Exact downfolding and its perturbative approximation

TL;DR

This work presents an exact downfolding formalism that derives a target-space action by partitioning degrees of freedom into a target space $\mathcal{T}$ and rest space $\mathcal{R}$ and integrating out $\mathcal{R}$ via a path-integral. It introduces a generating functional $\mathcal{G}[f,\bar{f}]$ whose Taylor expansion yields an exact sequence of $n$-particle interactions $G^{(n)}$, giving a formal action $S_{\mathrm{eff}}[f,\bar{f}] = S_{f}[f,\bar{f}] - \sum_{n=1}^\infty G^{(n)}_{1,\dots,n;\bar{n},\dots,\bar{1}} \bar{f}_{\bar{i}_1}\cdots \bar{f}_{\bar{i}_n} f_{i_n}\cdots f_{i_1}$. The paper then details practical approximations, criteria for truncation, and how cRPA emerges from this exact framework, highlighting the importance of kinetic (hybridization) and particle-hole couplings and their retarded nature. Analytic illustrations of the dominant contributions are provided, followed by ab initio analyses for Ni and SrCuO$_2$ that quantify the coupling hierarchies and show the critical role of basis choice. The results demonstrate a principled path to build faithful, controlled effective target-space models and lay groundwork for automated downfolding tools. An explicit, compact form of the exact target-space action and its diagrammatic interpretation are provided, enabling systematic comparison with existing downfolding methods and clarifying when a reduced two-body description suffices.

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.

Figures (9)

• Figure 1: Diagrammatic representation of $G^{(1)}$ (a) and $G^{(2)}$ (b--f). The gray background indicates rest space ($\color{bdiv2-3}\mathcal{R}$) variables which are summed over, while the white background indicates target space ($\color{bdiv2-6}\mathcal{T}$). White circles indicate ingoing and black circles indicate outgoing legs. Light cyan filled shapes in the rest space $\mathcal{R}$ indicate $g^{(k)}$ (see main text). The one-particle vertex $A^{1:1}$ is represented with a dark cyan line, while the two-particle vertices $A^{2:2}$, $A^{3:1/1:3}$, $B^{2:2}$ are given as dark cyan, orange, and purple squares, respectively. We only draw topologically distinct vertices, reducing the number of contributions to 25 for the two-particle case. The number of (imaginary) times that each diagram depends on is denoted in the top left corner as $N\tau$, under the assumption of instantaneous vertices.
• Figure 2: Graphical definition of connectedness for diagrams (b3), (c3) and (e2) (cf. \ref{['fig:diagrams']}) in terms of connected single-particle $g^{(1)}$ and two-particle $G^{(2)}_c$ Green's functions. We depict $G^{(2)}_c$ as a purple filled shape. The light cyan filled shape ($g^{(k)}$) is defined such that it encompasses only the set of possible decompositions in terms of connected Green's functions which lead to a connected diagram. The disconnected contributions (right, red box) are not included in $g^{(k)}$, as they cancel by virtue of the linked cluster theorem.
• Figure 3: Diagrams contained in cRPA in orders $(A^{n:m})^2$ and $(g^{(1)})^2$ and $(A^{n:m})^3$ and $(g^{(1)})^4$ under the assumption of orthogonal basis sets of the rest- and target space. Light blue lines denote rest space single-particle propagators, vertices are colored according to \ref{['fig:diagrams']}. The red dashed line represents a bare target space propagator $G^f_0$ that is required to construct a four-point vertex from the six-point contribution $A^{3:1} g^{(1)} A^{3:1}$ (indicated by the red arrow). The rest space vertex is defined by the pale-blue four-point vertex.
• Figure 4: Additional diagrams of order $(A^{n:m})^2$ in the case of non-orthogonal basis functions of the target and rest spaces. Light purple lines denote the rest-to-target propagator. Vertices are colored according to \ref{['fig:diagrams']}. The lowest row visualizes the diagrammatic content of cRPA, while the upper two rows graphically construct these starting from the exact downfolding.
• Figure 5: Diagrams in the target space effective theory up to four external $f$ fields under the approximations detailed in \ref{['sec:ph-interactions']}. (a) Single-particle term (two-point vertex) $\Sigma^f$, (b) Two-particle term (four-point vertex) $\Gamma^f$, (c) Six-point vertex $\Gamma^{f;(6)}$. The $c$ propagators are denoted with light cyan lines, and the $cf$ vertices with dark cyan rectangles/squares (as in \ref{['fig:diagrams']}).