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arXiv:1902.01176 · Physical review. B./Physical review. B

Position representation of effective electron-electron interactions in solids

Tor Jonas Sjöstrand, Fredrik Nilsson, Christoph Friedrich, F. Aryasetiawan

Published May 21, 2019• 4 citations
arXivHTMLPDF

Abstract

An essential ingredient in many model Hamiltonians, such as the Hubbard model, is the effective electron-electron interaction $U$, which enters as matrix elements in some localized basis. These matrix elements provide the necessary information in the model, but the localized basis is incomplete for describing $U$. We present a systematic scheme for computing the manifestly basis-independent dynamical interaction in position representation, $U(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})$, and its Fourier transform to time domain, $U(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\tau})$. These functions can serve as an unbiased tool for the construction of model Hamiltonians. For illustration we apply the scheme within the constrained random-phase approximation to the cuprate parent compounds ${\mathrm{La}}_{2}{\mathrm{CuO}}_{4}$ and ${\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4}$ within the commonly used one- and three-band models, and to nonsuperconducting ${\mathrm{SrVO}}_{3}$ within the ${t}_{2g}$ model. Our method is used to investigate the shape and strength of screening channels in the compounds. We show that the $\mathrm{O}\phantom{\rule{0.16em}{0ex}}2{p}_{x,y}\ensuremath{-}\mathrm{Cu}\phantom{\rule{0.16em}{0ex}}3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ screening gives rise to regions with strong attractive static interaction in the minimal (one-band) model in both cuprates. On the other hand, in the minimal (${t}_{2g}$) model of ${\mathrm{SrVO}}_{3}$ only regions with a minute attractive interaction are found. The temporal interaction exhibits generic damped oscillations in all compounds, and its time integral is shown to be the potential caused by inserting a frozen point charge at $\ensuremath{\tau}=0$. When studying the latter within the three-band model for the cuprates, short time intervals are found to produce a negative potential.

DOI
10.1103/physrevb.99.195136

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An essential ingredient in many model Hamiltonians, such as the Hubbard model, is the effective electron-electron interaction UUU, which enters as matrix elements in some localized basis. These matrix elements provide the necessary information in the model, but the localized basis is incomplete for describing UUU. We present a systematic scheme for computing the manifestly basis-independent dynamical interaction in position representation, U(r,r\ensuremath′;\ensuremathω)U(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\omega})U(r,r\ensuremath′;\ensuremathω), and its Fourier transform to time domain, U(r,r\ensuremath′;\ensuremathτ)U(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}};\ensuremath{\tau})U(r,r\ensuremath′;\ensuremathτ). These functions can serve as an unbiased tool for the construction of model Hamiltonians. For illustration we apply the scheme within the constrained random-phase approximation to the cuprate parent compounds La2CuO4{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}La2​CuO4​ and HgBa2CuO4{\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4}HgBa2​CuO4​ within the commonly used one- and three-band models, and to nonsuperconducting SrVO3{\mathrm{SrVO}}_{3}SrVO3​ within the t2g{t}_{2g}t2g​ model. Our method is used to investigate the shape and strength of screening channels in the compounds. We show that the O2px,y\ensuremath−Cu3dx2\ensuremath−y2\mathrm{O}\phantom{\rule{0.16em}{0ex}}2{p}_{x,y}\ensuremath{-}\mathrm{Cu}\phantom{\rule{0.16em}{0ex}}3{d}_{{x}^{2}\ensuremath{-}{y}^{2}}O2px,y​\ensuremath−Cu3dx2\ensuremath−y2​ screening gives rise to regions with strong attractive static interaction in the minimal (one-band) model in both cuprates. On the other hand, in the minimal (t2g{t}_{2g}t2g​) model of SrVO3{\mathrm{SrVO}}_{3}SrVO3​ only regions with a minute attractive interaction are found. The temporal interaction exhibits generic damped oscillations in all compounds, and its time integral is shown to be the potential caused by inserting a frozen point charge at \ensuremathτ=0\ensuremath{\tau}=0\ensuremathτ=0. When studying the latter within the three-band model for the cuprates, short time intervals are found to produce a negative potential.
Understanding which minimal effective model captures the essential physics of cuprates is a key step towards unraveling the mechanism behind high- T c superconductivity. Recent measurements of the dynamical spin structure factor (DSF) in cuprate ladder compounds have indicated the presence of an additional, attractive term in the single-band Hubbard model, potentially originating from electron-phonon interactions. Here, we demonstrate that similar DSF features can also be captured by t − J descriptions with a smaller attractive term. Motivated by this observation, we systematically investigate the strength and origin of different contributions to the single-band Hamiltonians by downfolding either from the three-band Emery model or the electron-phonon coupled Hubbard-Holstein model. For one-dimensional systems, we find that the extended versions of both single-band descriptions can reproduce the experimentally observed DSF signatures. Finally, we extend our analysis to two dimensions by comparing two-hole correlation functions for the different single-band models. Our results provide new insights into the long-standing question of which single-band Hamiltonian can capture the essential physics of cuprates.
Correlated electron physics is intrinsically a multiscale problem, since high-energy electronic states screen the interactions between the correlated electrons close to the Fermi level, thereby reducing the magnitude of the interaction strength and dramatically shortening its range. Thus the handling of screening is an essential ingredient in the first-principles modeling of correlated electron systems. Screening is an intrinsically dynamic process and the corresponding downfolding methods such as the constrained random phase approximation indeed produce a dynamic interaction. However, many low-energy methods require an instantaneous interaction as input, which makes it necessary to map the fully dynamic interaction to an effective instantaneous interaction strength. It is not clear if and when such an effective model can capture the physics of the one with dynamic interaction and how to best perform the mapping. Here, we provide a systematic benchmark relevant to correlated materials, in the form of the Anderson impurity model. Overall, we find that a static approximation can be valid and that the moment-based approach recently proposed by Scott and Booth can be a good tool to find the value of the static interaction. We also identify physical regimes, especially under doping, where an instantaneous interaction cannot capture all of the relevant physics.
In a previous work [Phys. Rev. B 77, 085122 (2008)], a procedure for constructing low-energy models of electrons in solids was proposed. The procedure starts with dividing the Hilbert space into two subspaces: the low-energy part (``ddd space'') and the rest of the space (``rrr space''). The low-energy model is constructed for the ddd space by eliminating the degrees of freedom of the rrr space. The thus derived model contains the strength of electron correlation expressed by a partially screened Coulomb interaction, calculated in the constrained random-phase approximation (cRPA), where screening channels within the ddd space, Pd{P}_{d}Pd​, are subtracted. One conceptual problem of this established downfolding method is that for entangled bands it is not clear how to cut out the ddd space and how to distinguish Pd{P}_{d}Pd​ from the total polarization. Here, we propose a simple procedure to overcome this difficulty. The ddd space is defined to be an isolated set of bands generated from a set of maximally localized Wannier basis, which consequently defines Pd{P}_{d}Pd​. The rrr subspace is constructed as the complementary space orthogonal to the ddd subspace, resulting in two sets of completely disentangled bands. Using these disentangled bands, the effective parameters of the ddd space are uniquely determined by the cRPA method. The method is successfully applied to 3d3d3d transition metals.
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Effective Hamiltonians offer a realistic and practical approach to the representation of low-lying electronic excitations in strongly correlated charge transfer insulators, such as the high-Tc parent La2CuO4. Not only do such Hamiltonians minimize the required number of orbital degrees of freedom, but it is suggested that the fundamental problem of charged excitations in these materials may be folded into renormalized Hamiltonian parameters which already reflect solid state screening, and that such parameters may easily be obtained from constrained local density functional calculations. Beneficial consequences of the renormalization include reasonable insulating gaps already from the Hartree-Fock one-electron spectra, and a restriction of configuration interaction to those essential correlations associated with the spin degrees of freedom. In large periodic clusters where states have well characterized crystal momentum, such spin correlations in the vicinity of particles doped into the parent insulator are still reminiscent of the total spin eigenstates found in impurity model solutions.
We propose a new, alternative method for ab-initio calculations of the electronic structure of solids, which has been specifically adapted to treat many-body effects in a more rigorous way than many existing ab-initio methods. We start from a standard band-structure calculation for an effective one-particle Hamiltonian approximately describing the material of interest. This yields a suitable set of one-particle basis functions, from which well localized Wannier functions can be constructed using a method proposed by Marzari and Vanderbilt. Within this (minimal) basis of localized Wannier functions the matrix elements of the non-interacting (one-particle) Hamiltonian as well as the Coulomb matrix elements can be calculated. The result is a many-body Hamiltonian in second quantization with parameters determined from first principles calculations for the material of interest. The Hamiltonian is in the form of a multi-band Hamiltonian in second quantization (a kind of extended, multi-band Hubbard model) such that all the standard many-body methods can be applied. We explicitly show how this approach can be solved in the simplest many-body approximation, the mean-field Hartree-Fock approximation (HFA), which takes into account exact exchange and corrects for self-interaction effects.