Theory of intervalley-coherent AFM order and topological superconductivity in tWSe$_2$

Abstract

The recent observation of superconductivity in the vicinity of Fermi surface reconstructed insulating or metallic states has established twisted bilayers of WSe\textsubscript{2} as an exciting platform to study the interplay of strong electron-electron interactions, broken symmetries and topology. In this work, we use a first-principles, material-specific theoretical treatment that is unbiased with respect to electronic instabilities to study the emergence of electronic ordering in twisted WSe\textsubscript{2} driven by gate-screened Coulomb interactions. We construct exponentially localized moiré Wannier orbitals that faithfully capture the bandstructure and topology of the system, project the gate-screened Coulomb interaction onto them and use unbiased functional renormalization group techniques to resolve the momentum and orbital structure of the leading instabilities and the relevant energy scales. We find an interplay between intervalley-coherent antiferromagnetic (IVC-AFM) order and chiral, mixed-parity $d/p$-wave superconductivity for carrier concentrations near a displacement field and twist-angle-tunable van-Hove singularity. Our microscopic approach establishes incommensurate IVC-AFM spin fluctuations as the dominant electronic mechanism driving the formation of superconductivity in $θ= 5.08^{\circ}$ twisted WSe\textsubscript{2} and explains key aspects of recent experiments including the asymmetric density dependence of the spin ordering with respect to the van-Hove line, the single and double-peak structure of the DOS in the ordered (hole-doped) IVC-AFM phase, the emergence of superconductivity as the density is varied across the van-Hove line and the evolution of the displacement field-density phase diagram with twist angles between $3.7^{\circ} \dots 5^{\circ}$.

Figures (9)

• Figure 1: Fermiology and Wannierization of $\theta = 5.08^{\circ}$ tWSe2 for an external displacement field $E_z=20\,\mathrm{meV}$. (a) Continuum bandstructure in valley $\nu=+1$ (black) and spectral overlap of its Bloch states with a three-orbital model comprising Wannier orbitals centered at the MM stacking regions ($T$, $s$-orbital) and XM/MX stacking regions ($H_1$, $p_+$-orbital)/($H_2$, $p_+$-orbital) of the moiré superlattice (tri-colored map and inset). The bandstructure of the effective Wannier model truncated after 4.5 moiré lattice vectors is indicated by the green dashed line. (b) Amplitude $|\psi_l\boldsymbol{(}r)|^2$ of the Wannier functions ($T, H_1, H_2$) on the two layers ($l=0,1$) that capture the spectral weight of the continuum model. (c) Density of states as function of holes per moiré unit cell $n/n_0$ and displacement field $E_z$. The upper panel shows snapshots of the Fermi surface along the van-Hove line and its orbital polarization using the same colormap as in (a) for valley $\nu=+1$. The Fermi surface of the opposite valley $\nu=-1$ is shown in gray. The green dashed line indicates the mini-Brillouin zone of tWSe2. The blue circle indicates the position of the higher-order VHS, where three van-Hove points merge at $K^{\nu}$ as demonstrated in the insets. The three van-Hove points are moved towards $\Gamma$ ($M$) for increasing (decreasing) value of the external displacement field along the van-Hove line as indicated by the blue triangle (square).
• Figure 2: FRG phase diagram of $\theta = 5.08^{\circ}$ tWSe2 resolving the interplay of IVC-AFM order and chiral $d/p$-wave superconductivity along the gate-tunable van-Hove singularity. (a) Main panel: FRG phase diagram as function of holes per moiré unit cell $n/n_0$ and external displacement field $E_z$. The phase diagram shows the critical scale $\Lambda_{\mathrm c}$ of the leading Fermi surface instability and indicates regimes of spin/charge density wave order (DW, red) and superconductivity (SC, blue). The density of states (DOS) is shown in the background (gray) to pinpoint the position of the van-Hove line. Close to the HOVHS (blue circle), the system features IVC-AFM order, while for lower values of the displacement field chiral $d/p$-wave superconductivity emerges on the van-Hove line. Inset: Zoom-in on the density-displacement field region where IVC-AFM order gives way to superconductivity. The colored arrows indicate different directions when moving along the van-Hove line (green arrows) or at constant displacement field cuts (grey, blue arrows). The critical scale $\Lambda_{\mathrm{c}}$ along these line cuts is shown in the panels on the left, indicating a maximal critical temperature of $\Lambda_{\mathrm{c}} \sim 500\,\mathrm{mK}$ that is reached in the immediate vicinity of the IVC-AFM phase. (b) Momentum structure and (in)commensurability of the IVC-AFM state along the van-Hove line. The subpanels show the absolute value of the inter-valley particle-hole susceptibility $\chi^{\text{FRG}}_{\uparrow \downarrow \uparrow \downarrow}(\boldsymbol{q})$ at scale $\Lambda_{\mathrm{c}}$ for the relevant orbital components involving the Wannier orbitals $T/H_2$. The leading momentum transfer $\boldsymbol{Q}_C$ locks to $K'$ at the HOVHS (blue circle) and successively shifts towards incommensurate momenta along the high symmetry path $K'$-$M$ when lowering the external displacement field along the van-Hove line (blue square). (c) Amplitude and phase dependence of the pairing instability at $E_z=25\,\mathrm{meV}$ (blue circle) decomposed into singlet $\psi_0(\boldsymbol{k})$ and triplet $d_z(\boldsymbol{k})$ components. The continuous colorbar indicates the relative phase, whereas the amplitude is encoded by the opacity. The SC order parameter is two-fold degenerate and likely minimizes its free energy by the chiral superposition $d+id/p+ip$.
• Figure 3: Asymmetry of particle-hole instabilities due to Fermi surface broadening induced nesting. (a) Critical (relative) interaction scale $\alpha_c$ of the RPA renormalized interaction for different scales $\Lambda$ along the van-Hove line. The leading momentum transfer $\boldsymbol{Q}_C$ along the high symmetry path $K^{\nu}$-$M$ is encoded by the colorbar shared among panels (a,b). The minimal value of $\alpha_c$ for each curve is indicated by the gray dashed line showing a clear asymmetry of particle-hole instabilities with respect to the position of the HOVHS (blue circle). The lower panel shows the ($\Lambda$-broadened) Fermi surface sheets of the spin-valley polarized sub-sectors, i.e., all eigenenergies that are within the energy shell $[E_F - \Lambda, E_F + \Lambda]$ around the Fermi energy $E_F$. (b) Same information as in panel (a), taken for the constant displacement field cut $E_z = 29\,\mathrm{meV}$ as indicated by the grey arrows in \ref{['fig:frg']} (a). The grey solid line indicates the DOS as the density $n/n_0$ is varied across the van-Hove line. For low scales, the minimal value of $\alpha_c$ (grey dashed line) is encountered on the van-Hove line (green arrow) and the leading momentum transfer $\boldsymbol{Q}_c$ is displaced from $K^{\nu}$. At larger scales, $\alpha^{\text{min}}_c$ is shifted to larger hole fillings and $\boldsymbol{Q}_c$ locks back to $K^{\nu}$.
• Figure 4: Spectral properties of the IVC-AFM domains and VHS splitting in $\theta = 5.08^{\circ}$ tWSe2 obtained by FRG+MF. (a) Momentum-resolved spectral function $A(\boldsymbol{k}, \omega)$ and partial DOS (PDOS) in the IVC-AFM phase along the irreducible path of the original BZ at $E_z = 32$ meV and $n/n_0 \sim -1.27$ (blue circle). The translational symmetry breaking of the IVC-AFM phase causes an effective $\sqrt{3} \times \sqrt{3}$ moiré supercell (green hexagon in inset on the upper right). The IVC-AFM order parameter causes the emergence of (partial) mini-gaps around $K^{\nu}$, leading to an asymmetric splitting of the original van-Hove peak and a drastic reduction of the hole carrier density at the Fermi level (grey dashed line) that becomes visible in the PDOS. (b) DOS and magnitude of the IVC-AFM order parameter in the $(E_z, n)$ phase diagram obtained for FRG+MF simulations at each point in phase space. The presence of IVC-AFM order leads to the emergence of a second van-Hove line (VHS2) associated to the hole-doped IVC-AFM state. The lower panel shows the amplitude of the IVC-AFM order parameter and the nearby SC regions predicted by the FRG. (c) Spatially-resolved spin spectral density $\rho^S_{l=0,1}(\boldsymbol{r})$ in the lower (upper) layer of tWSe2.
• Figure 5: IVC-AFM spin fluctuation mechanism to unconventional superconductivity in tWSe2 by commensurate to incommensurate transition. (a) Absolute value of the RPA-renormalized susceptibility $\chi^{\text{RPA}}_{\uparrow \downarrow \uparrow \downarrow}(\boldsymbol{q})$ at the critical interaction strength $\alpha_c U$ involving the dominant Wannier orbitals $T/H_2$. The leading momentum transfer $\boldsymbol{Q}_C$ locks to $K'$ at the HOVHS (blue circle) and successively shifts towards incommensurate momenta along the high symmetry path $K'$-$M$ when lowering the external displacement field along the van-Hove line (blue square). (b) Fluctuations around the (incommensurate) IVC-AFM order give rise to chiral $d/p$-wave superconductivity comprising an equal mixture of spin-singlet $\psi_0^{\text{RPA}}(\boldsymbol{k})$ and spin-triplet $d_z^{\text{RPA}}(\boldsymbol{k})$ weight in the order parameter. For $E_z \sim 25\,\mathrm{meV}$ (blue square), the phase of the resulting order parameter is indicated by the circular colorbar and the amplitude (projected on the Fermi surface) is encoded by the opacity. The superconducting order parameter only has notable weight on the triangular pockets around $K^{\nu}$, whereas the superconducting amplitude vanishes on the larger Fermi arcs connecting neighboring $K^{\nu}$ points.