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Journal of the Japan Society of Powder and Powder Metallurgy

Numerical Exact Solutions to the Many-body Problem Based on the Differential Forms

Shin-ichiro Kondo, Kazuyoshi Yoshimura

Published Jun 14, 2023• 2 citations
Publisher

Abstract

The Many-body problem is very important issue in physics. Usually perturbation calculations based on the Green's function have been used to analyze such a problem, however, infinite orders of perturbation calculations are required to obtain exact solutions, thus resulting in the analytical and numerical difficulties. To overhaul such difficulties, we propose the new calculation method based on the differential forms, which are able to evaluate exact solutions if Hamiltonian is composed of Fermi particles. Since our proposed method is based on time evolution equations, comparisons with the calculations derived from Feynman Kernel is possible, with showing complete agreement. Furthermore we applied this method to the simplest Anderson Hamiltonian to investigate the appearance of magnetic moment. The calculation results show that magnetic moment easily disappears with small Coulomb repulsive energy, possibly implying the spin fluctuations.

DOI
10.2497/jjspm.70.281

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The Many-body problem is very important issue in physics. Usually perturbation calculations based on the Green's function have been used to analyze such a problem, however, infinite orders of perturbation calculations are required to obtain exact solutions, thus resulting in the analytical and numerical difficulties. To overhaul such difficulties, we propose the new calculation method based on the differential forms, which are able to evaluate exact solutions if Hamiltonian is composed of Fermi particles. Since our proposed method is based on time evolution equations, comparisons with the calculations derived from Feynman Kernel is possible, with showing complete agreement. Furthermore we applied this method to the simplest Anderson Hamiltonian to investigate the appearance of magnetic moment. The calculation results show that magnetic moment easily disappears with small Coulomb repulsive energy, possibly implying the spin fluctuations.
Usually, expected values for various physical quantities, such as the number of electrons occupying certain states or the Coulomb interaction between different states of electrons, can be expressed in terms of integrals. In contrast, our method, based on differential forms, shows that expected values can be obtained by averaging over time. To confirm the validity of our method, we prepare the two cases: one is a very simple case with no many-body interaction, and the other is the case where the many-body term is included (the simplest Anderson Hamiltonian). Regarding the simple case without inclusion of many-body term, we prove analytically that the number of electrons occupying any state derived from our method is equivalent to the analytical one evaluated from the Greenʼs function method. When the many-body term is included, our results show good numerical agreement with the analytical ones derived from the Greenʼs function method. By the two cases, the calculation of expected values based on our method is considered valid.
Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on many-body perturbation theory. They can be obtained by iterating a set of functional differential equations (DEs) relating the one-particle Green's function (GF) to its functional derivative with respect to an external perturbing potential. In this work, we apply a linear response expansion in order to obtain insights into various approximations for GF calculations. The expansion leads to an effective screening while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations, we discretize them and retain only one point in space, spin and time for all variables. Within this one-point model we obtain an explicit solution for the GF, which allows us to explore the structure of the general family of solutions and to determine the specific solution that corresponds to the physical one. Moreover, we analyze the performances of established approaches like GW over the whole range of interaction strength, and we explore alternative approximations. Finally, we link certain approximations for the exact solution to the corresponding manipulations of the DE which produces them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the DE is known.
The paper addresses the many-body quasiparticle dynamics of the Anderson impurity model at finite temperatures in the framework of the equation-of-motion method. We find a new exact identity relating the one-particle and many-particle Green’s functions. Using this identity we present a consistent and general scheme for a construction of generalized mean fields (elastic scattering corrections) and self-energy (inelastic scattering) in terms of Dyson equation. A new approach for the complex expansion for the single-particle propagator in terms of Coulomb repulsion U and hybridization V is proposed. Using the exact identity, the essentially new many-body dynamical solution of SIAM has been derived. This approach offers a new way for the systematic construction of the approximative interpolating dynamical solutions of the strongly correlated electron systems.
To analyze the many-body problem based on differential forms, we apply the Heisenberg equation of motion to the Hamiltonian including the many-body term, focusing on the simplest electron–phonon interaction. Simultaneous differential equations are obtained after applying the Heisenberg equation of motion. To confirm our proposed method, we compare the numerical solutions to the simultaneous differential equations with the analytical results based on the Green's function method. Good agreement is observed between the results, indicating that the analysis based on the Heisenberg equation of motion is suitable for analyzing the steady state. To examine the non-steady state, we investigate weakly pulsed phonon excitation. The numerical calculations suggest that phonons may be conserved if the excitation period is very short. Furthermore, based on the Pauli exclusion principle, the number of simultaneous differential equations that exactly describe electron–electron interactions appears to be finite but becomes extremely large as the number of electrons increases.
A new many-body theory perturbationally consistent to all orders of the perturbation is presented and applied to the Hubbard problem as an example. Use is made of the recent Kim and Wilson commutator projection operator to derive the Dyson equation for the one-particle Green's function and an exact set of coupled differential equations for the damping or self-energy term which are solved self-consistently. The Hubbard III result follows in our first approximation and the Esterling-Lange (EL)-type result in the second. Since we keep the hopping finite throughout the calculation, no degeneracy problem is encountered as in EL.
By splitting a Hamiltonian into two parts, using the solvability of eigenvalue problem of one part of the Hamiltonian, proving a useful identity and deducing an expansion formula of power of operator binomials, we obtain an explicit and general form of time evolution operator in the representation of solvable part of the Hamiltonian. Further we find out an exact solution of Schrödinger equation in a general time-independent quantum system, and write down its concrete form when the solvable part of this Hamiltonian is taken as the kinetic energy term. Comparing our exact solution with the usual perturbation theory makes some features and significance of our solution clear. Moreover, through deriving out the improved forms of the zeroth, first, second and third order perturbed solutions including the partial contributions from the higher order even all order approximations, we obtain the improved transition probability. In special, we propose the revised Fermi's golden rule. Then we apply our scheme to obtain the improved forms of perturbed energy and perturbed state. In addition, we study an easy understanding example to illustrate our scheme and show its advantage. All of this implies the physical reasons and evidences why our exact solution and perturbative scheme are formally explicit, actually calculable, operationally efficient, conclusively more accurate. Therefore our exact solution and perturbative scheme can be thought of theoretical developments of quantum dynamics. Further applications of our results in quantum theory can be expected.