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arXiv:2308.14379 · Physical review. B./Physical review. B

Ward identity preserving approach for investigation of phonon spectrum with self-energy and vertex corrections

Sudhakar Pandey

Published Aug 23, 2024• 0 citations
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Abstract

We propose an approach for investigating the many-body effects on the phonon spectrum in an electron-phonon coupled system by taking into account the electron self-energy and vertex corrections which respect the Ward identity representing the conservation of electronic charge. Our approach provides a systematic diagrammatic expansion of the phonon self-energy ${\mathrm{\ensuremath{\Pi}}}_{\mathbf{q}}(\ensuremath{\omega})$ in powers of the electron-phonon scattering strength squared $({g}^{2})$. In this approach the many-body corrections to the phonon spectrum vanish identically for the $\mathbf{q}=0$ mode due to an exact cancellation between the contributions arising from electron self-energy and vertex corrections. This cancellation holds irrespective of the microscopic details of the system such as dimensionality, adiabaticity, strength of electron-phonon scattering, etc. Our results demonstrate that the contributions of electron self-energy and vertex corrections are not only comparable but they also tend to cancel each other so that the phonon spectrum remains nearly unaffected due to many-body effects in the regime of long-wavelength excitations ($\mathbf{q}\ensuremath{\rightarrow}0$). Our results provide another constraint for the applicability of the Migdal theorem.

DOI
10.1103/physrevb.110.064321

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Comment on “Towards exact solutions for the superconducting T c induced by electron-phonon interaction”

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In a series of recent papers starting with , Liu, Yang, Pan claim to have determined the exact dressed electron-boson vertex for a number of very general interacting many-body problems by inverting certain Ward-Takahashi identities. Here, we point out that their Ward-Takahashi identities are missing terms which make the inversion of the identities impossible. One therefore cannot formulate exact self-contained integral equations for the dressed electron propagator. In addition, the proposed vertex expressions do not reproduce well-established results for the leading perturbative corrections. Published by the American Physical Society 2024

arXiv:2404.02918•Physical review. B./Physical review. B•Jul 22, 2024•1 citation
We propose an approach for investigating the many-body effects on the phonon spectrum in an electron-phonon coupled system by taking into account the electron self-energy and vertex corrections which respect the Ward identity representing the conservation of electronic charge. Our approach provides a systematic diagrammatic expansion of the phonon self-energy \ensuremathΠq(\ensuremathω){\mathrm{\ensuremath{\Pi}}}_{\mathbf{q}}(\ensuremath{\omega})\ensuremathΠq​(\ensuremathω) in powers of the electron-phonon scattering strength squared (g2)({g}^{2})(g2). In this approach the many-body corrections to the phonon spectrum vanish identically for the q=0\mathbf{q}=0q=0 mode due to an exact cancellation between the contributions arising from electron self-energy and vertex corrections. This cancellation holds irrespective of the microscopic details of the system such as dimensionality, adiabaticity, strength of electron-phonon scattering, etc. Our results demonstrate that the contributions of electron self-energy and vertex corrections are not only comparable but they also tend to cancel each other so that the phonon spectrum remains nearly unaffected due to many-body effects in the regime of long-wavelength excitations (q\ensuremath→0\mathbf{q}\ensuremath{\rightarrow}0q\ensuremath→0). Our results provide another constraint for the applicability of the Migdal theorem.
We study the impact of the many-body effects on the phonon spectral function of Holstein polaron in one-dimension in the antiadiabatic regime by incorporating the contributions from the electron self-energy and vertex corrections within a weak-coupling approach that respects the charge-conserving Ward identity. We find that while the polaronic spectral weight is suppressed due to contribution from the electron self-energy, on the other hand, the same is enhanced due to contribution from the vertex corrections. While strength of both the contributions increases with increasing the wave vector ($\q$) of phonons, they nearly cancel each other for the small-$\q$ modes so that the polaronic spectral weight is weakly affected due to the many-body effects. For the large-$\q$ modes near the zone boundary, the net many-body correction is dominated by the contribution of the electron self-energy which increases faster in comparison to that of the vertex corrections with increasing the wave vector thereby resulting in a significant suppression of the polaronic spectral weight. We find that while the weak-coupling perturbative approach provides a reliable estimation of the impact of the many-body effects deep inside the antiadibatic regime, the renormalization of quasiparticle spectrum must be taken into account for an accurate estimation when the phonon energy approaches the electronic bandwidth.
We derive sum rules for the phonon self-energy and the electron-phonon contribution to the electron self-energy of the Holstein-Hubbard model in the limit of large Coulomb interaction UUU. Their relevance for finite UUU is investigated using exact diagonalization and dynamical mean-field theory. Based on these sum rules, we study the importance of vertex corrections to the electron-phonon interaction in a diagrammatic approach. We show that they are crucial for a sum rule for the electron self-energy in the undoped system while a sum rule related to the phonon self-energy of doped systems is satisfied even if vertex corrections are neglected. We provide explicit results for the vertex function of a two-site model.
On the basis of the self-consistent calculation scheme for the electron self-energy Σ with the use of the three-point vertex function Γ always satisfying the Ward identity, we find that the obtained quasi-particle dispersion in the normal state in gapped systems, such as semiconductors, insulators, and molecules, is well reproduced by that in the one-shot GW (or G0W0) approximation. In calculating the superconducting transition temperature Tc, we also find a similar situation; the result for Tc in the gauge-invariant self-consistent framework including the effect of Γ satisfying the Ward identity is different from that in the conventional Eliashberg theory (which amounts to the GW approximation for superconductivity) but is close to that in the G0W0 approximation. Those facts indicate that the G0W0 approximation actually takes proper account of both vertex and high-order self-energy corrections in a mutually cancelling manner and thus we can understand that the G0W0 approximation is better than the fully self-consistent GW one in obtaining some of physical quantities.
We lay down the {\em ab initio} many-body quantum theory of electrons and phonons in- and out-of-equilibrium at any temperature. We begin by addressing a fundamental issue concerning the {\em ab initio} Hamiltonian in the harmonic approximation, which we show must be determined {\em self-consistently} to avoid inconsistencies. After identifying the most suitable partitioning into a ``noninteracting'' and an ``interacting'' part we embark on the Green's function diagrammatic analysis. We single out key diagrammatic structures to carry on the expansion in terms of dressed propagators and screened interaction. The final outcome is the finite-temperature nonequilibrium extension of the Hedin equations, featuring the appearance of the time-local Ehrenfest diagram in the electronic self-energy. The Hedin equations have limited practical utility for real-time simulations of driven systems. We leverage the versatility of diagrammatic expansion to generate a closed system of integro-differential equations for the Green's functions and nuclear displacements. These are the Kadanoff-Baym equations for electrons and phonons. Another advantage of the diagrammatic derivation is the ability to use conserving approximations, which ensure the satisfaction of all fundamental conservation laws during the time evolution. As an example we show that the adiabatic Born-Oppenheimer approximation is not conserving whereas its dynamical extension is conserving provided that the electrons are treated in the Fan-Migdal approximation with a dynamically screened electron-phonon coupling. We also derive the formal solution of the Kadanoff-Baym equations in the long time limit and at the steady state. The expansion of the phononic Green's function around the quasi-phonon energies points to a possible correlation-induced splitting of the phonon dispersion in materials with no time-reversal invariance.
Abstract Determining the range of validity of Migdal’s approximation for electron-phonon (e-ph) coupled systems is a long-standing problem. Many attempts to answer this question employ the Holstein Hamiltonian, where the electron density couples linearly to local lattice displacements. When these displacements are large, however, nonlinear corrections to the interaction must also be included, which can significantly alter the physical picture obtained from this model. Using determinant quantum Monte Carlo and the self-consistent Migdal approximation, we compared superconducting and charge-density-wave correlations in the Holstein model with and without second-order nonlinear interactions. We find a disagreement between the two cases, even for relatively small values of the e-ph coupling strength, and, importantly, that this can occur in the same parameter regions where Migdal’s approximation holds. Our results demonstrate that questions regarding the validity of Migdal’s approximation go hand in hand with questions of the validity of a linear e-ph interaction.
In a series of recent papers starting with , Liu, Yang, Pan claim to have determined the exact dressed electron-boson vertex for a number of very general interacting many-body problems by inverting certain Ward-Takahashi identities. Here, we point out that their Ward-Takahashi identities are missing terms which make the inversion of the identities impossible. One therefore cannot formulate exact self-contained integral equations for the dressed electron propagator. In addition, the proposed vertex expressions do not reproduce well-established results for the leading perturbative corrections. Published by the American Physical Society 2024