Constrained Random Phase Approximation: the spectral method

TL;DR

This work introduces spectral cRPA (s-cRPA), a constrained RPA method that preserves electron number and selects active states per k-point via leverage-based screening, addressing negativity and stability issues seen in projector-based schemes. Compared to w-cRPA and p-cRPA, s-cRPA generally yields larger Hubbard interactions $U$ for Sc and Cu and removes more intra-d d-screening, while maintaining positivity and exact state count. Applied to CaFeO$_3$, s-cRPA gives onsite $U$ and exchange $J$ values closer to those required in DFT+$U$ to reproduce insulating behavior, though still not sufficient alone, indicating the need for beyond-RPA or moment-constrained enhancements. The paper also reports methodological advances, including multi-centre interaction calculations and a low-scaling, compressed Matsubara grid for full frequency-dependent interactions, enabling scalable studies of defects and spatial decay. Overall, s-cRPA emerges as a promising, more robust tool for computing effective Hubbard interactions in strongly correlated materials, with clear paths for further refinement and broader application.

Abstract

We present a constrained Random Phase Approximation (cRPA) method, termed spectral cRPA (s-cRPA), and compare it to established cRPA approaches for Scandium and Copper by varying the 3d shell filling. The s-cRPA method generally produces larger Hubbard U interaction values compared to conventional approaches. When applied to the realistic system CaFeO$_3$ , s-cRPA yields interaction parameters that align more closely with those required within DFT+U to reproduce the experimentally observed insulating state, addressing the metallic behaviour predicted by standard density functionals. We examine the issue of negative interaction values encountered in the projector cRPA method for filled d-shells. We show that s-cRPA provides improved numerical stability by preserving electron number conservation, a constraint that is violated in the projector cRPA method. The s-cRPA approach addresses some limitations of standard cRPA methods, particularly the tendency to underestimate U values, suggesting its potential utility for the community. Additionally, we have enhanced our implementation to include computation of multi-centre interactions for analysing spatial decay and developed an efficient low-scaling variant employing a compressed Matsubara grid to obtain full frequency-dependent interactions.

Figures (5)

• Figure 1: Bands of fcc Sc (a) and Cu (b) with d-character resolution obtained from Wannier90.Mostofi2008685 Red bands indicate strong d-character, blue bands indicate strong s-character. For reference, the original band structure obtained with VASPPhysRevB.59.1758 is shown in grey. Filling levels are indicated by horizontal lines (neutral compound at zero energy).
• Figure 2: Comparison of the quotient of on-site effective d-Coulomb repulsion $U$ and fully screened interaction $W$ for Sc and Cu at different fillings with the different disentanglement schemes presented in Sec. \ref{['sec:crpa']}.
• Figure 3: (solid) Frequency dependence of effective interaction $U$ for Sc$^{4+}$ determined with various cRPA methods obtained from analytic continuation using a compressed Matsubara grid of 24 points.Kaltak2020 For comparison data obtained from direct computation on the real-frequency axis is shown (dashed) Gray line corresponds to bare Coulomb interaction.
• Figure 4: Spatial decay of effective cRPA interactions scaled to one-site interaction $U(R)/U(R=0)$ for Sc$^{4-}$.
• Figure 5: On-site Hubbard-Kanamori parameter $U$ (\ref{['eq:Udef']}) as a function of k-point sampling for each cRPA method studied, along with the fully screened interaction in RPA for Sc (a) and Cu (b). Each series contains four plots, one for each method. In every plot, the x-axis represents the number of k-points per direction used to sample the first Brillouin zone, and the y-axis shows the calculated $U$ value. The blue line corresponds to data obtained without the long-wavelength limit correction; it exhibits a linear dependence on the inverse number of k-points, allowing for extrapolation to infinite k-point sampling.PhysRevB.111.195144 The extrapolated interaction value is indicated as a blue number within each plot. The orange line shows results including the long-wavelength limit correction, with the value at an $8\times8\times8$ k-point grid represented by an orange number.