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arXiv:1406.6515 · Physical Review B

Collective charge excitations of strongly correlated electrons, vertex corrections, and gauge invariance

Hartmut Hafermann, Erik G. C. P. van Loon, Mikhail I. Katsnelson, Alexander I. Lichtenstein, Olivier Parcollet

Published Dec 1, 2014• 68 citations
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Abstract

We consider the collective, long-wavelength charge excitations in correlated media in presence of short- and long-range forces. As an example for the case of a short-range interaction, we examine the two-dimensional Hubbard model within dynamical mean-field theory (DMFT). It is shown explicitly that the DMFT susceptibility including vertex corrections respects the Ward identity and yields a manifestly gauge-invariant response in finite dimensions. For computing the susceptibility, we use a different expression and establish its formal equivalence to the standard DMFT formula. It allows for a more stable analytical continuation. We find a zero-sound mode expected for short-range forces. The relation between the vertex corrections, gauge invariance, and the appearance of the collective modes is discussed. Long-range forces are treated within extended dynamical mean-field theory. In order to obtain a gauge-invariant response, it is necessary to additionally incorporate some nonlocal vertex corrections into the polarization. In doing so, we obtain plasmons in the three-dimensional Hubbard model. The plasma frequency is determined by the (single-particle) density distribution as a consequence of gauge invariance. We compare this result with the plasma frequency extracted from the analytical continuation of the susceptibility. It is in good agreement with the prediction from the gauge-invariance condition.

DOI
10.1103/physrevb.90.235105

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We consider the collective, long-wavelength charge excitations in correlated media in presence of short- and long-range forces. As an example for the case of a short-range interaction, we examine the two-dimensional Hubbard model within dynamical mean-field theory (DMFT). It is shown explicitly that the DMFT susceptibility including vertex corrections respects the Ward identity and yields a manifestly gauge-invariant response in finite dimensions. For computing the susceptibility, we use a different expression and establish its formal equivalence to the standard DMFT formula. It allows for a more stable analytical continuation. We find a zero-sound mode expected for short-range forces. The relation between the vertex corrections, gauge invariance, and the appearance of the collective modes is discussed. Long-range forces are treated within extended dynamical mean-field theory. In order to obtain a gauge-invariant response, it is necessary to additionally incorporate some nonlocal vertex corrections into the polarization. In doing so, we obtain plasmons in the three-dimensional Hubbard model. The plasma frequency is determined by the (single-particle) density distribution as a consequence of gauge invariance. We compare this result with the plasma frequency extracted from the analytical continuation of the susceptibility. It is in good agreement with the prediction from the gauge-invariance condition.
We present a fully self-consistent combined GW and dynamical mean field (DMFT) study of the extended two-dimensional Hubbard model. The inclusion of the local dynamical vertex stemming from the DMFT self-energy and polarization is shown to cure the known problems of self-consistent GW. We calculate momentum-resolved spectral functions, two-particle polarizations, and electron-loss spectra, as well as the effective dynamical interaction induced by nonlocal screening. The momentum-dependence introduced by GW into the extended DMFT description leads to a narrowing of the quasiparticle width and more pronounced Hubbard bands in the metallic regime as one approaches the charge-ordering transition. It further affects the shape of collective modes, giving rise to dispersive plasmon-like long-wavelength and stripe modes.
We consider the thermodynamic consistency of the charge response function in the (extended) Hubbard model. In dynamical mean-field theory (DMFT), thermodynamic consistency is preserved. We prove that the static, homogeneous DMFT susceptibility is consistent as long as vertex corrections obtained from the two-particle impurity correlation function are included. In the presence of a nonlocal interaction, the problem may be treated within extended DMFT (EDMFT), or its diagrammatic extension, the dual boson approach. We show that here maintaining thermodynamic consistency requires knowledge of three- and four-particle impurity correlation functions, which are typically neglected. Nevertheless, the dual boson approximation to the response is remarkably close to consistency. This holds even when two-particle vertex corrections are neglected. EDMFT is consistent only in the strongly correlated regime and near half-filling, where the physics is predominantly local.
In this thesis, we tackle various problems in strongly correlated electron systems, which can be addressed properly via the non-perturbative dynamical mean field theory (DMFT) approach using the continuous time quantum Monte Carlo method as an impurity solver. First, we revisit the old Nagaoka ferromagnetism problem in the U=oo Hubbard model and study the stability of the ferromagnetic state as a function of the temperature, the doping level, and the next-nearest-neighbor hopping t'. We then address the nature of the Mott transition in the two-dimensional Hubbard model at half-filling using cluster DMFT. Cluster DMFT can incorporate the short-range correlations beyond DMFT by extending the spatial range in which correlations are treated exactly to a finite cluster size. The non-local correlations reduce substantially the critical interaction U and modify the shape of the transition lines in the phase diagram. We then concentrate on the calculation of two-particle response functions from the ab initio perspective by means of computing the one-particle excitation spectrum using the combination of the density functional theory (DFT) and DMFT and extracting the two-particle irreducible vertex function from a local two-particle Green's function computed within DMFT. In particular, we derive the equations for calculating the magnetic/charge susceptibility and the pairing susceptibility in superconductivity. This approach is applied to the Hubbard model and the periodic Anderson model and we determine the phase diagram of magnetism and superconductivity in these models. We show that the superconducting phase is indeed stable near the magnetic phase where the pairing interaction mediated by spin fluctuations is dominantly enhanced. The non-local correlation effect to superconductivity is also discussed using the dual fermion approach and the dynamical vertex approximation. We finally apply the vertex function approach within DFT+DMFT to a Fe-based superconductor, BaFe2As2, and compute the dynamical magnetic susceptibility in this material. Our calculation results show a good agreement with the magnetic excitation spectra observed in a neutron scattering experiment. The response function calculation method derived in this thesis can capture both a localized and an itinerant nature of collective excitations in strongly correlated electron systems.
We study the charge-density dynamics within the two-dimensional extended Hubbard model in the presence of long-range Coulomb interaction across the metal-insulator transition point. To take into account strong correlations we start from self-consistent extended dynamical mean-field theory and include nonlocal dynamical vertex corrections through a ladder approximation to the polarization operator. This is necessary to fulfill charge conservation and to describe plasmons in the correlated state. The calculated plasmon spectra are qualitatively different from those in the random-phase approximation: they exhibit a spectral density transfer and a renormalized dispersion with enhanced deviation from the canonical √q behavior. Both features are reminiscent of interaction induced changes found in single-electron spectra of strongly correlated systems.
We reconsider derivation of Ward identities for spin stiffnesses, which determine the nonlinear sigma model of magnetic degrees of freedom of interacting electrons in the presence of antiferromagnetic or incommensurate correlations. We emphasize that in the approaches, which do not break explicitly spin symmetry of the action, the spatial components of gauge kernel, which are used to obtain spin stiffnesses, remain gauge invariant even in case of spontaneous spin symmetry breaking. We derive the corrected Ward identities, which account for this gauge invariance. We find that the frequency dependence of temporal spin stiffnesses is not fixed by the obtained identities, and show that the infinitesimally small external staggered field is crucially important to obtain finite static uniform transverse susceptibility. On the other hand, the spatial spin stiffnesses are determined by the gauge kernel of the Legendre transformed theory, which is in general different from the gauge kernel of the original theory and obtain an explicit expression for spatial spin stiffnesses through susceptibilities and current correlation functions. We verify numerically the obtained results within dynamic mean-field theory, and obtain doping dependencies of the resulting spin stiffnesses for antiferromagnetic and incommensurate phases.
In this article, we discuss the nontrivial collective charge excitations (plasmons) of the extended square lattice Hubbard model. Using a fully nonperturbative approach, we employ the hybrid Monte Carlo algorithm to simulate the system at half-filling. A modified Backus-Gilbert method is introduced to obtain the spectral functions via numerical analytic continuation. We directly compute the single-particle density of states which demonstrates the formation of Hubbard bands in the strongly correlated phase. The momentum-resolved charge susceptibility also is computed on the basis of the Euclidean charge-density-density correlator. In agreement with previous extended dynamical mean-field theory studies, we find that, at high strength of the electron-electron interaction, the plasmon dispersion develops two branches.