Abstract
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.
Highlights
The one-particle Green’s function (GF)[1,2,3] is a powerful quantity since it contains a wealth of information about a physical system, such as the expectation value of any single-particle operator over the ground state, the ground-state total energy, and the spectral function
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 Green’s function, 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
In this paper we explore several aspects of the set of first order nonlinear coupled differential equations which are conventionally solved perturbatively in order to calculate the one-particle Green’s function
Summary
The one-particle Green’s function (GF)[1,2,3] is a powerful quantity since it contains a wealth of information about a physical system, such as the expectation value of any single-particle operator over the ground state, the ground-state total energy, and the spectral function. One would need a new initial condition to completely define the desired solution of this differential equation, since the derivative δG δφ has been introduced Usually another route is taken: one includes the functional derivative in (5) in the definition of a self-energy[4]. A good starting point is obtained by reformulating the problem in terms of a coupled set of equations containing the one-particle Green’s function, the polarizability P , the self-energy Σ, the screened. We discretize Eq (5) and consider in a first instance only one point for each space, spin, and time variable: we will call this latter approximation the ”1-point model”, as opposed to the full functional problem The strategy underlying this procedure is the following: for the 1-point model, we can derive the exact explicit solution of the algebraic differential equation, and solve the initial value problem.
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