Itinerant Magnetism in Twisted Bilayer WSe$_2$ and MoTe$_2$

TL;DR

The work demonstrates that itinerant ferromagnetism in twisted bilayer WSe$_2$ and related TMDs arises from a Stoner-like instability within interaction-renormalized moiré bands, examined through a self-consistent Hartree–Fock framework. By constructing a continuum moiré theory in the symmetry-unbroken parent state, incorporating long-range Coulomb exchange, and folding the moiré Brillouin zone to accommodate various orders, the authors reproduce the Lifshitz boundary between layer-hybridized and layer-polarized regimes and map rich magnetic phase diagrams as functions of hole filling $\nu$, interlayer bias $V_z$, twist angle, and interaction strength. The results reveal distinct ground states across regimes: a layer-hybridized region hosting a valley-polarized ferromagnet (FM$_z$) at below half filling, a stripe AFM in the layer-polarized sector, a gapped $120^{\circ}$ AFM near half filling, and generalized Wigner crystals at fractional fillings; in MoTe$_2$ these tendencies are even more pronounced. Overall, long-range exchange in a renormalized, symmetry-unbroken parent band structure provides a broadly applicable framework for itinerant magnetism in moiré TMDs, with implications for tuning magnetic and topological properties via filling and electric fields.

Abstract

Using a self-consistent Hartree-Fock theory, we show that the recently observed ferromagnetism in twisted bilayer WSe$_2$ [Nat. Commun. 16, 1959 (2025)] can be understood as a Stoner-like instability of interaction-renormalized moiré bands. We quantitatively reproduce the observed Lifshitz transition as function of hole filling and applied electric field that marks the boundary between layer-hybridized and layer-polarized regimes. The former supports a ferromagnetic valley-polarized ground state below half-filling, developing a topological charge gap at half-filling for smaller twist angles. At larger twist angles, the system hosts a gapped triangular Néel antiferromagnet. On the other hand, the layer-polarized regime supports a stripe antiferromagnet below half-filling and a wing-shaped multiferroic ground state above half-filling. We map the evolution of these states as a function of filling factor, electric field, twist angle, and interaction strength. Our results demonstrate that long-range exchange in a symmetry-unbroken parent state with strongly renormalized moiré bands provides a broadly applicable framework to understand itinerant magnetism in moiré TMDs.

Figures (15)

• Figure 1: Interaction-renormalized moiré bands for the symmetry unbroken ("parent") Hartree-Fock state. Compared to the noninteracting theory (a), interactions shift the Van Hove singularity and soften the Lifshitz transition (b). This coincides with the change from layer-hybridized to layer-polarized regimes shown by the dotted line in (c). The red dashed line is an analytical result obtained in the absence of moiré potentials. (d) By matching the Lifshitz transition to experiment knuppel_correlated_2025 we can reproduce its shape with $\theta = 3.65^\circ$ and $\epsilon \approx 30$. Panels (e) and (f) illustrate how the linear boundary in (a) follows from the constant density of states of a 2DEG while the quadratic shape arises from the interplay between Fock interactions and moiré tunneling.
• Figure 2: Ground state magnetic order and layer polarization for $\theta = 3.65^\circ$ and $\epsilon = 25$. (a) Out-of-plane spin polarization corresponding to the FM$_z$ phase, which exists both near charge neutrality and in regions where Fock interactions enhance the DOS [Fig. \ref{['fig:fig1']}(b)]. The experimentally observed ferromagnetic region at small $V_z$ below half-filling ($\nu = 1$) is clearly visible. (b) In-plane spin polarization of the IVC stripe antiferromagnet, which exists only in a small $\nu$ window in layer-polarized regions. (c) Order parameter of the $120^\circ$ AFM: $\sum_{\bm k} f_{\bm k} \langle c_{\bm k,\uparrow}^\dag c_{\bm k+\bm q,\downarrow} \rangle$. (d) Order parameter of the generalized Wigner crystal state with $\sqrt{3}\times\sqrt{3}$, $2\times2$, and $2\times3$ reconstructions of the moiré unit cell.
• Figure 3: Orbital magnetization per moiré unit cell. Results are shown as a function of filling $\nu$ and interlayer potential $V_z$ for the FM$_z$ phase for tWSe$_2$ at $\theta = 3.65^\circ$ (left) and tMoTe$_2$ at $\theta = 3.5^\circ$ (right). Twisted WSe$_2$ peaks at about $0.4~\mu_B$ near half filling, while twisted MoTe$_2$ shows a stronger response of about $1.2~\mu_B$ that is further enhanced by finite $V_z$, reflecting the larger Berry curvature and narrower bands.
• Figure 4: Phase diagram, Fermi surface topology and ground state at half filling versus twist angle and interaction strength. (a) Phase diagram for $\theta=3.65^\circ$ and $\epsilon = 25$, shown together with a representative Fermi surface of the gapless phases. Red and blue indicate the Fermi surfaces of the two valleys. The $\nu=1$ phase diagram versus (b) interaction strength ($1/\epsilon$) and (c) twist angle, where the color gives the charge gap $\Delta$. The gapped IVC $120^\circ$ AFM persist even for weak interactions in regions where the Fermi energy is close to the VHS on the unbroken parent state (M).
• Figure S1: Illustration of the energy landscape of parallel-stacked bilayers of 2H TMDs as a function of the stacking configuration $\bm \phi$. The orange triangles gives all configurations not related by symmetry. The dots correspond to MM (red), MX/XM (blue), and DW (green) stacking.