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physica status solidi (b)

The Perturbative Treatment of the Equation of State in Clustered Many-Particle Systems

G. R�pke, H. Reinholz

Published Nov 1, 2002• 3 citations
Publisher

Abstract

Electron–hole plasmas in semiconductors, electron–ion plasmas or nucleonic matter are examples for many-fermion systems. Their quantum-statistical treatment will be performed using a thermodynamic Green function approach. The formation of bound states can be considered within a chemical picture. However, we will show that the systematic perturbative treatment of the self-energy seems to lead to the occurrence of a singularity. This can be resolved by considering the broadening of the bound state energy levels. Thus, it is possible to unify different approaches such as the model of the partially ionized plasma and the averaged atom model.

DOI
10.1002/1521-3951(200211)234:1<254::aid-pssb254>3.0.co;2-r

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Electron–hole plasmas in semiconductors, electron–ion plasmas or nucleonic matter are examples for many-fermion systems. Their quantum-statistical treatment will be performed using a thermodynamic Green function approach. The formation of bound states can be considered within a chemical picture. However, we will show that the systematic perturbative treatment of the self-energy seems to lead to the occurrence of a singularity. This can be resolved by considering the broadening of the bound state energy levels. Thus, it is possible to unify different approaches such as the model of the partially ionized plasma and the averaged atom model.
Abstract The equation of state for a dense system of interacting Fermions is derived using functional integral technique. To investigate the formation of bound states, a Hubbard‐Stratonovich transformation is applied which expresses the action as functional of pair fields. The evaluation of the partition function in lowest orders with respect to the pair fields leads to a result which can be interpreted as the contribution of two‐particle states, accounting for density corrections as Pauli blocking and self‐energy shift. Comparison is performed with results for the equation of state of hot dense matter (plasmas, nuclear matter) obtained within a Green's function approach.
A review of electronic dynamics of single-impurity and many-impurity Anderson models is contained in this report. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the non-perturbative technique for the thermodynamic Green functions. The method of the irreducible Green functions is used as the basic tool. This irreducible Green functions method allows one to describe the quasiparticle spectra with damping of the strongly correlated electron systems in a very general and natural way and to construct the relevant dynamical solution in a self-consistent way on the level of Dyson equation without decoupling the chain of the equations of motion for the Green functions. The subject matter includes the improved interpolating solution of the Anderson model. It was shown that an interpolating approximation, which simultaneously reproduces the weak-coupling limit up to second order in the interaction strength U and the strong-coupling limit up to second order in the hybridization V (and thus also fulfils the atomic limit) can be formulated self-consistently. This approach offers a new way for the systematic construction of approximate interpolation dynamical solutions of strongly correlated electron systems.
The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
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.