Ultra-Strong Spin-Orbit Coupling and Topological Moiré Engineering in Twisted ZrS₂ Bilayers

Twisted van der Waals heterostructures have recently emerged as an intriguing and highly tunable platform to realize unconventional electronic phases in two dimensions Balents et al. (2020); Kennes et al. (2021); Andrei and MacDonald (2020); Andrei et al. (2021) and more. Spurred by the discovery of Mott insulation and superconductivity in twisted bilayer graphene Cao et al. (2018a, b), remarkable progress in fabrication and twist-angle control has lead to observations of correlated insulating states or superconductivity in a variety of materials, including trilayer and double-bilayer graphene, homo- and hetero-bilayers of twisted transition metal dichalcogenides Lu et al. (2019a); Stepanov et al. (2020); Rozhkov et al. (2016); Shen et al. (2020); Kerelsky et al. (2021); Rubio-Verdú et al. (2020); Arora et al. (2020); Chen et al. (2020a); Liu et al. (2020a); Cao et al. (2020a); He et al. (2020); Wang et al. (2020), and heterostructures at a twist on hexagonal boron nitride substrates Chen et al. (2019a, b). At its heart, this rich phenomenology stems from electronic interference effects due to the moiré superlattice, which can selectively quench kinetic energy scales to realize almost dispersionless bands, permitting a twist angle controlled realization of regimes dominated by strong electronic interactions. At the same time, the drastic reduction of kinetic energy of the low-energy moiré bands implies straightforward gate-tunable access to a wide range of filling fractions, permitting wide-ranging experimental access to the phase diagrams of paradigmatic models of strongly-correlated electrons Kennes et al. (2021). Consequently, the putative realization of strongly-correlated electron physics in a tunable setting has garnered significant attention, resulting in growing experimental evidence for novel correlated phases, including unconventional superconductivity Andrei et al. (2021); Lu et al. (2019b); Cao et al. (2020b).

Notably, and despite negligible intrinsic spin-orbit coupling in graphene, these were found to include topological states of matter. Here, the realization of the interaction-induced quantum anomalous Hall effect without external magnetic fields in twisted bilayer Sharpe et al. (2019); Serlin et al. (2020); Wu et al. (2021) and trilayer Chen et al. (2020b) graphene has spurred numerous proposals for more exotic fractionalized topological states of matter Liu et al. (2020b); Abouelkomsan et al. (2019); Repellin and Senthil (2019); Ledwith et al. (2019), which however rely on a delicate interplay of spontaneous ferromagnetic order, valley polarization and substrate engineering effects to induce the requisite non-trivial band topology. Generalizations to twisted transition-metal dichalcogenides have focused on telluride-based group VI compounds with 2H structure in the monolayer which exhibit an intrinsic quantum spin Hall effect Wu et al. (2018), with the quantum anomalous Hall effect recently observed Li et al. (2021) and similarly expected to emerge from spontaneous valley polarization Zhang et al. (2021); Xie et al. (2021).

Central to the present work, we demonstrate for twisted bilayer ZrS₂ with 1T structure that the paradigm of twist-controlled suppression of the bare kinetic energy scales can be straightforwardly extended to instead promote spin-orbit coupling to constitute the dominant energy scale at low energies, opening up a new and exotic regime for experimental and theoretical investigation. Remarkably, we find that the two-component character of the valence band maximum in such two-dimensional group IV transition metal dichalcogenides enters in an essential manner, leading to the emergence of a clean moiré Kagomé lattice with almost dispersionless quantum spin Hall bands at small twist angles. We demonstrate that this tunable realization of a ZrS₂ moiré heterostructure with strong spin-orbit coupling and strong interactions can therefore provide a robust and novel platform to probe the profound interplay of non-trivial band topology and electronic correlations, and shed light on elusive quantum phases beyond the purview of conventional condensed matter systems.

ZrS₂ is an exfoliable semiconductor with 1T structure Zhang et al. (2015) in its ground state [Fig. 1(a)]. In contrast to group-VI transition metal dichalcogenides such as MoS₂ with 2H structure, the valence band maximum in ZrS₂ and other group-IV transition metal dichalcogenides is located at $\Gamma$ already in the monolayer and is composed of two-fold degenerate chalcogen $p_{x}$, $p_{y}$ orbitals. Spin-orbit coupling lifts their degeneracy and introduces a $\sim 100\textrm{meV}$ gap [Fig. 1(b)]. This property readily carries over to aligned bilayers with symmetric AA and AB stacking configurations [Figs. 1(c), (d)]; here, the valence band maximum at $\Gamma$ follows from antibonding combinations of the out-of-plane chalcogen $p_{x}$, $p_{y}$ orbitals. These are energetically separated from bonding combinations by $\sim 80-100$ meV [Fig. 1(c), (d)], with a secondary local valence band maximum of $p_{z}$ orbitals furthermore located close to $\Gamma$ and similarly detuned by $\sim 50$ meV for AA stacking.

In twisted bilayers, the atomic interlayer registry interpolates continuously between local AA, AB and BA alignment as a function of position and a moiré pattern with three-fold rotation symmetry forms [Fig. 1(e)]. At sufficiently small twist angles, the energetic considerations for aligned bilayers discussed above immediately suggest that the top-most (highest energy) moiré valence bands should be similarly composed of antibonding $p_{x}$, $p_{y}$ chalcogen orbitals. If spin-orbit coupling is neglected, these are degenerate at $\Gamma$ in both AA and AB regions [Fig. 1(e)] by virtue of rotation symmetry. However, the valence band edge differs between the two stackings, with the smooth interpolation between local alignments in the moiré unit cell encoded in an effective periodic scalar moiré potential $V(\mathbf{r})$ [Fig. 1(f)]. Minima of $V(\mathbf{r})$ are located at the AB and BA regions and form an effective honeycomb lattice. Notably, for the purposes of capturing the highest-energy moiré valence bands, the model retains to an excellent approximation the full six-fold rotation and mirror symmetries of the monolayer, even though the macroscopic crystal is chiral. This situation is in principle analogous to twisted bilayer MoS₂ Xian et al. (2020); Angeli and MacDonald (2021), which hosts a series of almost dispersionless bands of Mo $d_{z^{2}}$ orbital character on an emergent moiré honeycomb lattice.

Crucially, the loss of rotational symmetry away from local AA, AB stacking lifts the orbital degeneracy between $p_{x}$, $p_{y}$ antibonding orbitals (in the absence of spin-orbit coupling), introducing a second energy scale into the problem. In stark contrast to 2H TMD bilayers, the two-component character of the $\Gamma$ valley states enters in an essential manner. From symmetry considerations, their orbital splitting is expected to be maximal in three “domain wall” regions “X” per moiré unit cell in Fig. 1(e), in which the transition-metal atoms of both layers form “stripes”, with the rotational symmetry of the local stacking order reduced to C₂. Contrary to the scalar moiré potential, maxima of orbital $p_{x}$, $p_{y}$ splitting hence form a Kagomé pattern [Fig. 1(g)]. Remarkably, if the resulting energetic gain exceeds the scalar potential $V(\mathbf{r})$, it becomes favorable for charge to migrate from the honeycomb AB/BA regions to “X” regions, realizing an emergent Kagomé lattice of $s$-like moiré orbitals in a highly-tunable setting [Fig. 1(e)].

A minimal continuum model of this scenario readily follows from the above symmetry considerations as

$\displaystyle\hat{H}=\hat{H}_{0}+\hat{H}_{\textrm{soc}}+\hat{H}_{\textrm{pot}}$

(1)

where $\hat{H}_{0}$ describes the two-fold degenerate antibonding $p_{x}$, $p_{y}$ chalcogen states

$\displaystyle\hat{H}_{0}$

$\displaystyle=-\frac{\hbar^{2}}{2m^{\star}}\left\{(k_{x}^{2}+k_{y}^{2})\hat{\mathbf{1}}+\vphantom{\frac{\eta}{2}}\eta\left[(k_{x}^{2}-k_{y}^{2})\hat{\boldsymbol{\tau}}_{z}+2k_{x}k_{y}\hat{\boldsymbol{\tau}}_{x}\right]\right\}$

(2)

with the orbital degree of freedom represented via Pauli matrices $\hat{\boldsymbol{\tau}}$. Here, $m^{\star}$ denotes the effective average band mass, and $\eta=\tfrac{m_{+}-m_{-}}{m_{+}+m_{-}}$ parametrizes the ratio of light ($m_{-}$) and heavy ($m_{+}$) hole $p$ bands at $\Gamma$. Atomic spin-orbit interactions

$\displaystyle\hat{H}_{\textrm{soc}}=\frac{\lambda_{\textrm{soc}}}{2}~{}\hat{\boldsymbol{\tau}}_{y}\hat{\boldsymbol{\sigma}}_{z}$

(3)

lift the orbital degeneracy, opening up a gap at $\Gamma$ as discussed in detail below. Here, $\hat{\boldsymbol{\sigma}}_{z}$ acts on spin. Central to the emergence of the Kagomé lattice, the moiré potential acts non-trivially on the orbital pseudospin, and can generically be written as a Fourier expansion

$\displaystyle\hat{H}_{\textrm{pot}}=\sum_{n}V_{n}~{}f_{n}^{0}(\mathbf{r})~{}+~{}\sum_{n}V^{\prime}_{n}\left[\hat{\boldsymbol{\tau}}_{x}~{}f_{n}^{(x)}(\mathbf{r})+\hat{\boldsymbol{\tau}}_{z}~{}f_{n}^{(z)}(\mathbf{r})\right]$

(4)

Here, $n$ indexes the $n$-th moiré Brillouin zone. $V_{n}$ parameterizes the Fourier modes of the scalar potential in direct analogy to twisted WS₂ Angeli and MacDonald (2021), with $f_{n}^{(0)}(\mathbf{r})=\cos(\mathbf{b}_{n,1}\mathbf{r})+\cos(\mathbf{b}_{n,2}\mathbf{r})+\cos(\mathbf{b}_{n,3}\mathbf{r})$ chosen to retain the full six-fold rotation symmetry and $\mathbf{b}_{n,i}$ describing the three reciprocal lattice vectors $i=1,2,3$ (related via $C_{3}$ rotations) to the $n$-th Brillouin zone.

The pseudospin $\hat{\boldsymbol{\tau}}_{x}$, $\hat{\boldsymbol{\tau}}_{z}$ contributions to the potential are related in the presence of (approximate) mirror symmetry, with $f_{n}^{(x)}(\mathbf{r})=-\frac{\sqrt{3}}{2}\cos(\mathbf{b}_{n,1}\mathbf{r})+\frac{\sqrt{3}}{2}\cos(\mathbf{b}_{n,3}\mathbf{r})$ and $f_{n}^{(z)}(\mathbf{r})=\frac{1}{2}\cos(\mathbf{b}_{n,1}\mathbf{r})-\cos(\mathbf{b}_{n,2}\mathbf{r})+\frac{1}{2}\cos(\mathbf{b}_{n,3}\mathbf{r})$. The salient physics is encoded already in the lowest harmonic – with $\mathbf{b}_{1,1}=[2\pi,-2\pi/\sqrt{3}]/a_{0}$, $\mathbf{b}_{1,2}=[0,4\pi/\sqrt{3}]/a_{0}$, $\mathbf{b}_{1,3}=[2\pi,2\pi/\sqrt{3}]/a_{0}$ and $a_{0}$ the moiré lattice length, the scalar potential hosts two minima in the AB and BA regions at $\mathbf{r}=[1/2,\pm 1/(2\sqrt{3})]$. Conversely, the pseudospin potential that determines the splitting of $p_{x}$, $p_{y}$ orbitals has three maxima in the Kagomé X regions at $\mathbf{r}=[1/2,0]a_{0},~{}[1/4,\sqrt{3}/4]a_{0},~{}[3/4,\sqrt{3}/4]a_{0}$. Up to an overall energy scale, a minimal continuum model that includes only the first harmonic will therefore depend on just three dimensionless parameters $\eta$, $2Vm^{\star}a_{0}^{2}/\hbar^{2}$, $2V^{\prime}m^{\star}a_{0}^{2}/\hbar^{2}$, where $a_{0}\propto\theta^{-1}$ scales with the twist angle. Local lattice relaxation effects are encoded in the higher harmonics of the potential; as the local stacking of AB and BA regions arise energetic favorable, lattice relaxation results in large domains with almost uniform AB or BA stacking [see domains highlighted with red and purple dashed lines in Fig. 1 (e)] with the salient stacking variation in the X regions. These parameters can be obtained fitting the band structure obtained from DFT calculations as described below.

Fig. 2(a) depicts the structure of the resulting moiré bands without spin-orbit coupling, as a function of scalar $V\equiv V_{1}$and pseudospin $V^{\prime}\equiv V^{\prime}_{1}$ potentials. For $V^{\prime}=0$, the scalar potential $V$ localizes the hole charge density on a honeycomb lattice of AB/BA regions [Fig. 2(a), left column; Fig. 2(b) (I)], and an energetically well-separated set of honeycomb bands with Dirac points at $\mathbf{K}$, $\mathbf{K}^{\prime}$ emerges at the top of the valence band. These retain a two-fold orbital $p_{x}$, $p_{y}$ character, with the degeneracy of the bands weakly broken due to orbital anisotropy $\eta\neq 0$. This directly mirrors the low-energy band structure of twisted bilayer graphene, however with the two-orbital structure resulting from the $p_{x}$, $p_{y}$ degeneracy of the constituent states at $\Gamma$ as opposed to a valley degeneracy.

However, already the next lower in energy (fifth) moiré valence band reveals upon closer inspection a charge density distribution with a Kagomé pattern [Fig. 2(b), pattern (II)], localized in the “X” regions of the moiré unit cell [Fig. 1(e)]. These states gain energy from a finite pseudospin potential $V^{\prime}$, which lifts them to higher energies: Beyond a critical $V^{\prime}$, the fifth “Kagomé” band and the bottom $p_{x}$, $p_{y}$ honeycomb bands invert their energetic ordering at $\Gamma$. Consequently, the charge density distribution of the top $p_{x}$, $p_{y}$ bands shifts from AB/BA honeycomb regions to “X” Kagomé sites [Fig. 2(b), pattern (III)]. If the moiré potentials are sufficiently weak, the three resulting bands that constitute the emergent moiré Kagomé lattice couple to a fourth moiré orbital centered on the hexagons of the lattice, with a charge density distribution that forms a ring around the AA regions of the moiré unit cell [Fig. 2(b), pattern (IV)]. As the twist angle is further reduced, an energetically well-separated set of three Kagomé lattice bands emerges as the top-most set of moiré valence states [Fig. 2(b), patterns (V), (VI)].

The above behavior closely matches the results from large-scale ab initio calculations of the twisted moiré supercell, depicted in Fig. 2(d), left column, for three representative twist angles [also see Supplementary Information]. As the angle is reduced, a set of bands with a Kagomé charge distribution at $\Gamma$ splits off progressively from deeper valence bands. For the larger twist angles $\geq 2.28^{\circ}$ that are still within computational reach for density functional calculations, this energetic separation is not yet sufficient to completely separate the Kagomé bands of chalcogen antibonding $p_{x}$, $p_{y}$ character from states with $p_{z}$ or bonding $p_{x}$, $p_{y}$ character ($<50{\rm meV}$ below the band edge), not included in the continuum theory. Nevertheless, the top-most Kagomé bands of interest are already well-captured via the continuum model for the smallest twist angle [Fig. 2(d), bottom-left] upon accounting only for the lowest harmonic of the moiré potential.

Crucially, the inclusion of spin-orbit coupling [Eq. (3)] now opens up a gap at the Kagomé Dirac points and lifts the quadratic band touching degeneracy at $\Gamma$ [Fig. 2(c)], reflected in ab initio simulations with spin-orbit interactions [Fig. 2(d), right column]. As the top-most valence states originate from $p_{x}$, $p_{y}$ orbitals at small twist angles, spin-flip spin-orbit interactions are negligible and spin-$z$ remains a good quantum number. Remarkably, this results in three almost dispersionless moiré bands that realize a novel Kagomé topological quantum spin Hall insulator with spin Chern numbers $\mathcal{C}_{s}=\pm 1$ for the first and third flat band [Fig. 2(c), (d)]. In marked contrast to conventional topological materials however, while superlattice interference quenches the kinetic energy scales, spin-orbit coupling $\lambda_{\textrm{soc}}$ enters as a bare atomic scale and hence becomes the dominant energy scale that governs the low-energy physics of the moiré valence bands in ZrS₂. This highly-tunable materials realization of an “ultra-strong” spin-orbit interaction regime in a moiré heterostructure constitutes a central result of this paper.

To model the emergent top-most flat topological moiré band in twisted ZrS₂, we proceed with a fit of the pseudospin continuum theory [Eq. (1)] to the spin-orbit-coupled ab initio band structure for $\theta=2.64^{\circ}$ [Fig. 2(d), middle-right panel]. As the minimal model of Eq. (1) does not account for bonding $p_{x},p_{y}$ or $p_{z}$ states, the third-highest ab initio valence band ($-50{\rm meV}$ below the valence band edge) is composed primarily of bonding $p_{x},p_{y}$ and $p_{z}$ orbitals and is excluded from the fit. We note that this band separates energetically from the three Kagomé moiré bands at lower twist angles. We obtain excellent agreement for the top two bands of $p_{x}$, $p_{y}$ antibonding character using $\eta=0.33$, $m^{\star}=0.27m_{0}$, $\lambda_{\textrm{soc}}=57{\rm meV}$, $V_{1}=5.5{\rm meV}$, $V^{\prime}_{1}=-9.3{\rm meV}$, $V_{2}=11.5{\rm meV}$, $V^{\prime}_{2}=-5.1{\rm meV}$. Scaling with twist angle similarly matches the ab initio band structure at $2.45^{\circ}$ [Fig. 2(d), bottom-right panel]. As expected, the top-most band is topologically non-trivial with spin Chern number $\mathcal{C}_{s}=\pm 1$. Fig. 2(e) compares the corresponding charge density distributions at $\Gamma$ for ab initio and continuum model calculations; both exhibit comparable Kagomé patterns as well as a competing band at lower energies with a ring-shaped charge pattern around the $AA$ region, which similarly becomes energetically separated from Kagomé bands at lower twist angles [Fig. 2(a)].

A key advantage of the continuum theory is the possibility to study the behavior at small twist angles in a computationally feasible manner. Fig. 3(a) depicts the band width of the top-most topological moiré Kagomé band, as well as the single-particle gap to the next deeper valence band, as a function of twist angle $a_{0}\sim\theta^{-1}$. The band width of the top-most topological band decreases exponentially with twist angle, whereas the ratio between band width and band gap saturates below $\approx 2^{\circ}$ and approaches one. Below this twist angle, the three Kagomé bands become fully isolated in energy from deeper valence states [Fig. 3(g)]. This immediately suggests a fruitful tight-binding parameterization at ultra-small angles, presuming that local lattice relaxation effects remain manageable. Results are shown in Fig. 3(b) for a tight-binding model depicted schematically in (c), but including up to 8th-neighbor hopping to ensure a good fit over all angles [see Supplementary Information]. For small angles $\ll 2^{\circ}$, the top three bands become well-captured by a nearest-neighbor Kagomé tight-binding model with imaginary hoppings. Third-neighbor hopping $t_{\varhexagon}$ through the hexagons are leading corrections to this model and follow from the elliptical shapes of the charge density distribution at the Kagomé “X” sites.

The sizable imaginary nearest-neighbor hopping [Fig. 3(b)] is a direct consequence of the strong spin-orbit coupling limit and can be interpreted as a finite effective staggered magnetic flux through the elementary triangles of the Kagomé lattice. It lifts the quadratic touching of flat and dispersive Kagomé bands and opens up a gap at the Dirac points, realizing a time-reversal-invariant version of a parent model for fractional Chern insulators Tang et al. (2010); Wu et al. (2011). Here, uniformity of the Berry curvature is a key figure of merit Parameswaran et al. (2011) and determines, jointly with the ideal droplet condition for the Fubini-Study metric Claassen et al. (2015); Jackson et al. (2014); Lee et al. (2017), the propensity for flat Chern bands to host fractional quantum Hall phases. Figs. 3(d), (e) depict the Berry curvature for the top-most and third topological Kagomé moiré band, for two representative twist angles, with Berry curvature fluctuations $(\Delta\Omega)^{2}=\int_{\textrm{BZ}}[\Omega(\mathbf{k})-\sqrt{3}\mathcal{C}/8\pi^{2}]^{2}d\mathbf{k}$ quantified in (f), where $\mathcal{C}=\pm 1$ is the Chern number. The Berry curvature flattens monotonically as the twist angle is reduced, with fluctuations substantially suppressed for the third Kagomé valence band at small angles.

The tunable realization of isolated topological flat bands in twisted ZrS₂ is an ideal starting point for the stabilization of a host of correlated topological states of matter, ranging from the quantum anomalous Hall effect to elusive fractional Chern insulator and fractional topological insulator phases. To investigate the role of electronic correlations, we augment the effective tight-binding description [Fig. 3] via a screened Coulomb repulsion, constrained for simplicity to a local Hubbard ($U\sum_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}$) and nearest-neighbor density ($(U^{\prime}/2)\sum_{\left<ij\right>\sigma\sigma^{\prime}}\hat{n}_{i\sigma}\hat{n}_{j\sigma^{\prime}}$) interaction. Suppose first that the top-most topological moiré Kagomé band is tuned to half filling via electrostatic gating. A non-trivial spin Chern number precludes a straightforward Wannier tight-binding representation of the band. Instead, as deeper fully-filled valence bands are energetically separated, the low-energy behavior can be captured via projecting Coulomb interactions $U,U^{\prime}$ to the half-filled flat topological band, in direct analogy to lowest Landau level projections for the fractional quantum Hall effect. The resulting interacting problem is governed by an effective Hamiltonian

$\displaystyle\hat{H}_{\textrm{eff}}$

$\displaystyle=\sum_{\mathbf{k}\sigma}\epsilon_{\mathbf{k}}\hat{c}_{\mathbf{k}\sigma}^{{\dagger}}\hat{c}_{\mathbf{k}\sigma}^{\vphantom{{\dagger}}}+\frac{1}{L}\sum_{\begin{subarray}{c}\mathbf{k}\mathbf{k}^{\prime}\mathbf{q}\\ \sigma\sigma^{\prime}\end{subarray}}V_{\mathbf{k}\mathbf{k}^{\prime}\mathbf{q}}^{\sigma\sigma^{\prime}}~{}\hat{c}_{\mathbf{k},\sigma}^{\vphantom{{\dagger}}}\hat{c}_{\mathbf{k}^{\prime},\sigma^{\prime}}^{\vphantom{{\dagger}}}\hat{c}_{\mathbf{k}^{\prime}-\mathbf{q},\sigma^{\prime}}^{{\dagger}}\hat{c}_{\mathbf{k}+\mathbf{q},\sigma}^{{\dagger}}$

(5)

where $\hat{c}_{\mathbf{k}\sigma}^{{\dagger}},\hat{c}_{\mathbf{k}\sigma}^{\vphantom{{\dagger}}}$ create/annihilate electrons in the flat band, $\epsilon_{\mathbf{k}}$ denotes the residual band dispersion, $L$ is the system size, and

$\displaystyle V_{\mathbf{k}\mathbf{k}^{\prime}\mathbf{q}}^{\sigma\sigma^{\prime}}=\frac{1}{2}\sum_{\alpha\alpha^{\prime}}v_{\alpha\alpha^{\prime}}(\mathbf{q})~{}u_{\mathbf{k}}^{(\alpha\sigma)}u_{\mathbf{k}^{\prime}}^{(\alpha^{\prime}\sigma^{\prime})}\left[u_{\mathbf{k}^{\prime}-\mathbf{q}}^{(\alpha^{\prime}\sigma^{\prime})}u_{\mathbf{k}+\mathbf{q}}^{(\alpha\sigma)}\right]^{\star}$

(6)

is the Coulomb repulsion projected to the Bloch states $u_{\mathbf{k}}^{(\alpha\sigma)}$ of the top-most band, with

$\displaystyle v(\mathbf{q})=\left[\begin{array}[]{ccc}U&U^{\prime}\cos(\tfrac{\mathbf{k}\mathbf{a}_{1}}{2})&U^{\prime}\cos(\tfrac{\mathbf{k}\mathbf{a}_{2}}{2})\\ U^{\prime}\cos(\tfrac{\mathbf{k}\mathbf{a}_{1}}{2})&U&U^{\prime}\cos(\tfrac{\mathbf{k}\mathbf{a}_{3}}{2})\\ U^{\prime}\cos(\tfrac{\mathbf{k}\mathbf{a}_{2}}{2})&U^{\prime}\cos(\tfrac{\mathbf{k}\mathbf{a}_{3}}{2})&U\end{array}\right]$

(10)

Here, momenta $\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}$ are defined in the moiré Brillouin zone, $\mathbf{a}_{i}$ denote the Moiré lattice vectors, and $\alpha,\sigma$ denote the sublattice and spin degrees of freedom.

Since a sufficiently short-ranged interaction $U>U^{\prime}$ mainly imparts a local energetic penalty for electron pairs of opposite spin occupying the same Kagomé “X” sites, a flat-band ferromagnetic instability generically ensues Mielke (1991) at half filling of the top-most quantum spin Hall band, in direct analogy to quantum Hall ferromagnetism Sondhi et al. (1993). The resulting spontaneous spin-polarized state is gapped and aligned in the $z$ direction – it exhibits a quantum anomalous Hall effect by virtue of filling a quantum spin Hall band for one spin component, and becomes an exact zero-energy ground state in the absence of dispersion Neupert et al. (2011). Fig. 4(a) depicts the corresponding phase diagram as a function of twist angle and interaction strength $U$ vs band width $W$ of the top-most band, evaluated from exact diagonalization of Eq. (5) on a $4\times 4$ unit cell cluster. A robust quantum anomalous Hall state emerges for interactions on the order of four times the moiré band width and remains robust over a wide range of twist angles. Notably, the underlying mechanism is distinct from the observed quantum anomalous Hall effect in twisted bilayer graphene, relying instead on the intrinsic topologically non-trivial moiré band structure due to strong spin-orbit coupling and obviating the necessity for concurrent valley polarization and substrate effects.

Persistence of the ferromagnetic instability for fractional fillings of the Kagomeé flat bands naturally suggests the possibility to stabilize a variety of Abelian and non-Abelian fractional quantum Hall states in the absence of external magnetic fields Bergholtz and Liu (2013), by analogy to a fractionally-filled Landau level. To this end, we focus on the Laughlin $\nu=1/3$ state at $1/6$ hole doping, and study the interacting problem at small twist angles in exact diagonalization. Analogous to the half-filled case, electrons in the almost-flat band can avoid local Coulomb repulsion $U$ via spontaneous spin polarization, yielding a robust ferromagnetic instability as a function of $U$ [Fig. 4(b), right axis, dashed line] over all investigated twist angles. However, spontaneous spin polarization due to $U$ now leaves a single Chern band at $1/3$ hole doping, with the resulting electronic phase governed by longer-ranged Coulomb interactions $U^{\prime}$. To study the propensity to realize a Laughlin state, we numerically investigate the resulting phase diagram as function of bandwidth $W/U^{\prime}$ [Fig. 4(b), left axis]. For $W=0$, corresponding to the Landau level limit of a perfectly-flat Chern band, exact diagonalization calculations for $6\times 5$ unit cells indicate the robust stabilization of a fractional Chern insulator. This phase is characterized by a three-fold ground state degeneracy for periodic boundary conditions [Fig. 4(c)] with a gap to well-separated many-body excitations which persists as a function of system size. These ground states lie in three total momentum sectors that match the generalized Pauli principle for FCIs Regnault and Bernevig (2011), flow into each other upon adiabatic insertion of a magnetic flux through handles of the torus (periodic boundary conditions) and remain energetically separated from excitations, confirming the $\nu=1/3$ FCI Regnault and Bernevig (2011); Wu et al. (2011). These conclusions remain largely independent of the twist angle, and the FCI persists upon inclusion of finite band dispersion $W$ until the many-body excitation gap closes for $W/U^{\prime}\sim 0.3$ [Fig. 4(b), false color].

Having established a robust correlated quantum anomalous Hall phase at half filling and evidence for a $\nu=1/3$ fractional Chern insulator at one-sixth hole doping, an interesting follow-up question concerns the role of proximal deeper moiré valence bands, beyond the single-band approximation. For interactions that exceed the single-particle gap to other bands but remain smaller than the overall band width of the three Kagomé bands, the robustness of fractional Chern insulator phases has been well-documented Kourtis et al. (2014), in direct analogy to Landau level mixing in the conventional quantum Hall effect. A more substantial challenge however stems from details of possible longer-ranged electron interactions and exchange processes, which could serve to either enhance or suppress the stability of the fractionalized phases at different filling fractions. These processes sensitively depend on the screening environment and gating Throckmorton and Vafek (2011), and microscopic calculations present a substantial methodological obstacle for twisted materials Pizarro et al. (2019); Vanhala and Pollet (2020). Conversely, for sufficiently small twist angles, if the Coulomb repulsion exceeds the overall band width of the three Kagomé bands, sufficient screening could serve to form a local moment at overall half filling $\nu=3/2$. Such a Kagomé Mott insulator would constitute a Moiré realization of a paradigmatic frustrated magnetic model, which has been under intense scrutiny for the potential to host an elusive quantum spin liquid phase.

Beyond the (fractional) quantum anomalous Hall effect, the realization of flat-band quantum spin Hall insulators further opens up the possibility to realize a myriad of unconventional ordered states of matter with non-trivial topology, including time-reversal invariant fractionalized phases or topological superconductors. Consequently, twisted ZrS₂ bilayers constitute a promising and tunable materials platform for such investigations, granting access to a novel and exotic regime of ultra-strong spin-orbit coupling that is not readily realizable in conventional crystalline solid-state systems. More broadly, a natural question concerns the extension of similar ideas of pseudospin potential engineering and strong spin-orbit coupling to other transition-metal dichalcogenide heterostructures such as TiS₂ and HfS₂ with multi-component character of the valence band edge. At the same time, the emergence of a moiré Kagomé lattice from the fortuitous but robust interplay of geometry and interlayer coupling at small twist angles opens up a new pathway towards a moiré realization of magnetic phases in a paradigmatic frustrated system.