Comparing the effective enhancement of local and nonlocal spin-orbit couplings on honeycomb lattices due to strong electronic correlations

We investigate the interplay of electronic correlations and spin-orbit coupling (SOC) for a one-band and two-band honeycomb lattice models. The main difference between the two models concerning SOC is that in the one-band case the SOC is a purely nonlocal term in the basis of the $p_z$ orbitals, whereas in the two-band case with $p_x$ and $p_y$ as basis functions, it is purely local. In order to grasp the correlation effects on nonlocal spin-orbit coupling, we apply the TRILEX approach that allows to calculate nonlocal contributions to the self-energy approximately. For the two-band case, we apply dynamical mean-field theory. In agreement with previous studies, we find that for all parameter values in our study, the effect of correlations on the spin-orbit coupling strength is that the bare effective SOC parameter is increased. However, this increase is much weaker in the nonlocal than in the local SOC case. Concerning the TRILEX method, we introduce the necessary formulas for calculations with broken SU(2) symmetry.

Figure 1

Illustration of the sign ${\nu }_{ij}$ for the Kane-Mele model. Every right turn gives ${\nu }_{ij}$ the value $+1$ and every left turn the value $-1$.

Figure 2

Sketch of the lattice structure of the two honeycomb lattices. (Top) Single-band honeycomb lattice for the KMH model with nonlocal SOC. (Bottom) Two-orbital lattice, relevant for bismuthene with a local SOC term. The bonds with strongest hopping term in the models are indicated by $t$, and the SOC is also indicated by arrows.

Figure 3

(Top) Dispersion relation for the noninteracting Kane-Mele Model with ${\lambda }_{\mathrm{SO}}^{1B}=0.14t$. (Bottom) Dispersion relation for the noninteracting two-band model (bismuthene) with ${\lambda }_{\mathrm{SO}}^{2B}=0.21t$. For both band structures, we set the nearest-neighbor orbital-diagonal hopping to $t=1$, in order to set a common unit of energy.

Figure 4

Lattice self-energy $\mathrm{\Sigma }(\mathbf{k},i\omega )$ (top) and polarization $P(\mathbf{q},i\mathrm{\Omega })$ (bottom) within TRILEX: the electron-boson coupling vertex $\mathrm{\Lambda }$ is approximated by a local quantity in the Hedin equations, and $\alpha$ and $\beta$ denote sublattice indices.

Figure 5

Illustration of $\text{Im}{\mathrm{\Sigma }}^{\mathrm{SO}}(\mathbf{r}-{\mathbf{r}}^{\prime },i{\omega }_0)$. Predominantly next-nearest neighbors are contributing and right and left turns are associated with opposite signs (compare Fig. 1). $U=7.0t$ and $\alpha =0.33$

Figure 6

Frequency dependence of SOC-enhancement of the two-band model (top) and the one-band model (bottom) for different values of $U$ (and $J$).

Figure 7

Evolution of the ${\lambda }_{\text{eff}}^{1B}/t$ and the renormalization factor $Z$ with increasing interaction $U/t$. The dashed line denotes a regime where it becomes increasingly hard to stabilize a paramagnetic solution.

Figure 8

Evolution of the renormalization factor $Z$ and ${\lambda }_{\text{eff}}^{2B}/{\lambda }_{\text{eff}}^{2B,\infty }/{\lambda }_{\text{eff}}^{2B,\text{HF}}$ with increasing interaction $U$. ${\lambda }_{\text{eff}}^{2B,\text{HF}}$ is the self-consistent Hartree-Fock solution calculated from Eqs. (D1) and (D2). ${\lambda }_{\text{eff}}^{2B,\infty }$ denotes the high-frequency contribution to the effective SOC and is given by Eq. (D1) when the densities are taken from DMFT Eq. (D3).

Figure 9

Compare the increase of ${\lambda }_{\text{eff}}$ with respect to a decreasing renormalization factor $Z$ for both models.

Figure 10

General SU(2)-broken Hedin-Equation of the self-energy (top) and polarization (bottom).

Figure 11

Comparing $P^{ch,ch}$ to $P^{ch,z}$ at the $\mathbf{K}$ point for $U=7t$ and $\alpha =0.33$. All other contributions, such as $P_{AB}, \mathrm{Re}{P}^{z,ch}$, etc. vanish.

Figure 12

Frequency dependence of the SOC-enhancement for the two-band model for $U=6.7t$ and $J=1.3t$ on the imaginary axis. The high-frequency contribution is calculated from Eq. (D3).

Figure 13

Second-order expansion for single-site model with on-site interaction only. $G_0^{\sigma }$ is the noninteracting Green's function and $U$ the bare interaction.

Figure 14

Effective spin-orbit coupling ${\lambda }_{\mathrm{eff}}^{1B}$ for different parameters $U$ and $\alpha$. Comparing also TRILEX ${\mathrm{\Lambda }}^2$ (denoted as ${\mathrm{\Lambda }}^2$) and TRILEX without channel off-diagonal polarization (denoted as $\mathrm{\Lambda }$ (${\delta }_{\eta {\eta }^{\prime }}$)).

Figure 15

Frequency dependence of SOC enhancement of the one-band model (bottom) for $U=7.0t$ (like Fig. 6) for different $\alpha$.

Figure 16

Im $\frac{1}{2}({\mathrm{\Sigma }}_{+}^{\uparrow }+{\mathrm{\Sigma }}_{-}^{\uparrow })$ of the two-band model at $U=12t$ and $J=2t$, with and without SOC (top). ${\mathrm{\Sigma }}_{\text{imp}}$ of the one-band model at $U=4.8t$, for different $\alpha$ values with and without SOC (bottom).

Figure 17

$\text{Im}{\mathrm{\Sigma }}^d(\mathbf{k},i{\omega }_0)$ at $U=4.8t$ and $\alpha =0.33$ for ${\lambda }_{\mathrm{SO}}^{1B}=0$ (top) and ${\lambda }_{\mathrm{SO}}^{1B}=0.14t$ (bottom).

Figure 18

Difference of the two most distant points in $\text{Im}{\mathrm{\Sigma }}^d(\mathbf{k},i{\omega }_0) \mathrm{\Gamma }$ and $K$ as a measure of the $\mathbf{k}$ dependence of ${\mathrm{\Sigma }}^d$ for ${\lambda }_{\mathrm{SO}}^{1B}$ and different parameters of $U$ and $\alpha$. We again compare TRILEX ${\mathrm{\Lambda }}^2$ to TRILEX as in Fig. 14.