Low-energy effective interactions beyond the constrained random-phase approximation by the functional renormalization group

TL;DR

This paper introduces the Wick-ordered constrained functional renormalization group (cfRG) as an advanced method to compute low-energy effective interactions in solids, extending beyond the traditional cRPA approach by including all interaction channels.

Contribution

The study demonstrates how cfRG generalizes cRPA, revealing significant corrections in multi-band systems and showing potential impacts on magnetic properties in low-energy models.

Findings

cfRG corrections are suppressed in symmetric monolayer graphene models

Symmetry breaking enhances cfRG corrections without qualitative change

Significant qualitative corrections observed in CuO-like chain model

Abstract

In the derivation of low-energy effective models for solids targeting the bands near the Fermi level, the constrained random phase approximation (cRPA) has become an appreciated tool to compute the effective interactions. The Wick-ordered constrained functional renormalization group (cfRG) generalizes the cRPA approach by including all interaction channels in an unbiased way. Here we present applications of the cfRG to two simple multi-band systems and compare the resulting effective interactions to the cRPA. First we consider a multiband model for monolayer graphene, where we integrate out the $\sigma$-bands to get an effective theory for $\pi$-bands. It turns out that terms beyond cRPA are strongly suppressed by the different $xy$-plane reflection symmetry of the bands. In our model the cfRG-corrections to cRPA become visible when one disturbs this symmetry difference slightly, however without qualitative changes. This study shows that the embedding or layering of two-dimensional electronic systems can alter the effective interaction parameters beyond what is expected from screening considerations. The second example is a one-dimensional model for a diatomic system reminiscent of a CuO chain, where we consider an effective theory for Cu 3d-like orbitals. Here the fRG data shows relevant and qualitative corrections compared to the cRPA results. We argue that the new interaction terms affect the magnetic properties of the low-energy model.