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arXiv:2001.01353 · Physical review. B./Physical review. B

Extending the Gutzwiller approximation to intersite interactions

Garry Goldstein, Nicola Lanatà, Gabriel Kotliar

Published Jul 31, 2020• 4 citations
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Abstract

We develop an extension of the Gutzwiller approximation (GA) formalism that includes the effects of Coulomb interactions of arbitrary range (including density density, exchange, pair hopping, and Coulomb-assisted hopping terms). This formalism reduces to the ordinary GA formalism for the multiband Hubbard models in the presence of only local interactions. This is accomplished by combining the $1/z$ expansion---where $z$ is the coordination number, and only the leading-order terms contribute in the limit of infinite dimensions---with a ${P}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}{P}_{R}\ensuremath{-}I$ expansion, where ${P}_{R}$ is the Gutzwiller projector on the site $R$. The method is conveniently formulated in terms of a Gutzwiller Lagrange function. We apply our theory to the extended single-band Hubbard model. Similarly to the usual Brinkman-Rice mechanism, we find a Mott transition. A valence skipping transition is observed, where the occupation of the empty and doubly occupied states for the Gutzwiller wave function is enhanced with respect to the uncorrelated Slater determinant wave function.

DOI
10.1103/physrevb.102.045152

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We develop an extension of the Gutzwiller approximation (GA) formalism that includes the effects of Coulomb interactions of arbitrary range (including density density, exchange, pair hopping, and Coulomb-assisted hopping terms). This formalism reduces to the ordinary GA formalism for the multiband Hubbard models in the presence of only local interactions. This is accomplished by combining the 1/z1/z1/z expansion---where zzz is the coordination number, and only the leading-order terms contribute in the limit of infinite dimensions---with a PR\ifmmode†\else\textdagger\fiPR\ensuremath−I{P}_{R}^{\ifmmode\dagger\else\textdagger\fi{}}{P}_{R}\ensuremath{-}IPR\ifmmode†\else\textdagger\fi​PR​\ensuremath−I expansion, where PR{P}_{R}PR​ is the Gutzwiller projector on the site RRR. The method is conveniently formulated in terms of a Gutzwiller Lagrange function. We apply our theory to the extended single-band Hubbard model. Similarly to the usual Brinkman-Rice mechanism, we find a Mott transition. A valence skipping transition is observed, where the occupation of the empty and doubly occupied states for the Gutzwiller wave function is enhanced with respect to the uncorrelated Slater determinant wave function.
The main flaw of the well-known Brinkman-Rice description, obtained through the Gutzwiller approximation, of the paramagnetic Mott transition in the Hubbard model is in neglecting high-energy virtual processes that generate, for instance, the antiferromagnetic exchange $J\ensuremath{\sim}{t}^{2}/U$. Here, we propose a way to capture those processes by combining the Brinkman-Rice approach with a variational Schrieffer-Wolff transformation, and apply this method to study the single-band metal-to-insulator transition in a Bethe lattice with infinite coordination number, where the Gutzwiller approximation becomes exact. We indeed find for the Mott transition a description very close to the real one provided by the dynamical mean-field theory, an encouraging result in view of possible applications to more involved models.
Based on the variational Gutzwiller theory, we present a method for the computation of response functions for multiband Hubbard models with general local Coulomb interactions. The improvement over the conventional random-phase approximation is exemplified for an infinite-dimensional two-band Hubbard model where the incorporation of the local multiplet structure leads to a much larger sensitivity of ferromagnetism on the Hund coupling. Our method can be implemented into local-density approximation and Gutzwiller schemes and will therefore be an important tool for the computation of response functions for strongly correlated materials.
Within a Lagrangian formalism, we derive the time-dependent Gutzwiller approximation for general multi-band Hubbard models. Our approach explicitly incorporates the coupling between time-dependent variational parameters and a time-dependent density matrix from which we obtain dynamical correlation functions in the linear-response regime. Our results are illustrated for the one-band model where we show that the interacting system can be mapped to an effective problem of fermionic quasiparticles coupled to 'doublon' (double occupancy) bosonic fluctuations. The latter have an energy on the scale of the on-site Hubbard repulsion U in the dilute limit but become soft at the Brinkman–Rice transition, which is shown to be related to an emerging conservation law of doublon charge and the associated gauge invariance. Coupling with the boson mode produces a structure in the charge response and we find that a similar structure appears in dynamical mean-field theory.
Within a Lagrangian formalism, we derive the time-dependent Gutzwiller approximation for general multi-band Hubbard models. Our approach explicitly incorporates the coupling between time-dependent variational parameters and a time-dependent density matrix from which we obtain dynamical correlation functions in the linear-response regime. Our results are illustrated for the one-band model where we show that the interacting system can be mapped to an effective problem of fermionic quasiparticles coupled to 'doublon' (double occupancy) bosonic fluctuations. The latter have an energy on the scale of the on-site Hubbard repulsion U in the dilute limit but become soft at the Brinkman–Rice transition, which is shown to be related to an emerging conservation law of doublon charge and the associated gauge invariance. Coupling with the boson mode produces a structure in the charge response and we find that a similar structure appears in dynamical mean-field theory.
The description of electronic inhomogeneities in transition metal compounds requires a technique which is able to deal with the excitations of an inhomogeneous, strongly correlated system. In this talk I will outline the formalism of the timedependent Gutzwiller approximation [1] for the Hubbard model which allows for the computation of random phase approximation (RPA) like fluctuations on top of the Gutzwiller approximation (GA). The performance of the method is demonstrated by comparing static quantities like ground state energy and double occupancies as well as dynamical correlation functions in the charge, spin and pair channels with HartreeFock+RPA and exact results. I will further show that the method gives a good account of several spectroscopic quantities related to stripes in highTc superconductors. A major part of the talk will address the recent generalization of the method to the case of multiple bands [2,3]. Contrary to the standard RPA, the incorporation of the local multipletstructure within our approach leads to a much larger sensitivity of ferromagnetism on the the ratio between Hund coupling and intraorbital repulsion J/U. Besides, we find that the Brinkman Rice localization transition, which in the paramagnetic GA appears at integer doping is always masked by an antiferromagnetic or an orbitally ordered ground state. In addition, a quantum Lifshitz point is found at larger values of J/U which corresponds to the common boundary of ferro, para and incommensurate magnetic phases. [1] G. Seibold and J. Lorenzana, Phys. Rev. Lett. 86, 2605 (2001) [2] E. von Oelsen, G. Seibold, and J. Bunemann, New J. Phys. 13, 113031 (2011) [3] E. v. Oelsen, G. Seibold, and J. Bunemann, Phys. Rev. Lett. 107, 076402 (2011)
Using extended dynamical mean-field theory and its combination with the GWGWGW approximation, we compute the phase diagrams and local spectral functions of the single-band extended Hubbard model on the square and simple cubic lattices, considering long-range interactions up to the third nearest neighbors. The longer-range interactions shift the boundaries between the metallic, charge-ordered insulating, and Mott insulating phases, and lead to characteristic changes in the screening modes and local spectral functions. Momentum-dependent self-energy contributions enhance the correlation effects and thus compete with the additional screening effect from longer-range Coulomb interactions. Our results suggest that the influence of longer-range intersite interactions is significant, and that these effects deserve attention in realistic studies of correlated materials.