Atomic Schwinger-Dyson method in finite systems

Adam P. Szczepaniak

Yasumichi Aoki, Tatsumi Aoyama, Masafumi Kurachi B +6 more

Masahiro Nakano, Akira Hasegawa

A method is shown to derive a consistent Hamiltonian of nucleon and meson system in a finite nucleus from a Lorentz-scalar Lagrangian. For this purpose, the Schwinger-Dyson method is applied to a many-body problem of the interacting nucleons and mesons in a finite system. Using this method, closed equations of exact Green's functions for single-particle motions of a nucleon and a meson are derived. It is shown that the interactions of nucleons and mesons generate local dependent potentials in a finite system and that the potentials can be determined self-consistently on the basis of the many-body Schwinger-Dyson formalism. In the method, condensed meson fields are extracted from the meson fields and treated as c-number fields, so that the classical equations of the condensed fields become equivalent to those of the mean-field theory originally proposed by Walecka et al.$---- The present method is regarded as a natural extension of mean-field theory to a full quantum treatment for finite systems. The method provides a self-consistent description of the single-particle motions of nucleons and mesons and their mutual interactions. It allows us to include full quantum effects and to describe not only the ground state but also excited states in a systematic way.

Trond Digernes, E. Husstad, Veeravalli S. Varadarajan

Hiroyuki Matsuura

We proposed Atomic Schwinger–Dyson method (ASD method) in this paper, which was the nonperturbative and finite relativistic quantum field theory, and we treat many electron system and electronic matter. The ASD formalism consists of coupled Dyson equations of electrons and photons. Since, it includes self-energies in a nonperturbative way, higher-order correlations beyond Hartee–Fock approximation are taken into account. Some important differences between the ASD formalism for the system of finite electron density and SD formalism of zero electron density are shown. The main difference is due to the existence of condensed photon field, symmetry breaking, and what we call, Coulomb's potential. By paying special attention to the treatment of the condensed photon fields, the coupled Dyson equations of electron and photon are derived based on functional propagator method. It is shown that this treatment of the condensed fields naturally leads to tadpole energy, which cancels the Hartree energy. By using these photon propagators, explicit expression of ASD coupled equations and the energy density of matters are derived for numerical calculations in a subsequent paper. Similarities and differences between ASD and traditional methods such as the mean field theory or the Hartree–Fock method are discussed; it is shown that these traditional methods were included in our ASD formalism.

Chang Cheng, Wen Xin Ding

The “noncanonicality” of the underlying operators is the fundamental difficulty in formulating theories for most strong interaction systems such as the t-J model, the Hubbard-type model under the large U limit, and quantum spin etc. But B.S. Shastry proposed Schwinger-Dyson equations of motion to study the t-J model, which can rigorously solve this noncanonicality. In this article, we use the perturbation Schwinger-Dyson equation of motion method to study two spin 1/2 transverse field Ising models perturbed by longitudinal fields, and discuss potential generalizations of the corresponding lattice models.