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Bulletin of the American Physical Society

Higher order corrections to effective low-energy theories for strongly correlated electron systems

Sasha Chernyshev, Dimitrios Galanakis, Philip Phillips, A. M. Rozhkov, A.–M. S. Tremblay

Published Mar 24, 2005• 0 citations
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Three well-known perturbative approaches to deriving low-energy effective theories, the degenerate Brillouin-Wigner perturbation theory (projection method), the canonical transformation, and the resolvent methods, are compared. We use the Hubbard model as an example to show how, to fourth order in hopping ttt, all methods lead to the same effective theory, namely the $t\text{\ensuremath{-}}J$ model with ring exchange and various correlated hoppings. We emphasize subtle technical difficulties that make such a derivation less trivial to carry out for orders higher than second. We also show that in higher orders, different approaches can lead to seemingly different forms for the low-energy Hamiltonian. All of these forms are equivalent since they are connected by an additional unitary transformation whose generator is given explicitly. The importance of transforming the operators is emphasized and the equivalence of their transformed structure within the different approaches is also demonstrated.
Electron correlation is a measure of the errors that are inherent in the Hartree-Fock theory or orbital models. When the electron density is high, correlation is weak and the traditional electronic theory works well. However, at a low density of electrons correlation effects become strong and the traditional theory fails to describe the electron system correctly. Therefore, the electron correlation plays a radical role in such materials as high-temperature superconductors and heavy fermions, etc. To date, there is no agreement on how to deal with higher-order terms (correlation energy) in the series of electron’s ground state energy although a method that is termed diagrammatic iteration approach (DIA) was developed more than one decade ago by the authors of this article. That is why no consensus on the origin and mechanism of superconductivity has been engaged in superconductivity community. From the viewpoint of methodology, the DIA is indeed an approach to higher-order terms from the lower-order ones, i.e. it is a new method to show how to go beyond the random phase approximation (RPA) step by step by iteration. Here, we are logically presenting it to the community of modern physics with more analyses and hope to attract more attention to it and promote its applications.
Abstract First-principles approaches have been successful in solving many-body Hamiltonians for real materials to an extent when correlations are weak or moderate. As the electronic correlations become stronger often embedding methods based on first-principles approaches are used to better treat the correlations by solving a suitably chosen many-body Hamiltonian with a higher level theory. The success of such embedding theories, often referred to as second-principles, is commonly measured by the quality of self-energy Σ which is either a function of energy or momentum or both. However, Σ should, in principle, also modify the electronic eigenfunctions and thus change the real space charge distribution. While such practices are not prevalent, some works that use embedding techniques do take into account these effects. In such cases, choice of partitioning, of the parameters defining the correlated Hamiltonian, of double-counting corrections, and the adequacy of low-level Hamiltonian hosting the correlated subspace hinder a systematic and unambiguous understanding of such effects. Further, for a large variety of correlated systems, strong correlations are largely confined to the charge sector. Then an adequate nonlocal low-order theory is important, and the high-order local correlations embedding contributes become redundant. Here we study the impact of charge self-consistency within two example cases, TiSe2 and CrBr3, and show how real space charge re-distribution due to correlation effects taken into account within a first-principles Green’s function-based many-body perturbative approach is key in driving qualitative changes to the final electronic structure of these materials.