Reentrant superconductivity and Stoner boundaries in twisted WSe$_2$

TL;DR

This work addresses the microscopic origin of reentrant superconductivity observed in twisted bilayer WSe$_2$ at $\theta\sim5^\circ$ by linking spin–valley instabilities to pairing via a faithful three-orbital Wannier model and matrix-RPA analysis. The authors compute spin–valley susceptibilities and Stoner boundaries across the displacement-field–density phase diagram, identifying a Lifshitz transition near the van Hove singularity that strongly enhances spin–valley fluctuations. They find dominant interorbital spin–valley flip instabilities involving the $MM$, $XM$, and $MX$ orbitals, with the ordering vector $\boldsymbol{Q}$ switching from $\boldsymbol{Q}\approx\boldsymbol{0}$ on one side of the VHS to finite $\boldsymbol{Q}$ near $M$-points on the other side, reflecting a density-dependent spatial pattern of order. Crucially, fluctuations on the edges of the Stoner boundary favor superconductivity in nearby paramagnetic regions, producing a reentrant SC dome consistent with experimental observations and suggesting spin–valley fluctuations near the VHS as the mechanism behind reentrant superconductivity in twisted WSe$_2$.

Abstract

We investigate spin-valley instabilities and their connection to the reentrant superconducting states recently observed in the twisted bilayer dichalcogenide WSe$_2$ at a $5^o$ twist angle. Starting from an effective three-orbital faithful Wannier model for the spin-locked moiré bands, combined with orbital-dependent Hubbard interactions, we analyze the evolution of magnetic instabilities as a function of carrier density using the matrix random phase approximation (mRPA) approach. By computing the Stoner boundary lines from the spin-valley susceptibilities over the electric-field by hole filling phase diagram, we show that the spin-valley instabilities result in ordered states in the region close to the Lifshitz transition at the topmost moiré valence band, marked by crossing of the van Hove singularity in the density of states. These spin-valley ordered states are dominated by interorbital spin-valley-flips involving the $MM$ and $MX$ moiré orbitals and occur at different momenta in each side of the van Hove line, indicating a distinct spatial dependence of the spin-valley order parameter depending on the hole filling. Moreover, the corresponding Stoner boundaries exhibit strong fluctuations on its flanks, which can favor superconducting states in the regions close to the spin-valley-ordered ones. This mechanism provides a natural description for a reentrant superconducting dome consistent with the experimental results. As such, our results suggest spin-valley fluctuations near the van-Hove line as the microscopic origin of the reentrant superconductivity in twisted WSe$_2$.

Figures (4)

• Figure 1: Schematic of the non-interacting model we used, obtained by Wannierizing the tTMDh continuum model CrepelBridgingSmallLargeTwisted2024. Panel (a) shows a depiction of the real-space moiré superlattice for $\theta=5^o$ centered at an $MM$ sublattice site. This model considers two lattices, a triangular lattice of $MM$ sites (yellow lines), and a hexagonal lattice of $MX$ and $XM$ sites (green lines). Intralattice and interlattice hoppings are taken into account for distances up to $9$ unit cells. In panel (b), we show a representative band structure of the three-orbital model for electron filling $\nu=-1$ and electric field $E_z=20$ meV.
• Figure 2: Panel (a) shows a color map of the density of states in an electric field $E_z$ by electron filling $\nu$ diagram. A black vertical dashed line marks the $\nu=-1$ filling, which is relevant for the experimental phase diagram. The dark green region of the phase diagram marks the van Hove singularity (VHS). The yellow circle, green square and purple diamond symbols marks points of the phase diagram at constant electric field $E_z=20$ meV for which we show Fermi surfaces in panel (b). From top to bottom, panel (b) shows a Lifshitz transition and the crossing of the VHS by the Fermi energy, marked by a steep increase in the DOS. Orbital contributions to the Fermi surfaces are shown in colors, i.e. red for $MM$, blue for $XM$ and green for $MX$. Spin-valleys $(\uparrow+)$ and $(\downarrow-)$ are represented by thicker and thinner lines, respectively.
• Figure 3: Stoner phase diagram. Panel (a) shows the critical Stoner parameter $\alpha_c$ in the electric field $E_z$ by electron filling $\nu$ diagram at Hubbard interaction $U_{XM}=U_{MX}\approx41.3$ meV and $U_{MM}/U_{XM}\approx0.90$. The green color interpolates with black for $0.95\leq\alpha_c\leq1$ and the black colors characterizes $\alpha_c\geq1$. $\alpha_c\geq1$ means the Stoner criterion, marking the ordered state, has been achieved. Panels (b) and (c) shows constant electric field cuts for $E_z=23.22$ meV (green) and $E_z=18.71$ meV (purple) of the phase diagram. The constant $\alpha_c=1$ red line marks the Stoner boundary.
• Figure 4: Main eigenvalue of the spin-valley susceptibilities along the Brillouin zone high-symmetry directions. Line colors correspond to the same points in the phase diagram as the symbols in Fig. \ref{['fig:dos']}(a). For each curve, we set the main Hubbard interaction, $U_{MX}$, to be $0.99U_c$, where $U_c$ is the critical Hubbard interaction to trigger the Stoner criterion. The inset shows the homogeneous spin-valley susceptibility matrix elements for $[{\chi}(\boldsymbol{q})]^{p\xi,p\xi}_{p\xi,p\xi}$ at the van Hove singularity filling $\nu=-1.05$.