Exact downfolding and its perturbative approximation

TL;DR

This paper presents a rigorous formulation of the downfolding procedure for deriving effective models in many-electron systems, including exact and perturbative approaches, with applications to real materials.

Contribution

It introduces an exact downfolding formalism, conditions for perturbative truncation, and derives the constrained random phase approximation (cRPA) from first principles.

Findings

Derived an exact effective model by integrating out high-energy degrees of freedom.

Established conditions under which perturbative truncation of interactions is justified.

Applied the formalism to materials like fcc Nickel and SrCuO$_2$ to analyze effective models.

Abstract

Solving the many-electron problem, even approximately, is one of the most challenging and simultaneously most important problems in contemporary condensed matter physics with various connections to other fields. The standard approach is to follow a divide and conquer strategy that combines various numerical and analytical techniques. A crucial step in this strategy is the derivation of an effective model for a subset of degrees of freedom by a procedure called downfolding, which often corresponds to integrating out energy scales far away from the Fermi level. In this work we present a rigorous formulation of this downfolding procedure, which complements the renormalization group picture put forward by Honerkamp [PRB 85, 195129 (2012)}]. We derive an exact effective model in an arbitrarily chosen target space (e.g. low-energy degrees of freedom) by explicitly integrating out the the rest space (e.g. high-energy degrees of freedom). Within this formalism we state conditions that justify a perturbative truncation of the downfolded effective interactions to just a few low-order terms. Furthermore, we utilize the exact formalism to formally derive the widely used constrained random phase approximation (cRPA), uncovering underlying approximations and highlighting relevant corrections in the process. Lastly, we detail different contributions in the material examples of fcc Nickel and the infinite-layer cuprate SrCuO$_2$. Our results open up a new pathway to obtain effective models in a controlled fashion and to judge whether a chosen target space is suitable.