Stripe antiferromagnetism and chiral superconductivity in tWSe$_2$

TL;DR

This work develops a DFT-informed moiré continuum model for twisted WSe$_2$ that includes c-axis relaxation, yielding a minimal two-band description and enabling path-integral integration of higher-energy orbitals. Hartree-Fock analysis at hole filling $\nu=1$ and zero displacement field identifies competing states, notably stripe SDW and layer AFM alongside a QAHI, with the stripe SDW favored at moderate dielectric screening and certain twist angles. Building an effective $t$–$J$–$U$ framework from antiferromagnetic interactions, the authors show that second-neighbor superexchange $J_2$ drives intra-layer, next-nearest-neighbor pairing, giving a chiral superconducting state that breaks time-reversal symmetry and contains a mixture of singlet and triplet components (dominant singlet). The results provide a correlated, mechanism-based explanation for superconductivity in twisted WSe$_2$ linked to a proximal stripe SDW insulating state and offer a broadly applicable methodology for moiré TMDs.

Abstract

The layer-dependent Hamiltonians of parallel-stacked MoTe$_2$ and WSe$_2$ homobilayer moiré materials are topologically non-trivial, both in real space and in momentum space, and have been shown to support integer and fractional quantum anomalous Hall states, as well as antiferromagnetic and superconducting states. Here, we address the interplay between the antiferromagnetic and superconducting states observed in tWSe$_2$ when the Fermi level is close to its $M$-point van Hove singularity and the displacement field is small. We combine DFT with path-integrals to construct a minimal moiré band model that accounts for lattice relaxation along the $c$-axis and perform Hartree-Fock calculations to identify competing charge and spin ordered states. For tWSe$_2$ at $θ=2.7^\circ$ and $θ=3.65^\circ$, we find that a layer antiferromagnet (AFM), a stripe spin-density-wave (SDW), and the ferromagnetic Chern insulator (FM) are the primary candidates for the ground state at zero displacement field, and argue that antiferromagnetic spin interactions on the next neighbor bond $J_2$ can induce a time-reversal symmetry breaking chiral superconducting state.

Figures (5)

• Figure 1: (a) Schematic illustration of the displacement $\vec{d}$ and the metal atom separation $Z$ along the $c$-axis. (b) Variation of $Z$ across the $\vec{d}$-grid after $c$-axis relaxation for homobilayer MoTe$_2$. ($\vec{d}$ is defined to be zero at AA stacking.) $\Delta Z(\vec{d})\equiv Z(\vec{d}) - Z_m$, where $Z_m=7.279$Å is the spatial average $Z(\vec{d})$. The same plot for WSe$_2$ for which $Z_m=6.726$Å. The layer separations are minimized at the equilibrium metal on chalcogen (MX) and chalcogen on metal (XM) stacking points.
• Figure 2: (a) The three lowest energy bands of tMoTe$_2$ and tWSe$_2$ can be described by three band tight-binding models with the Wannier functions defined in Ref. [PhysRevX.13.041026], which are localized on sublattices A (red), B (blue), and C (black). The A and B Wannier wavefunctions are polarized to the top and bottom layers, respectively, while the sublattice C (black) Wannier wavefunction is an equal weight linear combination of top and bottom layer components. The A and B sublattices are more strongly weighted in the first two bands, and a two-band model is often sufficient. (b) Pictorial sketches of layer AFM, stripe SDW, and ferromagnetic (QAHI) states, described in terms of two-band models with A sublattice states in one layer and B sublattice states in the other layer. (c) Mean-field energies of the Layer AFM and stripe SDW states relative to the QAHI state ($\Delta E \equiv \langle \hat{H}\rangle-\langle\hat{H}\rangle_{QAHI}$) as a function of dielectric constant $\epsilon$ at twist angles $\theta=2.7^\circ$ and $\theta=3.65^\circ$. (d) Mean-field energy of the stripe SDW state relative to the QAHI state for tWSe$_2$ at $\theta=2.7^\circ$ with $\epsilon=10,14$ as a function of displacement field $E$.
• Figure 3: (a) Band structure of WSe$_2$ at $\theta=3.65^\circ$ based on continuum and approximate tight binding models. The fitted hopping parameters are $(t_1,t_2,t_3,t_4,t_5,t_6) = (5.46,\,4.93,\,2.18,\,-0.56,\,-0.74,\,0.26)\,\text{meV}$ and the phase magnitude for $t_2$ is $\phi=0.65\pi$. (b) The Fermi surface has a high-density of states where it is closest to $M$-points (centers of the Brillouin-zone edges). These favor SDW states at ordering wavevectors near $\vec{Q}_1=\vec{b}_1/2$, $\vec{Q}_2=\vec{b}_2/2$ and $\vec{Q}_3=\vec{Q}_1+\vec{Q}_2$. (c) The arrow directions represent the positive phase winding of the complex $t_2$ hopping amplitude for $\uparrow$ spins. For $\downarrow$ spins, the phase winds in the opposite direction.
• Figure 4: Schematic illustration of pairing amplitude $\Delta_{ij}$ phases along different bonds in the nearest-neighbor (NN) and the next-nearest-neighbor (NNN) chiral pairing states ($\omega = \exp(i2\pi/3)$).
• Figure 5: (a) Condensation energy per unit cell $= E_{S}-E_{N}$ for uniform and chiral representations is plotted against onsite repulsive interaction. (b) Magnitude of singlet and triplet order parameter of NNN pairing and their ratio vs.$J_2$. $\Delta^2_{s/t}=\langle a_{j\downarrow}a_{i\uparrow}\mp a_{j\uparrow}a_{i\downarrow}\rangle$. (c) The phase diagram for $U=35\,\mathrm{meV}$ characterized by the mean-field energy difference $\Delta E=E_{NN}-E_{NNN}$. The star corresponds to the values $J_1 = 3.4\,\mathrm{meV}$ and $J_2 = 2.78\,\mathrm{meV}$ estimated by $J_n=4t^2_n/U$. Grey area corresponds to the range of $(J_1,J_2)$ over which superconductivity was not found.