Spin-orbital magnetism in moiré Wigner molecules

Abstract

The interplay of spin and orbital degrees of freedom offers a versatile playground for the realization of a variety of correlated phases of matter. However, the types of spin-orbital interactions are often limited and challenging to tune. Here, we propose and analyze a new platform for spin-orbital interactions based upon a lattice of Wigner molecules in moiré transition metal dichalcogenides (TMDs). Leveraging the spin-orbital degeneracy of the low-energy Hilbert space of each Wigner molecule, we demonstrate that TMD materials can host a general spin-orbital Hamiltonian that is tunable via the moiré superlattice spacing and dielectric environments. We study the phase diagram for this model, revealing a rich landscape of phases driven by spin-orbital interactions, ranging from ferri-electric valence bond solids to a helical spin liquid. Our work establishes moiré Wigner molecules in TMD materials as a prominent platform for correlated spin-orbital phenomena.

Figures (13)

• Figure 1: (a) Charge density in a lattice of Wigner molecules. Each molecule (dotted circle) hosts three electrons localized at the vertices of an equilateral triangle, collectively forming a breathing kagome lattice. Inset: schematic energy spectrum a single molecule showing that the ground state manifold is composed of four degenerate levels: $(\eta^z,S^z) = (\pm 1, \pm 1/2)$. (b) Schematic of the local spin and orbital degrees of freedom interacting on a triangular lattice.
• Figure 2: (a) The point group symmetries of $\mathcal{H}$ on the triangular lattice, with definitions of the lattice vectors $\bm{a}_i$, and bond-dependent phase factors $\nu_{ij}$. (b) Schematic of direct Coulomb repulsion $J_{\rm direct}$ between two Wigner molecules, and their exchange interaction $J_{\rm ex}$ from wave-function overlap. (c) Super-exchange $J_{\rm super-ex}$ arising from virtual tunneling of electrons to excited states with $\nu \neq 3$. Also shown are the charge gap $\Delta E(\lambda)$ (upper panel) and the drastic modification of Wigner molecular shapes at $\nu =2,4$ (lower panel).
• Figure 3: (a) Phase diagram obtained by minimizing the energy over classical (pseudo)spin configurations at $(\phi, d_g) = (45^\circ, 20\,\mathrm{ nm})$. The classical phases are shown on the left; the Stripe and FM phases are also present in the quantum phase diagram. Purple arrows denote in-plane $\ev{\bm \eta_\parallel}$, purple circles ($\circ$) denote positive $\ev{\eta^z}$, and black dots ($\odot$) and crosses ($\otimes$) denote positive and negative $\ev{S^z}$, respectively. Our results are inconclusive in the cross-hatched region, where the minimum-energy states have large unit cells, suggesting high degeneracy or incommensurate order. (b) Corresponding quantum phase diagram from DMRG, showing the emergence of new phases induced by quantum fluctuations. The in-plane orbital polarization $\langle \bm{\eta}_\parallel \rangle$ is shown by purple arrows, and spin-spin correlations on nearest neighbor bonds are indicated by color (blue represents antiferromagnetic correlations). Note that $\bm\eta$ does not transform like a vector under the reflection $M_y$, so the physical electrical polarization is given by $\hat{z} \times \langle \bm{\eta}_\| \rangle$ (see Table \ref{['tab:sym']} and Fig. \ref{['fig5']}).
• Figure 4: Characterization of the HSL state found at $(a_M,\varepsilon_r) = (8.0 ~\mathrm{nm},7.0)$. (a) The exponential decay of the correlation functions $\ev{S^+_0 S^-_0}$, $\ev{\eta^z_0 \eta^z_r}$, and $\ev{S^z_0 S^z_r}$ indicates the lack of long-range order of either $\bm{S}$ or $\eta^z$. By contrast, $\langle C^z_0 C^z_r \rangle$ quickly approaches a constant plateau, signifying the ordering of the spin-currents. (b) The spin- and momentum-resolved entanglement spectrum exhibits two low-lying modes of opposite momentum and spin, indicating spin-momentum locking. (c) Spin current pattern $C_{ij}$, which is of equal magnitude and sign on each bond. Since time-reversal is respected, along each bond, the spins of opposite orientation exhibit opposite currents of equal magnitude (blue and red arrows). (d) The expectation value of vector orbital polarization $\bm{P}_{ij}$ (green arrows) is non-zero and $\mathcal{C}_{3v}$ symmetric.
• Figure 5: (a) Charge density distribution of the correlated phases that have local electrical polarization, i.e., $\hat{z} \times \langle \bm{\eta}_\parallel \rangle_i \neq 0$, subtracted from a state with no local electrical polarization ($\langle \bm{\eta}_\parallel \rangle_i = 0$) and normalized by the maximum density of this unpolarized state. Note that the electrical polarization is always rotated by 90$^\circ$ relative to the in-plane $\langle \bm{\eta}_\parallel \rangle_i$ indicated by purple arrows. Each phase has a distinct polarization pattern that serves as its unique fingerprint in STM.