Rashba spin-orbit coupling in the square-lattice Hubbard model: A truncated-unity functional renormalization group study

The Rashba-Hubbard model on the square lattice is the paradigmatic case for studying the effect of spin-orbit coupling, which breaks spin and inversion symmetry, in a correlated electron system. We employ a truncated-unity variant of the functional renormalization group which allows us to analyze magnetic and superconducting instabilities on equal footing. We derive phase diagrams depending on the strengths of Rasbha spin-orbit coupling, real second-neighbor hopping, and electron filling. We find commensurate and incommensurate magnetic phases which compete with $d$-wave superconductivity. Due to the breaking of inversion symmetry, singlet and triplet components mix; we quantify the mixing of $d$-wave singlet pairing with $f$-wave triplet pairing.

Figure 1

Diagrammatic representation of the FRG flow equation. The two-particle nodes represent the effective interaction at scale ${\mathrm{\Gamma }}_{\mathrm{\Lambda }}^{\left(4\right)}$ while the connecting lines are a propagator $G^{\mathrm{\Lambda }}$ and its derivative $\frac{d}{d\mathrm{\Lambda }}{G}^{\mathrm{\Lambda }}$ denoted by the line intersecting the loops. We indicate the three different two-particle irreducible diagram classes (channels): particle-particle $P$, crossed particle-hole $C$, and direct particle-hole $D$. Each is associated with one distinct transfer momentum.

Figure 2

Schematic phase diagram of the square lattice Rashba-Hubbard model as a function of filling factor $\nu$ and Rashba-SOC $\alpha$ for three different nearest-neighbor hopping strengths: $t^{\prime }=0$ (a), $t^{\prime }=-0.15$ (b), and $t^{\prime }=-0.3$ (c). The results were obtained for the weak-to-intermediate coupling regime $U=3$. Superconducting (SC) regions are colored in blue, spin density wave (SDW) instabilities in grey, incommensurate SDWs (iSDW) in red, and Fermi liquid (FL) behavior in white. The range of the filling is chosen to exclude wide regions of FL behavior.

Figure 3

Phase diagram of the Rashba-Hubbard model as a function of filling $\nu$ and SOC strength $\alpha$ at $t^{\prime }=0$ (a) and $t^{\prime }=-0.15$ (b). The critical scale ${\mathrm{\Lambda }}_{\mathrm{c}}$ roughly corresponds to a transition temperature and is encoded as transparency of the data points. Superconducting phases are encoded in blue, density wave instabilities are classified according to the distance of the leading transfer momentum $\mathit{q}$ to the $M=(\pi ,\pi )$ point of the Brillouin zone: commensurate instabilities ($\parallel M-\mathit{q}\parallel =0$) obtain a gray color and incommensurate ones ($\parallel M-\mathit{q}\parallel >0$) are increasingly red with greater distance to the $M$-point. The position of the van Hove singularities is marked as thin dotted line. Additionally, we annotate the specific points 1–6 for which the Fermi surface is visualized in Fig. 4.

Figure 4

Nesting analysis of SDW instabilities for $t^{\prime }=0$ (upper row, panels 1–3) and $t^{\prime }=-0.15$ (lower row, panels 4–6). The panels correspond to the points in $\nu ,\alpha ,t^{\prime }$ space marked in Fig. 3. The upper row shows the evolution of nesting for increasing $\alpha$ in the case $t^{\prime }=0$ to explain the return of the commensurate SDW. Blue arrows correspond to $M$-nesting and orange arrows to a $\mathit{q}\ne M$-nesting. In the lower row, we visualize the Fermi-surface and nesting vectors for three selected points in the $t^{\prime }=-0.15$ phase diagram.

Figure 5

Evolution of possible magnetization vectors in the $\mathit{q}=M$ phase at $t=-0.15$ [along the line towards point 5 in Fig. 3]. We show the magnetization direction as an evolution of increasing Rashba coupling $\alpha$. From the initial threefold degeneracy ($n=3$) at $\alpha =0$ we observe an easy-plane SDW (in the $xy$ plane, $n=2$) which turns into an easy-axis SDW (along the $z$ axis, $n=1$) for higher $\alpha$.

Figure 6

Relative weight of the singlet contribution to the superconducting order parameter for $t^{\prime }=0$ (a), $t^{\prime }=-0.15$ (b), and $t^{\prime }=-0.3$ (c). We observe a purely singlet order parameter at $\alpha =0$, with an increase in mixing as we increase the Rashba coupling for all values of $t^{\prime }$ under consideration. Note that the (i)SDW phases are grayed out for visual clarity. We do not observe a strong dependence of the singlet-triplet mixing on the filling $\nu$ for any value of $\alpha$.

Figure 7

Comparison to Ref. [37] for $\alpha =0$. Note that we have globally rescaled the critical scales of our data. This global offset is easily explained by the differences in implementations, the reference used a sharp cutoff as well as constant self-energies, which are absent here. It is nevertheless apparent that the results have the same features, with slightly shifted transition fillings between phases. cAFM (iAFM) corresponds to our SDW (iSDW) phase and dSC stands for $d$-wave superconductivity.

Figure 8

We show linecuts through the plots of Figs. 3 and 6 with an increased resolution in the filling $\nu$. This serves to both confirm the main features of the above calculation as well as establish the width of the nesting lines.