Construction of the spectral function from noncommuting spectral moment matrices

The $\text{LDA}+U$ method is widely used to study the properties of realistic solids with strong electron correlations. One of its main shortcomings is that it does not provide direct access to the temperature dependence of material properties such as the magnetization, the magnetic anisotropy energy, the Dzyaloshinskii-Moriya interaction, the anomalous Hall conductivity, and the spin-orbit torque. While the method of spectral moments allows us in principle to compute these quantities directly at finite temperatures, the standard two-pole approximation can be applied only to Hamiltonians that are effectively of single-band type. We do a first step to explore if the method of spectral moments may replace the $\text{LDA}+U$ method in first-principles calculations of correlated solids with many bands in cases where the direct assessment of the temperature dependence of equilibrium and response functions is desired: The spectral moments of many-band Hamiltonians of correlated electrons do not commute and therefore they do not possess a system of common eigenvectors. We show that nevertheless the spectral function may be constructed from the spectral moments by solving a system of coupled nonlinear equations. Additionally, we show how to compute the anomalous Hall conductivity of correlated electrons from this spectral function. We demonstrate the method for the Hubbard-Rashba model, where the standard two-pole approximation cannot be applied because spin-orbit interaction (SOI) couples the spin-up and the -down bands. In the quest for new quantum states that arise from the combination of SOI and correlation effects, the Hartree-Fock approximation is frequently used to obtain a first approximation for the phase diagram. We propose that using the many-band generalization of the self-consistent moment method instead of Hartree-Fock in such exploratory model calculations may improve the accuracy significantly, while keeping the computational burden low.

Figure 1

Dependence of the magnetic moment on the Hubbard parameter $U$ at temperature $T=23$ K. Two values of SOI strength are considered: $t^{\mathrm{R}}=0.01$ eV (blue circles) and $t^{\mathrm{R}}=0.02$ eV (red squares). The magnetic moment is given in units of the Bohr magneton (${\mu }_{\mathrm{B}}$).

Figure 2

Band structure for $U=2$ eV at $T=23$ K when $t^{\mathrm{R}}=0.02$ eV. For this value of $U$ the system is nonmagnetic. The Rashba splitting is larger for the upper Hubbard bands than for the lower ones.

Figure 3

Band structure for $U=4$ eV at $T=23$ K when $t^{\mathrm{R}}=0.02$ eV. For this value of $U$ the system is ferromagnetic.

Figure 4

Temperature dependence of the magnetic moment at $U=4$ eV for SOI strength $t^{\mathrm{R}}=0.01$ (circles) and 0.02 eV (squares). Stronger SOI suppresses the Curie temperature.

Figure 5

Temperature dependence of the intrinsic AHE conductivity at $U=4$ eV.

Figure 6

Dependence of the intrinsic AHE conductivity on Hubbard $U$ at $T=23$ K.