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Effect of propagator renormalization on the band gap of solids.

Sergei Iskakov, Alexander A. Rusakov, Dominika Zgid, Emanuel Gull

Published Dec 17, 2018• 0 citations
OpenAlex

Abstract

We present momentum-resolved spectral functions and band gaps from bare and self-consistent second order perturbation theory for insulating periodic solids. We establish that, for systems with large gap sizes, both bare and self-consistent perturbation theory yield reasonable gaps. However, smaller gap sizes require a self-consistent adjustment of the propagator. In contrast to results obtained within a quasi-particle formalism used on top of bare second order perturbation theory, no unphysical behaviour of the band gap is observed. We establish that the renormalization or `screening' of interaction terms in addition to propagators is not required to obtain accurate gaps in insulators. Our implementation of a fully self-consistent, $\Phi$-derivable and thermodynamically consistent finite temperature diagrammatic perturbation theory forms a framework on which embedding theories such as the dynamical mean field theory or self-energy embedding theories can be implemented.

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Effect of propagator renormalization on the band gap of insulating solids

Sergei Iskakov, Alexander A. Rusakov, Dominika Zgid +1 more

We present momentum-resolved spectral functions and band gaps from bare and self-consistent second-order perturbation theory for insulating periodic solids. We establish that, for systems with large gap sizes, both bare and self-consistent perturbation theory yield reasonable gaps. However, smaller gap sizes require a self-consistent adjustment of the propagator. In contrast to results obtained within a quasiparticle formalism used on top of bare second-order perturbation theory, no unphysical behavior of the band gap is observed. Our implementation of a fully self-consistent, $\mathrm{\ensuremath{\Phi}}$-derivable, and thermodynamically consistent finite-temperature diagrammatic perturbation theory forms a framework on which embedding theories such as the dynamical mean-field theory and self-energy embedding theories can be implemented.

arXiv:1812.07027•Physical review. B./Physical review. B•Aug 5, 2019•29 citations

Self-consistent density-functional approach to the correlated ground states and an unrestricted many-body perturbation theory

Behnam Farid

We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.

Philosophical Magazine B•Aug 1, 1997•18 citations

Self-consistent G W + extended dynamical mean field theory for semiconductors and insulators

Viktor Christiansson, Francesco Petocchi, Philipp Werner

Theoretical studies of semiconductors and band insulators are usually based on variants of the $GW$ method without full self-consistency, like single-shot $G^0W^0$ or quasiparticle self-consistent $GW$. Fully self-consistent $GW$ provides a poor description of the gap size and electronic structure due to the lack of vertex corrections. While it is hard to predict at which order corrections can be neglected, local vertex corrections to all orders can be consistently included by combining $GW$ with extended dynamical mean field theory (EDMFT). Here, we show that \textit{ab initio} multitier $GW$+EDMFT calculations, which achieve full self-consistency in a suitably defined low-energy space, provide an accurate description of semiconductors and band insulators, comparable to the established methods which are typically used for this class of materials. Our results demonstrate that the range of applicability of $GW$+EDMFT extends to weakly correlated systems, and they imply that despite the weak correlations, local vertex corrections are an important ingredient in the diagrammatic treatment of semiconductors and band insulators.

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Self-consistent density-functional approach to the correlated ground states and an unrestricted many-body perturbation theory

Behnam

We introduce a scheme through which the ground-state electronic density n and total energy Ey[n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called G W diagram, where W is the dynamically screened electron4ectron interaction function) for the selfenergy operator C. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the G W framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-selfconsistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40-65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 1 4 1 7 % . The same line of argument leads us to conclude that, in our model, 2635% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of C, which is entirely absent in the GW scheme. 5 1 . INTRODUCTION The ground-state (GS) density-functional theory (DFT) due to Hohenberg and Kohn (1964) provides a rigorous foundation for determining properties of systems in their ground states (for reviews see Lundqvist and March (1983), Jones and Gunnarsson (1989) and Dreizler and Gross (1990)). The central position in this theory is taken by the total-energy functional EY[n] which, owing to its unique absolute minimum a t the GS electronic density n, can be used to obtain not only the GS energy but also the GS charge density. tPresent address: Max-Planck-Institut fur Festkorperforschung, HeisenbergstraRe 1, 70569 Stuttgart, Federal Republic of Germany. 0141-8637/97 $12 .00

•Jan 1, 2006•0 citations

Unified Approach to Polarons and Phonon-Induced Band Structure Renormalization

Jon Lafuente-Bartolome, Chao Lian, Weng Hong Sio +3 more

Ab initio calculations of the phonon-induced band structure renormalization are currently based on the perturbative Allen-Heine theory and its many-body generalizations. These approaches are unsuitable to describe materials where electrons form localized polarons. Here, we develop a self-consistent, many-body Green's function theory of band structure renormalization that incorporates localization and self-trapping. We show that the present approach reduces to the Allen-Heine theory in the weak-coupling limit, and to total energy calculations of self-trapped polarons in the strong-coupling limit. To demonstrate this methodology, we reproduce the path-integral results of Feynman and diagrammatic Monte Carlo calculations for the Fröhlich model at all couplings, and we calculate the zero point renormalization of the band gap of an ionic insulator including polaronic effects.

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Dynamical density-matrix renormalization group for the Mott–Hubbard insulator in high dimensions

Satoshi Nishimoto, Florian Gebhard, Eric Jeckelmann

We study the Hubbard model at half band-filling on a Bethe lattice with infinite coordination number in the paramagnetic insulating phase at zero temperature. We use the dynamical mean-field theory (DMFT) mapping to a single-impurity Anderson model with a bath whose properties have to be determined self-consistently. For a controlled and systematic implementation of the self-consistency scheme we use the fixed-energy (FE) approach to the DMFT. In FE-DMFT the onset and the width of the Hubbard bands are adjusted self-consistently but the energies of the bath levels are kept fixed relatively to both band edges during the calculation of self-consistent hybridization strengths between impurity and bath sites. Using the dynamical density-matrix renormalization group method (DDMRG) we calculate the density of states with a resolution ranging from 3% of the bare bandwidth $W=4t$ at high energies to 0.5% in the vicinity of the gap. The DDMRG resolution and accuracy for the density of states and the gap is superior to those obtained with other numerical methods in previous DMFT investigations. We find that the Mott gap closes at a critical coupling $U_{\rm c}/t=4.45 \pm 0.05$. At $U=4.5t$, we observe prominent shoulders near the onset of the Hubbard bands. They are the remainders of the quasi-particle resonance in the metallic phase which appears to split when the gap opens at $U_{\rm c}$.

arXiv:cond-mat/0406666•Journal of Physics Condensed Matter•Sep 21, 2004•61 citations
We present momentum-resolved spectral functions and band gaps from bare and self-consistent second order perturbation theory for insulating periodic solids. We establish that, for systems with large gap sizes, both bare and self-consistent perturbation theory yield reasonable gaps. However, smaller gap sizes require a self-consistent adjustment of the propagator. In contrast to results obtained within a quasi-particle formalism used on top of bare second order perturbation theory, no unphysical behaviour of the band gap is observed. We establish that the renormalization or `screening' of interaction terms in addition to propagators is not required to obtain accurate gaps in insulators. Our implementation of a fully self-consistent, Φ\PhiΦ-derivable and thermodynamically consistent finite temperature diagrammatic perturbation theory forms a framework on which embedding theories such as the dynamical mean field theory or self-energy embedding theories can be implemented.
We present momentum-resolved spectral functions and band gaps from bare and self-consistent second-order perturbation theory for insulating periodic solids. We establish that, for systems with large gap sizes, both bare and self-consistent perturbation theory yield reasonable gaps. However, smaller gap sizes require a self-consistent adjustment of the propagator. In contrast to results obtained within a quasiparticle formalism used on top of bare second-order perturbation theory, no unphysical behavior of the band gap is observed. Our implementation of a fully self-consistent, $\mathrm{\ensuremath{\Phi}}$-derivable, and thermodynamically consistent finite-temperature diagrammatic perturbation theory forms a framework on which embedding theories such as the dynamical mean-field theory and self-energy embedding theories can be implemented.
We introduce a scheme through which the ground-state electronic density n and total energy Ev [n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called GW diagram, where W is the dynamically screened electron-electron interaction function) for the self-energy operator Σ. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the GW framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-self-consistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40–65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 14–17%. The same line of argument leads us to conclude that, in our model, 26–35% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of Σ, which is entirely absent in the GW scheme.
Theoretical studies of semiconductors and band insulators are usually based on variants of the GWGWGW method without full self-consistency, like single-shot G0W0G^0W^0G0W0 or quasiparticle self-consistent GWGWGW. Fully self-consistent GWGWGW provides a poor description of the gap size and electronic structure due to the lack of vertex corrections. While it is hard to predict at which order corrections can be neglected, local vertex corrections to all orders can be consistently included by combining GWGWGW with extended dynamical mean field theory (EDMFT). Here, we show that \textit{ab initio} multitier GWGWGW+EDMFT calculations, which achieve full self-consistency in a suitably defined low-energy space, provide an accurate description of semiconductors and band insulators, comparable to the established methods which are typically used for this class of materials. Our results demonstrate that the range of applicability of GWGWGW+EDMFT extends to weakly correlated systems, and they imply that despite the weak correlations, local vertex corrections are an important ingredient in the diagrammatic treatment of semiconductors and band insulators.
We introduce a scheme through which the ground-state electronic density n and total energy Ey[n] of inhomogeneous interacting systems can be determined in a self-consistent (SC) way. It is based on a SC perturbation expansion for the single-particle Green function G. Our employed expansion is argued to be valid under all conditions provided that the ground-state charge density of the interacting system is pure-state non-interacting v representable. The formalism discussed in this paper has been applied to a quasi-one-dimensional periodic system. It has even been possible to carry out SC calculations with the dynamically screened exchange diagram (the so-called G W diagram, where W is the dynamically screened electron4ectron interaction function) for the selfenergy operator C. A coherent mode of interpretation of a number of our results entails that the overwhelming success of the G W framework in relatively accurately reproducing the experimental energy gaps of a number of semiconductors and insulators may be a direct consequence of non-selfconsistency effects; whereas a non-self-consistent calculation in our tests increases the value of the local-density approximation (LDA) gap in the range of 40-65% of the ‘experimental’ gaps, self-consistency effects reduce this range to 1 4 1 7 % . The same line of argument leads us to conclude that, in our model, 2635% of the LDA error in the gap energies is due to the absence of a well known discontinuity in the LDA exchange-correlation potential, and the remaining part is due to the Kohn-Sham eigenvalues not being identifiable with one-particle excitation energies. A natural conclusion that may be drawn from these observations is that a correct description of excitation energies (e.g. of the gap energy) cannot be achieved without including, at least approximately, the vertex part of C, which is entirely absent in the GW scheme. 5 1 . INTRODUCTION The ground-state (GS) density-functional theory (DFT) due to Hohenberg and Kohn (1964) provides a rigorous foundation for determining properties of systems in their ground states (for reviews see Lundqvist and March (1983), Jones and Gunnarsson (1989) and Dreizler and Gross (1990)). The central position in this theory is taken by the total-energy functional EY[n] which, owing to its unique absolute minimum a t the GS electronic density n, can be used to obtain not only the GS energy but also the GS charge density. tPresent address: Max-Planck-Institut fur Festkorperforschung, HeisenbergstraRe 1, 70569 Stuttgart, Federal Republic of Germany. 0141-8637/97 $12 .00
Ab initio calculations of the phonon-induced band structure renormalization are currently based on the perturbative Allen-Heine theory and its many-body generalizations. These approaches are unsuitable to describe materials where electrons form localized polarons. Here, we develop a self-consistent, many-body Green's function theory of band structure renormalization that incorporates localization and self-trapping. We show that the present approach reduces to the Allen-Heine theory in the weak-coupling limit, and to total energy calculations of self-trapped polarons in the strong-coupling limit. To demonstrate this methodology, we reproduce the path-integral results of Feynman and diagrammatic Monte Carlo calculations for the Fröhlich model at all couplings, and we calculate the zero point renormalization of the band gap of an ionic insulator including polaronic effects.
We study the Hubbard model at half band-filling on a Bethe lattice with infinite coordination number in the paramagnetic insulating phase at zero temperature. We use the dynamical mean-field theory (DMFT) mapping to a single-impurity Anderson model with a bath whose properties have to be determined self-consistently. For a controlled and systematic implementation of the self-consistency scheme we use the fixed-energy (FE) approach to the DMFT. In FE-DMFT the onset and the width of the Hubbard bands are adjusted self-consistently but the energies of the bath levels are kept fixed relatively to both band edges during the calculation of self-consistent hybridization strengths between impurity and bath sites. Using the dynamical density-matrix renormalization group method (DDMRG) we calculate the density of states with a resolution ranging from 3% of the bare bandwidth W=4tW=4tW=4t at high energies to 0.5% in the vicinity of the gap. The DDMRG resolution and accuracy for the density of states and the gap is superior to those obtained with other numerical methods in previous DMFT investigations. We find that the Mott gap closes at a critical coupling Uc/t=4.45±0.05U_{\rm c}/t=4.45 \pm 0.05Uc​/t=4.45±0.05. At U=4.5tU=4.5tU=4.5t, we observe prominent shoulders near the onset of the Hubbard bands. They are the remainders of the quasi-particle resonance in the metallic phase which appears to split when the gap opens at UcU_{\rm c}Uc​.