Abstract
We predict that twisted bilayers of 1TZrS_{2} realize a novel and tunable platform to engineer twodimensional topological quantum phases dominated by strong spinorbit interactions. At small twist angles, ZrS_{2} heterostructures give rise to an emergent and twistcontrolled moiré Kagome lattice, combining geometric frustration and strong spinorbit coupling to give rise to a moiré quantum spin Hall insulator with highly controllable and nearlydispersionless bands. We devise a generic pseudospin theory for groupIV transition metal dichalcogenides that relies on the twocomponent character of the valence band maximum of the 1T structure at Γ, and study the emergence of a robust quantum anomalous Hall phase as well as possible fractional Chern insulating states from strong Coulomb repulsion at fractional fillings of the topological moiré Kagome bands. Our results establish groupIV transition metal dichalcogenide bilayers as a novel moiré platform to realize stronglycorrelated topological phases in a twisttunable setting.
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Introduction
Twisted van der Waals heterostructures have recently emerged as an intriguing and highly tunable platform to realize unconventional electronic phases in two dimensions^{1,2,3,4}. Spurred by the discovery of Mott insulation and superconductivity in twisted bilayer graphene^{5,6}, remarkable progress in fabrication and twistangle control has led to observations of correlated insulating states or superconductivity in a variety of materials, including trilayer and doublebilayer graphene, homo and heterobilayers of twisted transition metal dichalcogenides (TMDs)^{7,8,9,10,11,12,13,14,15,16,17,18}, and heterostructures at a twist on hexagonal boron nitride substrates^{19,20}. 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 lowenergy moiré bands implies straightforward gatetunable access to a wide range of filling fractions, permitting wideranging experimental access to the phase diagrams of paradigmatic models of stronglycorrelated electrons^{2}. Consequently, the putative realization of stronglycorrelated electron physics in a tunable setting has garnered significant attention, resulting in growing experimental evidence for novel correlated phases, including unconventional superconductivity^{4,21,22}.
Notably, and despite negligible intrinsic spin–orbit coupling in graphene, these were found to include topological states of matter. Here, the realization of the interactioninduced quantum anomalous Hall effect without external magnetic fields in twisted bilayer^{23,24,25} and trilayer^{26} graphene has spurred numerous proposals for more exotic fractionalized topological states of matter^{27,28,29,30}, which however rely on a delicate interplay of spontaneous ferromagnetic order, valley polarization, and substrate engineering effects to induce the requisite nontrivial band topology. Generalizations to twisted transitionmetal dichalcogenides have focused on telluridebased group VI compounds with 2H structure in the monolayer which exhibit an intrinsic quantum spin Hall effect^{31}, with the quantum anomalous Hall effect recently observed^{32} and similarly expected to emerge from spontaneous valley polarization^{33,34}.
Central to the present work, we demonstrate for twisted bilayer ZrS_{2} with 1T structure that the paradigm of twistcontrolled 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 twocomponent character of the valence band maximum in such twodimensional group IV transition metal dichalcogenides enters in an essential manner, leading to the emergence of a clean moiré Kagome lattice with almost dispersionless quantum spin Hall bands at small twist angles. We demonstrate that this tunable realization of a ZrS_{2} moiré heterostructure with strong spinorbit coupling and strong interactions can therefore provide a robust and novel platform to probe the profound interplay of nontrivial band topology and electronic correlations, and shed light on elusive quantum phases beyond the purview of conventional condensed matter systems.
Results
Emergent Kagome moiré pattern in twisted ZrS_{2} bilayers
ZrS_{2} is a group IV transition metal dichalcogenide with an exfoliable layered structure. Different from group VI TMDs such as Mo_{2} and WS_{2} that normally adopt a 2H layered structure, ZrS_{2} has a stable 1T structure^{35} in its ground state as shown in Fig. 1a, without distorting into the 1T’ structure^{36,37}. Bulk ZrS_{2} is a semiconductor with a band gap of 1.80 eV^{38,39} and it remains semiconducting when thinned down to the monolayer^{35,40}. In contrast to groupVI transition metal dichalcogenides such as MoS_{2} with 2H structure, the valence band maximum in ZrS_{2} and other groupIV transition metal dichalcogenides is located at Γ already in the monolayer and is composed of twofold degenerate chalcogen p_{x}, p_{y} orbitals. Spin–orbit coupling lifts their degeneracy and introduces a ~100 meV gap [Fig. 1b]. This property readily carries over to aligned bilayers with symmetric AA and AB stacking configurations [Fig. 1c, d]; here, the valence band maximum at Γ follows from antibonding combinations of the outofplane chalcogen p_{x}, p_{y} orbitals. These are energetically separated from bonding combinations by ~80−100 meV [Fig. 1c, d], with a secondary local valence band maximum of p_{z} orbitals furthermore located close to Γ and similarly detuned by ~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 threefold rotation symmetry forms [Fig. 1e]. At sufficiently small twist angles, the energetic considerations for aligned bilayers discussed above immediately suggest that the topmost (highest energy) moiré valence bands should be similarly composed of antibonding p_{x}, p_{y} chalcogen orbitals. If spinorbit coupling is neglected, these are degenerate at Γ in both AA and AB regions [Fig. 1e] 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(r) [Fig. 1f]. Minima of V(r) is located at the AB and BA regions and form an effective honeycomb lattice. Notably, for the purposes of capturing the highestenergy moiré valence bands, the potential retains to an excellent approximation of the full sixfold rotation and mirror symmetries of the monolayer, even though the macroscopic crystal is chiral. This situation is in principle analogous to twisted bilayer MoS_{2}^{41,42}, 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 spinorbit coupling), introducing a second energy scale into the problem. In stark contrast to 2H TMD bilayers, the twocomponent character of the Γ 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. 1e, in which the transitionmetal atoms of both layers form “stripes”, with the rotational symmetry of the local stacking order reduced to C_{2}. Contrary to the scalar moiré potential, maxima of orbital p_{x}, p_{y} splitting hence form a Kagome pattern [Fig. 1g]. Remarkably, if the resulting energetic gain exceeds the scalar potential V(r), it becomes favorable for charge to migrate from the honeycomb AB/BA regions to “X” regions, realizing an emergent Kagome lattice of slike moiré orbitals in a highlytunable setting [Fig. 1e].
Continuum model of twisted ZrS_{2}
A minimal continuum model of this scenario readily follows from the above symmetry considerations as
where \({\hat{H}}_{0}\) describes the twofold degenerate antibonding p_{x}, p_{y} chalcogen states
with the orbital degree of freedom represented via Pauli matrices \(\hat{{{{{{{{\boldsymbol{\tau }}}}}}}}}\). Here, m^{⋆} denotes the effective average band mass, and \(\eta =\frac{{m}_{+}{m}_{}}{{m}_{+}+{m}_{}}\) parametrizes the ratio of light (m_{−}) and heavy (m_{+}) hole p bands at Γ. Atomic spin–orbit interactions
lift the orbital degeneracy, opening up a gap at Γ as discussed in detail below. Here, \({\hat{{{{{{{{\boldsymbol{\sigma }}}}}}}}}}_{z}\) acts on spin. Central to the emergence of the Kagome lattice, the moiré potential acts nontrivially on the orbital pseudospin, and can generically be written as a Fourier expansion
Here, n indexes the nth moiré Brillouin zone. V_{n} parameterizes the Fourier modes of the scalar potential in direct analogy to twisted WS_{2}^{42}, with \({f}_{n}^{(0)}({{{{{{{\bf{r}}}}}}}})=\cos ({{{{{{{{\bf{b}}}}}}}}}_{n,1}{{{{{{{\bf{r}}}}}}}})+\cos ({{{{{{{{\bf{b}}}}}}}}}_{n,2}{{{{{{{\bf{r}}}}}}}})+\cos ({{{{{{{{\bf{b}}}}}}}}}_{n,3}{{{{{{{\bf{r}}}}}}}})\) chosen to retain the full sixfold rotation symmetry and b_{n,i} describing the three reciprocal lattice vectors i = 1, 2, 3 (related via C_{3} rotations) to the nth 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 \({f}_{n}^{(x)}({{{{{{{\bf{r}}}}}}}})=\frac{\sqrt{3}}{2}\cos ({{{{{{{{\bf{b}}}}}}}}}_{n,1}{{{{{{{\bf{r}}}}}}}})+\frac{\sqrt{3}}{2}\cos ({{{{{{{{\bf{b}}}}}}}}}_{n,3}{{{{{{{\bf{r}}}}}}}})\) and \({f}_{n}^{(z)}({{{{{{{\bf{r}}}}}}}})=\frac{1}{2}\cos ({{{{{{{{\bf{b}}}}}}}}}_{n,1}{{{{{{{\bf{r}}}}}}}})\cos ({{{{{{{{\bf{b}}}}}}}}}_{n,2}{{{{{{{\bf{r}}}}}}}})\;+\frac{1}{2}\cos ({{{{{{{{\bf{b}}}}}}}}}_{n,3}{{{{{{{\bf{r}}}}}}}})\). The salient physics is encoded already in the lowest harmonic—with \({{{{{{{{\bf{b}}}}}}}}}_{1,1}=[2\pi ,2\pi /\sqrt{3}]/{a}_{0}\), \({{{{{{{{\bf{b}}}}}}}}}_{1,2}=[0,4\pi /\sqrt{3}]/{a}_{0}\), \({{{{{{{{\bf{b}}}}}}}}}_{1,3}=[2\pi ,2\pi /\sqrt{3}]/{a}_{0}\) and a_{0} the moiré lattice length, the scalar potential hosts two minima in the AB and BA regions at \({{{{{{{\bf{r}}}}}}}}=[1/2,\pm 1/(2\sqrt{3})]\). Conversely, the pseudospin potential that determines the splitting of p_{x}, p_{y} orbitals has three maxima in the Kagome X regions at \({{{{{{{\bf{r}}}}}}}}=[1/2,0]{a}_{0},\,[1/4,\sqrt{3}/4]{a}_{0},\,[3/4,\sqrt{3}/4]{a}_{0}\). Up to an overall energy scale, a minimal continuum model that includes only the first harmonic will therefore depend on just three dimensionless parameters η, \(2V{m}^{\star }{a}_{0}^{2}/{\hslash }^{2}\), \(2{V}^{\prime}{m}^{\star }{a}_{0}^{2}/{\hslash }^{2}\), where a_{0} ∝ θ^{−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. 1e] with the salient stacking variation in the X regions. These parameters can be obtained by fitting the band structure obtained from DFT calculations as described below.
Figure 2a depicts the structure of the resulting moiré bands without spinorbit coupling, as a function of scalar V ≡ V_{1} and pseudospin \({V}^{\prime}\equiv {V}_{1}^{\prime}\) potentials. For \({V}^{\prime}=0\), the scalar potential V localizes the hole charge density on a honeycomb lattice of AB/BA regions [Fig. 2a, left column; Fig. 2b (I)], and an energetically wellseparated set of honeycomb bands with Dirac points at K, \({{{{{{{\bf{K}}}}}}}}^{\prime}\) emerges at the top of the valence band. These retain a twofold orbital p_{x}, p_{y} character, with the degeneracy of the bands weakly broken due to orbital anisotropy η ≠ 0. This directly mirrors the lowenergy band structure of twisted bilayer graphene, however with the twoorbital structure resulting from the p_{x}, p_{y} degeneracy of the constituent states at Γ 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 Kagome pattern [Fig. 2b, pattern (II)], localized in the “X” regions of the moiré unit cell [Fig. 1e]. These states gain energy from a finite pseudospin potential \({V}^{\prime}\), which lifts them to higher energies: Beyond a critical \({V}^{\prime}\), the fifth “Kagome” band and the bottom p_{x}, p_{y} honeycomb bands invert their energetic ordering at Γ. Consequently, the charge density distribution of the top p_{x}, p_{y} bands shifts from AB/BA honeycomb regions to “X” Kagome sites [Fig. 2b, pattern (III)]. If the moiré potentials are sufficiently weak, the three resulting bands that constitute the emergent moiré Kagome 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. 2b, pattern (IV)]. As the twist angle is further reduced, an energetically wellseparated set of three Kagome lattice bands emerges as the topmost set of moiré valence states [Fig. 2b, patterns (V), (VI)].
Ab initio characterization
The above behavior closely matches the results from largescale ab initio calculations of the twisted moiré supercell, depicted in Fig. 2d, left column, for three representative twist angles [see Supplementary Note 1]. As the angle is reduced, a set of bands with a Kagome charge distribution at Γ splits off progressively from deeper valence bands. For the larger twist angles ≥2.28^{∘} that are still within computational reach for density functional calculations, this energetic separation is not yet sufficient to completely separate the Kagome bands of chalcogen antibonding p_{x}, p_{y} character from states with p_{z} or bonding p_{x}, p_{y} character (<50 meV below the band edge), not included in the continuum theory. Nevertheless, the topmost Kagome bands of interest are already wellcaptured via the continuum model for the smallest twist angle [Fig. 2d, bottomleft] upon accounting only for the lowest harmonic of the moiré potential.
Crucially, the inclusion of spinorbit coupling [Eq. (3)] now opens up a gap at the Kagome Dirac points and lifts the quadratic band touching degeneracy at Γ [Fig. 2c], reflected in ab initio simulations with spinorbit interactions [Fig. 2d, right column]. As the topmost valence states originate from p_{x}, p_{y} orbitals at small twist angles, spinflip spinorbit interactions are negligible and spinz remains a good quantum number. Remarkably, this results in three almost dispersionless moiré bands that realize a novel Kagome topological quantum spin Hall insulator with spin Chern numbers \({{{{{{{{\mathcal{C}}}}}}}}}_{s}=\pm \!1\) for the first and third flat band [Fig. 2c, d]. In marked contrast to conventional topological materials, however, while superlattice interference quenches the kinetic energy scales, spinorbit coupling λ_{soc} enters as a bare atomic scale and hence becomes the dominant energy scale that governs the lowenergy physics of the moiré valence bands in ZrS_{2}. This highlytunable materials realization of an “ultrastrong” spinorbit interaction regime in a moiré heterostructure constitutes a central result of this paper.
To model the emergent topmost flat topological moiré band in twisted ZrS_{2}, we proceed with a fit of the pseudospin continuum theory [Eq. (1)] to the spinorbitcoupled ab initio band structure for θ = 2.64^{∘} [Fig. 2d, middleright panel]. As the minimal model of Eq. (1) does not account for bonding p_{x}, p_{y}, or p_{z} states, the thirdhighest ab initio valence band (−50 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 Kagome moiré bands at lower twist angles. We obtain excellent agreement for the top two bands of p_{x}, p_{y} antibonding character using η = 0.33, m^{⋆} = 0.27m_{0}, λ_{soc} = 57 meV, V_{1} = 5.5 meV, \({V}_{1}^{\prime}=9.3\;{{{{{{{\rm{meV}}}}}}}}\), V_{2} = 11.5 meV, \({V}_{2}^{\prime}=5.1\;{{{{{{{\rm{meV}}}}}}}}\). Scaling with twist angle similarly matches the ab initio band structure at 2.45^{∘} [Fig. 2d, bottomright panel]. As expected, the topmost band is topologically nontrivial with spin Chern number \({{{{{{{{\mathcal{C}}}}}}}}}_{s}=\pm \!1\). Figure 2e compares the corresponding charge density distributions at Γ for ab initio and continuum model calculations; both exhibit comparable Kagome patterns as well as a competing band at lower energies with a ringshaped charge pattern around the AA region, which similarly becomes energetically separated from Kagome bands at lower twist angles [Fig. 2a].
Tightbinding description of emergent moiré Kagome bands
A key advantage of the continuum theory is the possibility to study the behavior at small twist angles in a computationally feasible manner. Figure 3a depicts the bandwidth of the topmost topological moiré Kagome band, as well as the singleparticle gap to the next deeper valence band, as a function of twist angle a_{0} ~ θ^{−1}. The bandwidth of the topmost topological band decreases exponentially with twist angle, whereas the ratio between bandwidth and band gap saturates below ≈ 2^{∘} and approaches one. Below this twist angle, the three Kagome bands become fully isolated in energy from deeper valence states [Fig. 3g]. This immediately suggests a fruitful tightbinding parameterization at ultrasmall angles, presuming that local lattice relaxation effects remain manageable. Results are shown in Fig. 3b for a tightbinding model depicted schematically in (c), but including up to 8thneighbor hopping to ensure a good fit over all angles [see Supplementary Note 2]. For small angles ≪ 2^{∘}, the top three bands become wellcaptured by a nearestneighbor Kagome tightbinding model with imaginary hoppings. Thirdneighbor hopping t_{δ} through the hexagons are leading corrections to this model and follow from the elliptical shapes of the charge density distribution at the Kagome “X” sites.
The sizable imaginary nearestneighbor hopping [Fig. 3b] 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 Kagome lattice. It lifts the quadratic touching of flat and dispersive Kagome bands and opens up a gap at the Dirac points, realizing a timereversalinvariant version of a parent model for fractional Chern insulators^{43,44}.
Electronic interactions and spontaneous quantum anomalous Hall effect
The tunable realization of isolated timereversal symmetric topological flat bands in twisted ZrS_{2} is an ideal starting point for the stabilization of a host of correlated topological states of matter, ranging from interactioninduced quantum anomalous Hall effects^{45} to elusive fractional Chern and topological insulators^{46,47,48,49}. To investigate the role of electronic interactions and propensity for correlated topological phases in twisted ZrS_{2}, we now study the topmost moiré Kagome band at fractional fillings and augment the effective threeband Kagome tightbinding description [Fig. 3]—derived from the continuum theory, and a continuous function of the twist angle—via a screened Coulomb repulsion. The interaction is constrained for simplicity to a local Hubbard (\(U{\sum }_{i}{\hat{n}}_{i\uparrow }{\hat{n}}_{i\downarrow }\)) and nearestneighbor density (\(({U}^{\prime}/2){\sum }_{ < ij > \sigma {\sigma }^{\prime}}\hat{n}_{i\sigma} \hat{n}_{j{\sigma }^{\prime}}\)) interaction, expected to be a good approximation for screening due to metallic gates^{50,51}. Suppose first that the topmost topological moiré Kagome band is tuned to half filling via electrostatic gating. A nontrivial spin Chern number precludes a straightforward Wannier tightbinding representation of this individual band. Instead, as deeper fullyfilled valence bands are energetically separated, the lowenergy behavior can be captured starting Kagome tightbinding model via projecting Coulomb interactions \(U,{U}^{\prime}\) onto the Bloch states of a single fractionallyfilled 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
where \({\hat{c}}_{{{{{{{{\bf{k}}}}}}}}\sigma }^{{{{\dagger}}} },\hat{c}_{{{{{{{{\bf{k}}}}}}}}\sigma}\) create/annihilate electrons in the flat band with Bloch momenta k, ϵ_{k} denotes the residual band dispersion, L is the system size, and
is the Coulomb repulsion projected to the Bloch states \({u}_{{{{{{{{\bf{k}}}}}}}}}^{(\alpha \sigma )}\) of the topmost band, derived from the tightbinding model, with
Here, momenta \({{{{{{{\bf{k}}}}}}}},{{{{{{{{\bf{k}}}}}}}}}^{\prime},{{{{{{{\bf{q}}}}}}}}\) are defined in the moiré Brillouin zone, a_{i} denote the moiré lattice vectors, and α, σ denote the sublattice and spin degrees of freedom. Since a sufficiently shortranged interaction \(U \; > \;{U}^{\prime}\) mainly imparts a local energetic penalty for electron pairs of opposite spin occupying the same Kagome “X” sites, a flatband ferromagnetic instability generically ensues^{52} at half filling of the topmost quantum spin Hall band, in direct analogy to quantum Hall ferromagnetism^{53}. The resulting spontaneous spinpolarized 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 only; this fully spinpolarized state is an exact gapped zeroenergy ground state in the absence of dispersion^{45} and entails a quantized Hall conductivity. To study its robustness to the finite residual dispersion of the moiré band, we evaluate the phase diagram via exact diagonalization of Eq. (5) on a 4 × 4 unit cell cluster. Figure 4a depicts the phase diagram as a function of twist angle (which parameterizes the lowenergy electronic band structure and Bloch states) and interaction strength U vs bandwidth W of the topmost band. A robust quantum anomalous Hall state emerges for interactions on the order of four times the moiré bandwidth 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 nontrivial moiré band structure due to strong spinorbit coupling and obviating the necessity for concurrent valley polarization and substrate effects.
Fractional Chern insulators at fractional fillings
Having established a readilyaccessible interactioninduced quantum anomalous Hall state at half filling, we now turn to the possibility of more exotic topologicallyordered phases at fractional fillings. Here, previous theoretical works established numerous examples of topological tightbinding models^{46,54,55,56,57} that a fractionally filled flat band with nonzero Chern number can in principle behave analogous to a fractionally filled Landau level and realize Abelian and nonAbelian fractional quantum Hall states in the absence of an external magnetic field. Uniformity of the Berry curvature is a key figure of merit^{58} and determines, jointly with the ideal droplet condition for the FubiniStudy metric^{59,60,61}, the propensity for flat Chern bands to host fractional quantum Hall phases. Figure 3f quantifies Berry curvature fluctuations \({({{\Delta }}\Omega )}^{2}={\int}_{{{{{{{{\rm{BZ}}}}}}}}}{[{{\Omega }}({{{{{{{\bf{k}}}}}}}})\sqrt{3}{{{{{{{\mathcal{C}}}}}}}}/8{\pi }^{2}]}^{2}d{{{{{{{\bf{k}}}}}}}}\) for the Kagome moiré bands, where \({{{{{{{\mathcal{C}}}}}}}}=\pm \!1\) is the spin Chern number. One finds that the Berry curvature flattens monotonically as the twist angle is reduced, with fluctuations substantially suppressed for the third Kagome valence band at small angles.
To demonstrate twisted ZrS_{2} as a candidate to observe FQH physics with external magnetic fields, we focus on the conceptually simplest Laughlin ν = 1/3 state at 1/6 hole doping, and study the interacting problem at small twist angles in exact diagonalization. Analogous to the halffilled case, electrons in the almostflat band can avoid local Coulomb repulsion U via spontaneous spin polarization, yielding a robust ferromagnetic instability as a function of U [Fig. 4b, 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 longerranged 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. 4b, left axis]. For W = 0, corresponding to the Landau level limit of a perfectlyflat Chern band, our exact diagonalization calculations for 6 × 5 unit cells reveal a threefold ground state degeneracy for periodic boundary conditions [Fig. 4c] with a gap to wellseparated manybody 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^{56}, 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 ν = 1/3 FCI^{44,56}. Combined, these results indicate the robust stabilization of a fractional Chern insulator. The conclusions remain largely independent of the twist angle, and the FCI persists upon inclusion of finite band dispersion W until the manybody excitation gap closes for \(W/{U}^{\prime} \sim 0.3\) [Fig. 4b, false color].
Discussion
Having established a robust correlated quantum anomalous Hall phase at half filling and evidence for a ν = 1/3 fractional Chern insulator at onesixth hole doping, an interesting followup question concerns the role of proximal deeper moiré valence bands, beyond the singleband approximation. For interactions that exceed the singleparticle gap to other bands but remain smaller than the overall bandwidth of the three Kagome bands, the robustness of fractional Chern insulator phases has been welldocumented^{62}, in direct analogy to Landau level mixing in the conventional quantum Hall effect. A more substantial challenge however stems from details of possible longerranged 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^{50}, and microscopic calculations present a substantial methodological obstacle for twisted materials^{63,64}. Conversely, analyzing the potential stability of more exotic yet more fragile nonAbelian quantum Hall states remains an interesting topic for future investigation. Furthermore, for sufficiently small twist angles, if the Coulomb repulsion exceeds the overall bandwidth of the three Kagome bands, sufficient screening could serve to form a local moment at overall half filling ν = 3/2. Such a Kagome 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 flatband quantum spin Hall insulators further opens up the possibility to realize a myriad of unconventional ordered states of matter with nontrivial topology, including timereversal invariant fractionalized phases, or topological superconductors. Consequently, twisted ZrS_{2} bilayers constitute a promising and tunable materials platform for such investigations, granting access to a novel and exotic regime of ultrastrong spin–orbit coupling that is not readily realizable in conventional crystalline solidstate systems. More broadly, a natural question concerns the extension of similar ideas of pseudospin potential engineering and strong spinorbit coupling to other transitionmetal dichalcogenide heterostructures such as TiS_{2} and HfS_{2} with a multicomponent character of the valence band edge. At the same time, the emergence of a moiré Kagome 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.
Methods
Firstprinciples calculations
Ab initio calculations are performed with the Vienna Ab initio Simulation Package (VASP)^{65} based on density functional theory (DFT). Planewave basis sets are employed with an energy cutoff of 450 eV. The pseudopotentials are constructed with the projector augmented wave method^{66} and the exchangecorrelation functionals are treated within the generalized gradient approximation (GGA)^{67}. Only the Γ point is considered in the calculations due to the large size of the moiré supercells. A vacuum region larger than 15 Angstrom along the zaxis is applied to eliminate artificial interactions between periodic slab images. All atoms are relaxed until the forces on each atom are less than 0.01 eV/Angstrom. Van der Waals corrections are applied with the TkatchenkoScheffler method^{68} during the relaxation. The figures for the atomic structures and the charge density distributions are generated with the VESTA code^{69}.
In this work, DFT calculations are only used to provide a reliable singleparticle description of the top moiré valence bands of twisted bilayer ZrS_{2}. The role of strong electronic interactions within the flat band is subsequently investigated in detail starting from the continuum theory and the effective tightbinding models described in the main text (with parameters extracted from the DFT calculations), using largescale manybody exact diagonalization calculations (see below). Although the band gaps in the systems to the conduction band are underestimated by the DFT calculations at the GGA level, the conduction band lies at high energies and remains empty, hence does not affect the emergent manybody state at small hole doping of the flat moiré valence bands. Only the band dispersion and the shape of the moiré bands are relevant in this work and these are well captured by GGA. As shown in refs. ^{38,70}, manybody corrections to GGA for 2D transitional metal dichalcogenides mainly appear as a rigid shift of the bands such that band gap is enlarged.
Exactdiagonalization calculations
Fractional Chern insulating phases and the spontaneous quantum anomalous Hall effect are studied using exact diagonalization calculations of the manybody ground state of the projected interaction Hamiltonian [Eq. (5)]. Calculations are performed for L_{1} × L_{2} unit cell clusters with periodic boundary conditions and discrete momenta k = n_{1}b_{1}/L_{1} + n_{2}b_{2}/L_{2}, with n_{i} = 0, …, L_{i} − 1 and reciprocal lattice vectors \({{{{{{{{\bf{b}}}}}}}}}_{1,2}={[2\pi ,\pm 2\pi /\sqrt{3}]}^{\top }\). Electron spin is explicitly included. Simulations of the interactioninduced quantum anomalous Hall effect at half filling are performed for 4 × 4 clusters. Results for ν = 1/3 fractional Chern insulators at 1/6 hole doping are obtained for 24site (6 × 4) and 30site (6 × 5) clusters.
Data availability
The raw data sets used for the presented analysis within the current study are available from the corresponding authors on reasonable request.
Code availability
Custom codes used in this work can be provided by the corresponding author on reasonable request. Ab initio calculations were performed with VASP (version 5.4.4).
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Acknowledgements
This work is supported by the European Research Council (ERC2015AdG694097), Grupos Consolidados (IT124919), and SFB925. M.C. is supported by a startup grant from the University of Pennsylvania. A.R. is supported by the Flatiron Institute, a division of the Simons Foundation. We acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under RTG 1995, within the Priority Program SPP 2244 “2DMP”, under Germany’s Excellence Strategy  Cluster of Excellence and Advanced Imaging of Matter (AIM) EXC 2056  390715994 and RTG 2247. L.X. acknowledges the support from Distinguished Junior Fellowship program by the South Bay Interdisciplinary Science Center in the Songshan Lake Materials Laboratory and the KeyArea Research and Development Program of Guangdong Province of China (Grants No. 2020B0101340001). We acknowledge computational resources provided by the Simons Foundation Flatiron Institute, the Max Planck Computing and Data Facility, and the Platform for DataDriven Computational Materials Discovery of the Songshan Lake laboratory. This work was supported by the Max PlanckNew York City Center for Nonequilibrium Quantum Phenomena.
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M.C. and L.X. conceived the project and coordinated the research. M.C. devised the theoretical models and performed the manybody calculations. L.X. performed the ab initio calculations and analysis. M.C., L.X., D.M.K., and A.R. discussed and analyzed the results and contributed to writing the manuscript.
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Claassen, M., Xian, L., Kennes, D.M. et al. Ultrastrong spin–orbit coupling and topological moiré engineering in twisted ZrS_{2} bilayers. Nat Commun 13, 4915 (2022). https://doi.org/10.1038/s4146702231604w
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DOI: https://doi.org/10.1038/s4146702231604w
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