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arXiv:1809.10111 · physica status solidi (b)

A Scalable Numerical Approach to the Solution of the Dyson Equation for the Non‐Equilibrium Single‐Particle Green's Function

Natale Walter Talarico, Sabrina Maniscalco, Nicola Lo Gullo

Published Mar 8, 2019• 16 citations
arXivHTMLPDF

Abstract

A numerical method to solve the set of Dyson‐like equations in the framework of non‐equilibrium Green's functions is presented. The technique is based on the self‐consistent solution of the Dyson equations for the interacting single‐particle Green's function once a choice for the self‐energy, functional of the single‐particle Green's function itself, is made. The authors briefly review the theory of the non‐equilibrium Green's functions in order to highlight the main point useful in discussing the proposed technique. Then, the numerical implementationis presented and discussed, which is based on the distribution of the Keldysh components of the Green's function and the self‐energy on a grid of processes. It is discussed how the structure of the considered self‐energy approximations influences the distribution of the matrices in order to minimize the communication time among processes and which should be considered in the case of other approximations. The authors give an example of the application of this technique to the case of quenches in ultracold gases and to the single impurity Anderson model, also discussing the convergence and the stability features of the approach.

DOI
10.1002/pssb.201800501

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A numerical method to solve the set of Dyson‐like equations in the framework of non‐equilibrium Green's functions is presented. The technique is based on the self‐consistent solution of the Dyson equations for the interacting single‐particle Green's function once a choice for the self‐energy, functional of the single‐particle Green's function itself, is made. The authors briefly review the theory of the non‐equilibrium Green's functions in order to highlight the main point useful in discussing the proposed technique. Then, the numerical implementationis presented and discussed, which is based on the distribution of the Keldysh components of the Green's function and the self‐energy on a grid of processes. It is discussed how the structure of the considered self‐energy approximations influences the distribution of the matrices in order to minimize the communication time among processes and which should be considered in the case of other approximations. The authors give an example of the application of this technique to the case of quenches in ultracold gases and to the single impurity Anderson model, also discussing the convergence and the stability features of the approach.
An approximate solution of a Dyson equation satisfied by a single particle Green's function in many-impurity problems in solids is obtained for an extended impurity perturbation potential. We assume that there is no overlapping of any two impurity potentials and that the range of the impurity potential is arbitrary. The coordinate representation of the Dyson equation is solved by transforming an iterative solution of the equation into a form with no repetitive summation indices appearing in any term of given order. By neglecting fluctuations of the impurity distribution, a set of equations are derived to determine the Green's function and the self-energy in a self-consistent manner. A group-theoretic consideration of the self-energy and some sum rules are given.
We present a nonequilibrium Green's-functional approach to study the dynamics following a quench in weakly interacting Bose-Hubbard model (BHM). The technique is based on the self-consistent solution of a set of equations which represents a particular case of the most general set of Hedin's equations for the interacting single-particle Green's function. We use the ladder approximation as a skeleton diagram for the two-particle scattering amplitude useful, through the self-energy in the Dyson equation, for finding the interacting single-particle Green's function. This scheme is then implemented numerically by a parallelized code. We exploit this approach to study the correlation propagation after a quench in the interaction parameter, for one and two dimensions. In particular, we show how our approach is able to recover the crossover from the ballistic to the diffusive regime by increasing the boson-boson interaction. Finally we also discuss the role of a thermal initial state on the dynamics both for one- and two-dimensional BHMs, finding that, surprisingly, at high temperature a ballistic evolution is restored.
The zero-temperature single-particle Green's function of correlated fermion models with moderately large Hilbert-space dimensions can be calculated by means of Krylov-space techniques. The conventional Lanczos approach consists of finding the ground state in a first step, followed by an approximation for the resolvent of the Hamiltonian in a second step. We analyze the character of this approximation and discuss a numerically exact variant of the Lanczos method which is formulated in the time domain. This method is extended to obtain the nonequilibrium single-particle Green's function defined on the Keldysh-Matsubara contour in the complex time plane which describes the system's nonperturbative response to a sudden parameter switch in the Hamiltonian. The proposed method will be important as an exact-diagonalization solver in the context of self-consistent or variational cluster-embedding schemes. For the recently developed nonequilibrium cluster-perturbation theory, we discuss its efficient implementation and demonstrate the feasibility of the Krylov-based solver. The dissipation of a strong local magnetic excitation into a non-interacting bath is considered as an example for applications.
It is shown that the nonequilibrium self-energy of an interacting lattice-fermion model has a unique Lehmann representation. Based on the construction of a suitable noninteracting effective medium, we provide an explicit and numerically practicable scheme to construct the Lehmann representation for the self-energy, given the Lehmann representation of the single-particle nonequilibrium Green's function. This is of particular importance for an efficient numerical solution of Dyson's equation in the context of approximations where the self-energy is obtained from a reference system with a small Hilbert space. As compared to conventional techniques to solve Dyson's equation on the Keldysh contour, the effective-medium approach allows us to reach a maximum propagation time, which can be several orders of magnitude longer. This is demonstrated explicitly by choosing the nonequilibrium cluster-perturbation theory as a simple approach to study the long-time dynamics of an inhomogeneous initial state after a quantum quench in the Hubbard model on a $10\ifmmode\times\else\texttimes\fi{}10$ square lattice. We demonstrate that the violation of conservation laws is moderate for weak Hubbard interaction and that the cluster approach is able to describe prethermalization physics.
Perturbation theory using self-consistent Green's functions is one of the most widely used approaches to study many-body effects in condensed matter. On the basis of general considerations and by performing analytical calculations for the specific example of the Hubbard atom, we discuss some key features of this approach. We show that when the domain of the functionals that are used to realize the map between the noninteracting and the interacting Green's functions is properly defined, there exists a class of self-energy functionals for which the self-consistent Dyson equation has only one solution, which is the physical one. We also show that manipulation of the perturbative expansion of the interacting Green's function may lead to a wrong self-energy as a functional of the interacting Green's function, at least for some regions of the parameter space. These findings confirm and explain numerical results of Kozik et al. for the widely used skeleton series of Luttinger and Ward [Phys. Rev. Lett. 114, 156402 (2015)]. Our study shows that it is important to distinguish between the maps between sets of functions and the functionals that realize those maps. We demonstrate that the self-consistent Green's functions approach itself is not problematic, whereas the functionals that are widely used may have a limited range of validity.
We present an elementary and fairly detailed review of several Green’s function methods for treating nuclear and other many-body systems. We first treat the single-particle Green’s function, by way of which some details concerning linked diagram expansion, rules for evaluating Green’s function diagrams and solution of the Dyson’s integral equation for Green’s function are exhibited. The particle-particle hole-hole (pphh) Green’s function is then considered, and a specific time-blocking technique is discussed. This technique enables us to have a one-frequency Dyson’s equation for the pphh and similarly for other Green’s functions, thus considerably facilitating their calculation. A third type of Green’s function considered is the particle-hole Green’s function. RPA and high order RPA are treated, along with examples for setting up particle-hole RPA equations. A general method for deriving a model-space Dyson’s equation for Green’s functions is discussed. We also discuss a method for determining the normalization of Green’s function transition amplitudes based on its vertex function. Some applications of Green’s function methods to nuclear structure and recent deep inelastic lepton-nucleus scattering are addressed.