Photoemission Spectra from Reduced Density Matrices: the Band Gap in Strongly Correlated Systems

We present a method for the calculation of photoemission spectra in terms of reduced density matrices. We start from the spectral representation of the one-body Green's function G, whose imaginary part is related to photoemission spectra, and we introduce a frequency-dependent effective energy that accounts for all the poles of G. Simple approximations to this effective energy give accurate spectra in model systems in the weak as well as strong correlation regime. In real systems reduced density matrices can be obtained from reduced density-matrix functional theory. Here we use this approach to calculate the photoemission spectrum of bulk NiO: our method yields a qualitatively correct picture both in the antiferromagnetic and paramagnetic phases, contrary to mean-field methods, in which the paramagnet is a metal

Exploring new exchange-correlation kernels in the Bethe-Salpeter equation: a study of the asymmetric Hubbard dimer

The Bethe-Salpeter equation (BSE) is the key equation in many-body perturbation theory based on Green's functions to access response properties. Within the $GW$ approximation to the exchange-correlation kernel, the BSE has been successfully applied to several finite and infinite systems. However, it also shows some failures, such as underestimated triplet excitation energies, lack of double excitations, ground-state energy instabilities in the dissociation limit, etc. In this work, we study the performance of the BSE within the $GW$ approximation as well as the $T$-matrix approximation for the excitation energies of the exactly solvable asymmetric Hubbard dimer. This model allows one to study various correlation regimes by varying the on-site Coulomb interaction $U$ as well as the degree of the asymmetry of the system by varying the difference of potential $\Delta v$ between the two sites. We show that, overall, the $GW$ approximation gives more accurate excitation energies than $GT$ over a wide range of $U$ and $\Delta v$. However, the strongly-correlated (i.e., large $U$) regime still remains a challenge.Comment: 11 pages, 9 figure

The three channels of many-body perturbation theory: GWGW, particle-particle, and electron-hole TT-matrix self-energies

We derive the explicit expression of the three self-energies that one encounters in many-body perturbation theory: the well-known $GW$ self-energy, as well as the particle-particle and electron-hole $T$-matrix self-energies. Each of these can be easily computed via the eigenvalues and eigenvectors of a different random-phase approximation (RPA) linear eigenvalue problem that completely defines their corresponding response function. For illustrative and comparative purposes, we report the principal ionization potentials of a set of small molecules computed at each level of theory.Comment: 12 pages, 5 figure

Green functions and self-consistency: insights from the spherium model

We report an exhaustive study of the performance of different variants of Green function methods for the spherium model in which two electrons are confined to the surface of a sphere and interact via a genuine long-range Coulomb operator. We show that the spherium model provides a unique paradigm to study electronic correlation effects from the weakly correlated regime to the strongly correlated regime, since the mathematics are simple while the physics is rich. We compare perturbative GW, partially self-consistent GW and second-order Green function (GF2) methods for the computation of ionization potentials, electron affinities, energy gaps, correlation energies as well as singlet and triplet neutral excitations by solving the Bethe-Salpeter equation (BSE). We discuss the problem of self-screening in GW and show that it can be partially solved with a second-order screened exchange correction (SOSEX). We find that, in general, self-consistency deteriorates the results with respect to those obtained within perturbative approaches with a Hartree-Fock starting point. Finally, we unveil an important problem of partial self-consistency in GW: in the weakly correlated regime, it can produce artificial discontinuities in the self-energy caused by satellite resonances with large weights.Comment: 11 pages, 7 figure

Potential energy surfaces without unphysical discontinuities: the Coulomb-hole plus screened exchange approach

In this work we show the advantages of using the Coulomb-hole plus screened-exchange (COHSEX) approach in the calculation of potential energy surfaces. In particular, we demonstrate that, unlike perturbative $GW$ and partial self-consistent $GW$ approaches, such as eigenvalue-self-consistent $GW$ and quasi-particle self-consistent $GW$, the COHSEX approach yields smooth potential energy surfaces without irregularities and discontinuities. Moreover, we show that the ground-state potential energy surfaces (PES) obtained from the Bethe-Salpeter equation, within the adiabatic connection fluctuation dissipation theorem, built with quasi-particle energies obtained from perturbative COHSEX on top of Hartree-Fock (BSE@COHSEX@HF) yield very accurate results for diatomic molecules close to their equilibrium distance. When self-consistent COHSEX quasi-particle energies and orbitals are used to build the BSE equation the results become independent of the starting point. We show that self-consistency worsens the total energies but improves the equilibrium distances with respect to BSE@COHSEX@HF. This is mainly due to changes in the screening inside the BSE

Unphysical and physical solutions in many-body theories: from weak to strong correlation

International audienceMany-body theory is largely based on self-consistent equations that are constructed in terms of the physical quantity of interest itself, for example the density. Therefore, the calculation of important properties such as total energies or photoemission spectra requires the solution of nonlinear equations that have unphysical and physical solutions. In this work we show in which circumstances one runs into an unphysical solution, and we indicate how one can overcome this problem. Moreover, we solve the puzzle of when and why the interacting Green's function does not unambiguously determine the underlying system, given in terms of its potential, or non-interacting Green's function. Our results are general since they originate from the fundamental structure of the equations. The absorption spectrum of lithium fluoride is shown as one illustration, and observations in the literature for some widely used models are explained by our approach. Our findings apply to both the weak and strong-correlation regimes. For the strong-correlation regime we show that one cannot use the expressions that are obtained from standard perturbation theory, and we suggest a different approach that is exact in the limit of strong interaction

Approximations for many-body Green's functions: insights from the fundamental equations

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 relating the one-particle Green's function to its functional derivative with respect to an external perturbing potential. In the present work we apply a linear response expansion in order to obtain insights in various approximations for Green's functions 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 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. 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 for the differential equation which produce them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the differential equation is known.Comment: 17 pages, 7 figure