. As a stationary point is not a minimum in general and does not provide a bound for the exact grand potential, however, it is a priori unclear how to judge on the relative quality of two different approximations. By analyzing the Euler equation of the SFT variational principle, it is shown that a stationary point of the functional on a subspace given by a reference system composed of decoupled subsystems is also a stationary point in case of the coupled reference system. On this basis a strategy is suggested which generates a sequence of systematically improving approximations. The discussion is actually relevant for any variational approach that is not based on wave functions and the Rayleigh-Ritz principle. 1.
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.
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.
The equilibrium state of a system consisting of a large number of strongly interacting electrons can be characterized by its density operator. This gives a direct access to the ground-state energy or, at finite temperatures, to the free energy of the system as well as to other static physical quantities. Elementary excitations of the system, on the other hand, are described within the language of Green's functions, i.e. time- or frequency-dependent dynamic quantities which give a direct access to the linear response of the system subjected to a weak time-dependent external perturbation. A typical example is angle-revolved photoemission spectroscopy which is linked to the single-electron Green's function. Since usually both, the static as well as the dynamic physical quantities, cannot be obtained exactly for lattice fermion models like the Hubbard model, one has to resort to approximations. Opposed to more ad hoc treatments, variational principles promise to provide consistent and controlled approximations. Here, the Ritz principle and a generalized version of the Ritz principle at finite temperatures for the static case on the one hand and a dynamical variational principle for the single-electron Green's function or the self-energy on the other hand are introduced, discussed in detail and compared to each other to show up conceptual similarities and differences. In particular, the construction recipe for non-perturbative dynamic approximations is taken over from the construction of static mean-field theory based on the generalized Ritz principle. Within the two different frameworks, it is shown which types of approximations are accessible, and their respective weaknesses and strengths are worked out. Static Hartree-Fock theory as well as dynamical mean-field theory are found as the prototypical approximations.
We derive the self-energy functional theory for bosonic lattice systems with broken U(1) symmetry by parametrizing the bosonic Baym-Kadanoff effective action in terms of one- and two-point self-energies. The formalism goes beyond other approximate methods such as the pseudoparticle variational cluster approximation, the cluster composite boson mapping, and the Bogoliubov+U theory. It simplifies to bosonic dynamical-mean-field theory when constraining to local fields, whereas when neglecting kinetic contributions of noncondensed bosons, it reduces to the static mean-field approximation. To benchmark the theory, we study the Bose-Hubbard model on the two- and three-dimensional cubic lattice, comparing with exact results from path integral quantum Monte Carlo. We also study the frustrated square lattice with next-nearest-neighbor hopping, which is beyond the reach of Monte Carlo simulations. A reference system comprising a single bosonic state, corresponding to three variational parameters, is sufficient to quantitatively describe phase boundaries and thermodynamical observables, while qualitatively capturing the spectral functions, as well as the enhancement of kinetic fluctuations in the frustrated case. On the basis of these findings, we propose self-energy functional theory as the omnibus framework for treating bosonic lattice models, in particular, in cases where path integral quantum Monte Carlo methods suffer from severe sign problems (e.g., in the presence of nontrivial gauge fields or frustration). Self-energy functional theory enables the construction of diagrammatically sound approximations that are quantitatively precise and controlled in the number of optimization parameters but nevertheless remain computable by modest means.
The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix 〈m1m2∣A∣p1p2〉. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix A. Hole-state energies are given self-consistently in terms of the matrix A, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.