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arXiv:1607.08336 · Journal of the Physical Society of Japan

Model-Mapped RPA for Determining the Effective Coulomb Interaction

Hirofumi Sakakibara, Seung Woo Jang, Hiori Kino, Myung Joon Han, Kazuhiko Kuroki, Takao Kotani

Published Mar 31, 2017• 13 citations
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Abstract

We present a new method to obtain interaction part of a model Hamiltonian from the result of the first-principles calculation. The effective interaction contained in the model is determined based on the random phase approximation (RPA). In contrast to previous methods such as projected RPA or constrained RPA, the new method takes into account the long-range part of the polarization effect when determining the interaction in the model. After we discuss problems in previous RPA methods, we will give the formulation of the new method, and show how it works for the single-band Hubbard model of HgBa$_2$CuO$_4$.

DOI
10.7566/jpsj.86.044714

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We present a new method to obtain interaction part of a model Hamiltonian from the result of the first-principles calculation. The effective interaction contained in the model is determined based on the random phase approximation (RPA). In contrast to previous methods such as projected RPA or constrained RPA, the new method takes into account the long-range part of the polarization effect when determining the interaction in the model. After we discuss problems in previous RPA methods, we will give the formulation of the new method, and show how it works for the single-band Hubbard model of HgBa2_22​CuO4_44​.
We have applied the model-mapped random phase approximation (RPA) [H. Sakakibara et al., J. Phys. Soc. Jpn. 86, 044714 (2017)] to the cuprate superconductors La2CuO4{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}La2​CuO4​ and HgBa2CuO4{\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4}HgBa2​CuO4​, resulting in two-orbital Hubbard models. All the model parameters are determined based on first-principles calculations. For the model Hamiltonians, we perform fluctuation exchange calculation. Results explain relative height of Tc{T}_{c}Tc​ observed in experiment for La2CuO4{\mathrm{La}}_{2}{\mathrm{CuO}}_{4}La2​CuO4​ and HgBa2CuO4{\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4}HgBa2​CuO4​. In addition, we give some analyses for the interaction terms in the model, especially comparisons with those of the constrained RPA.
The accuracy of the constrained random phase approximation(cRPA) method is examined in multi-orbital Hubbard models containing all possible on-site density-density interactions. Using DMFT, we show that the effective model constructed using cRPA fails to reproduce the spectral properties of the original full model in a wide parameter range. By comparing quantities such as the density of states and quasiparticle residues of the full and the effective models, we show that cRPA systematically overestimates the screening of Hubbard U for DMFT impurity solvers. We instead propose a new method to investigate the screening mechanism in the system using the local polarization, which is highly successful in reproducing spectra and also shows that the true screening is far less than that predicted by RPA. Furthermore, we compare the fully screened interaction WWW given by RPA and our new method and show that the RPA WWW is also overscreened and misses the signatures of local screening, which are clearly present in our new method.
We have applied the model-mapped RPA [H. Sakakibara et al., J. Phys. Soc. Jpn. 86, 044714 (2017)] to the cuprate superconductors La2CuO4 and HgBa2CuO4, resulting two-orbital Hubbard models. All the model parameters are determined based on first-principles calculations. For the model Hamiltonians, we perform fluctuation exchange calculation. Results are consistent with experimental Tc. In addition, we give some analyses for the interaction terms in the model, especially comparisons with those of the constrained RPA.
In this chapter, we discuss how low-energy models can be derived from first-principle calculations. The main quantity to be determined is the effective Coulomb interaction between the electrons in the low-energy subspace, the so-called Hubbard U. We begin by giving a brief historical introduction to different ways to determine U. However, the main focus in the chapter lies on the constrained random-phase approximation (cRPA), which is a method to compute U based on the concept of partial screening. At the end of the chapter, we put the cRPA in a broader perspective by introducing the multi-tier approach, which is a general approach for downfolding that naturally encompasses the cRPA in the first step.
The many-body theory of interacting electrons poses an intrinsically difficult problem that requires simplifying assumptions. For the determination of electronic screening properties of the Coulomb interaction, the random phase approximation (RPA) provides such a simplification. Here we explicitly show that this approximation is justified for band structures with sizable band gaps. This is when the electronic states responsible for the screening are energetically far away from the Fermi level, which is equivalent to a short electronic propagation length of these states. The RPA contains exactly those diagrams in which the classical Coulomb interaction covers all distances, whereas neglected vertex corrections involve quantum tunneling through the barrier formed by the band gap. Our analysis of electron-electron interactions provides a real-space analogy to Migdal's theorem on the smallness of vertex corrections in electron-phonon problems. An important application is the increasing use of constrained RPA calculations of effective interactions. We find that their usage of Kohn-Sham energies accounts for the leading local (excitonic) vertex correction in insulators.
Working within an exactly solvable 3 level model, we discuss am extension of the Random Phase Approximation (RPA) based on a boson formalism. A boson Hamiltonian is defined via a mapping procedure and its expansion truncated at four-boson terms. RPA-type equations are then constructed and solved iteratively. The new solutions gain in stability with respect to the RPA ones. We perform diagonalizations of the boson Hamiltonian in spaces containing up to four-phonon components. Approximate spectra exhibit an improved quality with increasing the size of these multiphonon spaces. Special attention is addressed to the problem of the anharmonicity of the spectrum.