Assessing the performance of the random phase approximation for exchange and superexchange coupling constants in magnetic crystalline solids

The random phase approximation (RPA) for total energies has previously been shown to provide a qualitatively correct description of static correlation in molecular systems, where density functional theory (DFT) with local functionals are bound to fail. This immediately poses the question of whether the RPA is also able to capture the correct physics of strongly correlated solids such as Mott insulators. Due to strong electron localization, magnetic interactions in such systems are dominated by superexchange, which in the simplest picture can be regarded as the analog of static correlation for molecules. In this paper, we investigate the performance of the RPA for evaluating both superexchange and direct exchange interactions in the magnetic solids NiO, MnO, ${\mathrm{Na}}_3{\mathrm{Cu}}_2{\mathrm{SbO}}_6,{\mathrm{Sr}}_2{\mathrm{CuO}}_3,{\mathrm{Sr}}_2{\mathrm{CuTeO}}_6$, and a monolayer of ${\mathrm{CrI}}_3$, which were chosen to represent a broad variety of magnetic interactions. It is found that the RPA can accurately correct the large errors introduced by Hartree-Fock, independent of the input orbitals used for the perturbative expansion. However, in most cases, accuracies similar to RPA can be obtained with DFT+U, which is significantly simpler from a computational point of view.

Figure 1

$J_1$ and $J_2$ parameters for NiO calculated with EXX and RPA calculated as a function of $U$ used in the input PBE+U calculations. The experimentally determined values are shown as a horizontal grey line.

Figure 2

$J_1$ and $J_2$ for NiO calculated as a function of $U$ using LDA+U, PBE+U, HSE06@PBE+U, and RPA@PBE+U. The experimental values [54] are indicated with grey horizontal lines.

Figure 3

$J_1$ and $J_2$ for MnO calculated as a function of $U$ using LDA+U, PBE+U, HSE06@PBE+U, and RPA@PBE+U. The experimental values [55] are indicated with grey horizontal lines.

Figure 4

$J_1$ and $J_2$ for ${\mathrm{Na}}_3{\mathrm{Cu}}_2{\mathrm{SbO}}_6$ calculated as a function of $U$ using LDA+U, PBE+U, HSE06@PBE+U, and RPA@PBE+U. The experimental values [58] are indicated with grey horizontal lines.

Figure 5

Nearest neighbor coupling $J$ for ${\mathrm{Sr}}_2{\mathrm{CuO}}_3$ calculated as a function of U using LDA+U, PBE+U, HSE06@PBE+U, and RPA@PBE+U. The experimental span of values is indicated by the grey area. (Lower bound: inelastic Neutron scattering [60]. Upper bound: susceptibility measurements [61].)

Figure 6

$J_1$ and $J_2$ for ${\mathrm{Sr}}_2{\mathrm{CuTeO}}_6$ calculated as a function of $U$ using LDA+U, PBE+U, HSE06@PBE+U, and RPA@PBE+U. The experimental values [63] are indicated with grey horizontal lines.

Figure 7

Nearest neighbor coupling $J$ for a monolayer of ${\mathrm{CrI}}_3$ calculated as a function of $U$ using LDA+U, PBE+U, EXX@PBE+U, HSE06@PBE+U, and RPA@PBE+U.