Moiré materials based on M-point twisting

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Introduction

Moiré heterostructures have recently emerged as versatile quantum simulators of archetypal condensed matter models^1,2. When two identical or nearly identical monolayers are twisted, the resulting moiré modulation of the interlayer potential gives rise to an effective moiré discrete translation symmetry. In the moiré BZ, the moiré-modulated interlayer hybridization opens gaps in the folded band structure, quenching the kinetic energy of the monolayer electrons¹⁰. The moiré system thus enters an interaction-dominated regime, providing a tunable platform for simulating various prototypical condensed matter systems. A notable example is twisted bilayer graphene⁵, which hosts unconventional superconductors¹¹ and correlated insulators¹² near the magic angle and has recently been shown to simulate topological heavy-fermions^13,14. Transition-metal dichalcogenide (TMD) heterobilayers can emulate the Hubbard model on a triangular lattice^7,15, whereas twisted WTe₂ exhibits signatures of a one-dimensional Luttinger liquid, although its theoretical description remains challenging owing to the complex monolayer band structure¹⁶. Beyond these examples, a growing body of theoretical and experimental work has explored other exotic phases in TMDs^{17,18,19,20,21,22}. Furthermore, both integer and fractional Chern insulator states have been reported in moiré TMD^{23,24,25,26,27,28,29,30}, graphene^31,32,33 and graphene–boron nitride heterostructures^34,35,36,37.

Until now, nearly all moiré heterostructures have been based on twisting monolayers with triangular lattices and low-energy states near the Γ (refs. ^3,4) or K (refs. ^5,6,7,8,9) points, leading to systems with one or two valleys (in the two-valley case, time-reversal exchanges the valley). This work introduces a new family of moiré materials by twisting monolayers with triangular lattices and low-energy states around the M point of the BZ. These M-point moiré systems feature three time-reversal-preserving valleys related by C_3z rotation symmetry. Building on extensive ab initio calculations, we propose (among others³⁸) experimentally exfoliable twisted SnSe₂ and ZrS₂ as promising platforms for realizing M-point moiré heterostructures. We develop quantitative simplified models for these systems and perform a detailed analysis of the band structure, topology and charge density of the flat bands at the predicted small twist angles. We show analytically that M-point moiré Hamiltonians exhibit a new type of symmetry, termed momentum-space non-symmorphic^{39,40,41,42,43}. In crystallography, space groups are symmorphic or non-symmorphic, depending on whether they include symmetry operations that translate the origin by a fraction of the lattice vectors. Although in real space conventional crystalline groups can feature both symmorphic and non-symmorphic operations, in momentum space, all conventional crystalline groups exhibit only symmorphic operations. M-point moiré systems are the first experimentally realizable non-magnetic systems to exhibit momentum-space non-symmorphic symmetries, all without requiring an applied magnetic field in the range of thousands of Tesla^39,40,41. In a single valley, these non-symmorphic symmetries can render the system effectively one-dimensional at the single-particle level, making M-point moiré systems prime candidates for Luttinger-liquid simulators^16,44. With all three valleys considered, they can realize a multi-orbital triangular lattice Hubbard model (H. Hu et al., to be published), in which valley-spin local moments couple differently along the three C_3z-related directions, in a manner reminiscent of Kitaev’s honeycomb model⁴⁵.

M-point moiré models

For triangular monolayer lattices, the moiré lattice is also triangular, generated by the reciprocal lattice vectors $b_{M_{1}}$ and $b_{M_{2}}$ (see Supplementary Information Section IV). These vectors span the moiré reciprocal lattice $Q = Z b_{M_{1}} + Z b_{M_{2}}$ , as depicted in Fig. 1a. In general, the single-particle Hamiltonians of moiré systems take the form of a hopping model in momentum space. This arises because the moiré potential breaks the monolayer translation symmetry and couples momentum states that are connected by reciprocal moiré vectors. The single-particle moiré Hamiltonian can be written $H = \sum_{k, Q, Q^{'}, i, j} {[h_{Q, Q^{'}} (k)]}_{i j} {\hat{c}}_{k, Q, i}^{†} {\hat{c}}_{k, Q^{'}, j}$ , in which ${\hat{c}}_{k, Q, i}^{†}$ denotes the moiré plane-wave operators at moiré momentum k, and i denotes a combined index comprising orbital, spin, valley, layer or other further degrees of freedom.

**Fig. 1: Momentum-space Q-lattices for twisted triangular lattice monolayers.**

When the low-energy fermions of the monolayer are located at the Γ point^3,4, the operators ${\hat{c}}_{k, Q, i}^{†}$ carry total momentum k − Q and the Q-vectors lie on the triangular lattice shown in Fig. 1a. In the case of a monolayer with low-energy states located at the K point^5,6,7, the moiré fermions carry an extra valley index η = ±, in which the moiré Hamiltonian is diagonal. The Q-vectors form a honeycomb lattice, as illustrated in Fig. 1b. The moiré and monolayer operators are related by ${\hat{c}}_{k, Q, η, i}^{†} = {\hat{a}}_{η K_{K}^{l} + k - Q, l, i}^{†}$ for $Q \in Q_{η l}^{'}$ , in which ${\hat{a}}_{p, l, i}^{†}$ represents the monolayer operators from layer l = ± at momentum p and $K_{K}^{l}$ is the K-point momentum of layer l.

Distinctly, in M-point moiré materials, the Q-vectors form a kagome lattice, as shown in Fig. 1c. To be specific, the moiré operators in layer l—which, for the present case, include only an extra spin s = ↑,↓ index—are related to the monolayer ones according to ${\hat{c}}_{k, Q, s, l}^{†} = {\hat{c}}_{C_{3 z}^{η} K_{M}^{l} + k - Q, s, l}^{†}$ , for $Q \in Q_{η + l}$ , in which $K_{M}^{l}$ is the momentum of the monolayer M point. The three C_3z-related valleys indexed by η = 0, 1 and 2 are implicitly encoded by the kagome sublattice to which Q belongs: the valley-η fermions are supported on the $Q_{η \pm 1}$ sublattices (in which η + l is taken modulo 3), as derived in Supplementary Information Section IV. As we will show, the kagome Q-lattice leads to substantially different properties of M-point moiré materials.

Materials realizations

We now turn to 1T-SnSe₂ and 1T-ZrS₂ as experimentally exfoliable monolayers for realizing M-point moiré heterostructures (see Supplementary Information Section II). The monolayer crystal structure of both materials is shown in Fig. 2a,b and belongs to the $P \bar{3} m 1 1^{'}$ group, which is generated by translations, C_3z rotations, in-plane twofold rotations C_2x, inversion $I$ and time-reversal $T$ symmetries. The Sn (Zr) atoms form a triangular lattice, with the Se (S) atoms being located at the other C_3z-invariant Wyckoff positions above and below the Sn (Zr) plane. The ab initio band structures of monolayer SnSe₂ and ZrS₂ shown in Fig. 2c,d reveal two insulators for which the conduction band minimum is located at the M point. The first isolated Kramers-degenerate conduction band of SnSe₂ is atomic, being spanned by an effective s-like molecular orbital centred on the Sn atom. For ZrS₂, the low-energy M-point states are contributed primarily by the $d_{z^{2}}$ orbitals of Zr.

**Fig. 2: Exfoliable monolayers for M-point moiré materials.**

Moiré Hamiltonians

Because the SnSe₂ and ZrS₂ monolayers lack twofold out-of-plane rotation symmetry (C_2z), there are two distinct ways to stack and subsequently twist them by an angle θ to achieve a large-scale moiré periodicity. In the so-called AA-stacking configuration, the top (l = +1) and bottom (l = −1) layers are stacked directly on top of each other and then twisted by the layer-dependent angle $\frac{l θ}{2}$ . By contrast, for AB-stacking, the bottom layer is first rotated by 180° around the $\hat{z}$ axis, before applying the $\frac{l θ}{2}$ twist. As discussed in Supplementary Information Section III, the two configurations have different crystalline symmetries. Although both stackings feature C_3z and $T$ symmetries, they differ in the direction of the in-plane twofold rotation symmetry: the AA (AB)-stacking arrangement has C_2x (C_2y) symmetry.

We perform large-scale ab initio calculations (which include relaxation effects) at commensurate twist angles 13.17° ≥ θ ≥ 3.89° (see Methods and Supplementary Information Sections III and VIII) and construct two types of moiré Hamiltonian model for each angle and stacking configuration according to the method outlined in Supplementary Information Section IX. The first is a numerically exact model, which accurately reproduces a large set of spinful bands (at least the first five in each valley) in both energy and wavefunction. The second is an analytical approximate continuum model capturing the dispersion and wavefunction of the first or first two (depending on the angle) lowest-energy spinful gapped bands (and, qualitatively, the higher-energy spectrum) in each valley. The comprehensive results at all angles are presented in Supplementary Information Section XI. Unlike the case of Γ-point or K-point twisting, ab initio simulations are crucial for obtaining even the correct qualitative moiré Hamiltonian. The two-centred first-monolayer harmonic approximation incorrectly predicts continuous translation symmetry along one direction (for example, along the $C_{3 z}^{η} \hat{y}$ direction in valley η) and an overall gapless spectrum, as shown in Supplementary Information Section VI.

Figure 3 summarizes the ab initio results for twisted AA-stacked and AB-stacked SnSe₂ and ZrS₂ at low twist angle. Both stacking configurations exhibit approximate spin SU(2) symmetry (see Supplementary Information Section IX) and feature two sets of spinful gapped bands in each of the three C_3z-related valleys, as shown in Fig. 3a–d. The lowest-energy set of bands has a narrow bandwidth of around 10 meV. The charge density distribution (CDD) for the lowest two bands in valley η = 0, shown in Fig. 3e–h, reveals that these moiré systems have approximate spatial symmetries beyond the exact valley-preserving C_2x and C_2y symmetries expected in the AA-stacked and AB-stacked configurations, respectively. For instance, the CDD of the first set of spinful bands in AA-stacked SnSe₂, as well as the first two sets of bands in twisted ZrS₂, feature an approximate twofold rotation symmetry (the second set of spinful bands in AA-stacked SnSe₂ exhibits this symmetry to a lesser extent). In the AB-stacked configuration, the centre of the approximate C_2z symmetry aligns with the unit cell origin, whereas in the AA-stacked case, the effective ${\tilde{C}}_{2 z}$ rotation centre is shifted away from the unit cell origin and will be specified below. Moreover, the CDD suggests the presence of an approximate in-plane mirror symmetry, ${\tilde{M}}_{z}$ . These effective symmetries (whose origin is explained below and in Supplementary Information Section VB) prompt us to construct simplified analytical continuum models that can capture and explain these features.

**Fig. 3: Ab initio results for M-point moiré SnSe₂ and ZrS₂.**

In valley η = 0, the simplified M-point moiré Hamiltonian can be expressed as

(1)

in which m_x and m_y are the anisotropic effective masses of SnSe₂ and ZrS₂ (see Methods). As shown in Fig. 4a,b, the moiré potential takes the form of a hopping model on two of the three sublattices of the kagome M-point Q-lattice. Explicitly, the simplified Hermitian moiré potential tensor exhibits spin SU(2) symmetry and includes only interlayer terms, given by ${[T_{Q, Q^{'}}^{A A}]}_{l s; (- l) s} = (\pm i w_{1}^{A A} + w_{2}^{A A}) δ_{Q \pm q_{0}, Q^{'}} + w_{3}^{' A A} δ_{Q \pm (q_{1} - q_{2}), Q^{'}}$ and ${[T_{Q, Q^{'}}^{A B}]}_{l s; (- l) s} = w_{2}^{A B} δ_{Q \pm q_{0}, Q^{'}} + w_{4}^{' A B} δ_{Q \pm (q_{1} - q_{2}), Q^{'}}$ . The interlayer hopping parameters, obtained by fitting to the ab initio band structure, are listed in Methods. The band structure of the simplified model for AA-stacked SnSe₂ is shown in Fig. 4c, indicating excellent qualitative agreement with the ab initio results for such a small number of parameters. In the simplified models, for both SnSe₂ and ZrS₂, the overlap between the fitted and ab initio bands is larger than 95% (85%) with the first (second) set of spinful bands, as we show in Supplementary Information Section XI.

**Fig. 4: Analytical continuum M-point moiré models.**

Momentum-space non-symmorphic symmetries

The approximate symmetries inferred from the layer-resolved CDD of the M-point moiré Hamiltonian are exact symmetries in the simplified moiré models from equation (1) (see detailed discussion in Supplementary Information Section VI). Specifically, the centre of the effective twofold rotation symmetry ${\tilde{C}}_{2 z}$ for the AA-stacked Hamiltonian is located at $\frac{q_{η}}{| q_{η} |^{2}} \arg (i w_{1}^{A A} + w_{2}^{A A})$ in valley η. By contrast, the simplified AB-stacked moiré Hamiltonian exhibits C_2z symmetry, with its rotation centre aligned with the origin of the moiré unit cell. Because both models are effectively spinless (owing to atomistic arguments presented in Supplementary Information Section XI) and exhibit either ${\tilde{C}}_{2 z} T$ or $C_{2 z} T$ symmetry in each valley, the Berry curvature of any gapped set of bands is exactly zero. Consequently, the first two sets of bands of both the AA-stacked and AB-stacked moiré Hamiltonians are topological trivial and, hence, Wannierizable. This is also consistent (and the result of) the bands being flat and exhibiting a large (40 meV) gap from one another. However, the physics of these Hubbard (with interaction) bands is far from trivial in this system, as shown below.

Unlike the C_2z and ${\tilde{C}}_{2 z}$ symmetries, the effective mirror ${\tilde{M}}_{z}$ symmetry has an unconventional action on the momentum-space moiré fermions. Specifically, ${\tilde{M}}_{z}$ acts non-symmorphically in momentum space, with ${\tilde{M}}_{z} {\hat{c}}_{k, Q, s, l}^{†} {\tilde{M}}_{z}^{- 1} = {\hat{c}}_{k + q_{η}, Q + q_{η}, s, - l}^{†}$ for $Q \in Q_{η + l}$ . Because $q_{0} = \frac{b_{M_{1}}}{2}$ , the action of ${\tilde{M}}_{z}$ can only be made conventional by folding the moiré BZ along $q_{η}$ , which would break the moiré translation symmetry. The non-symmorphic action of the ${\tilde{M}}_{z}$ symmetry originates from the moiré fermions realizing a projective representation of the symmetry group of the system. Letting $T_{a_{M_{1, 2}}}^{'}$ denote the two moiré translation operators for valley η = 0 along the direct moiré lattice vectors $a_{M_{1, 2}}$ (with $a_{M_{i}} \cdot b_{M_{j}} = 2 π δ_{i j}$ ), we find that $[T_{a_{M_{2}}}^{'}, {\tilde{M}}_{z}] = {T_{a_{M_{1}}}^{'}, {\tilde{M}}_{z}} = 0$ (contrasting with a conventional mirror M_z symmetry, which would commute with both $T_{a_{M_{1}}}^{'}$ and $T_{a_{M_{2}}}^{'}$ ).

It is important to note that the effective ${\tilde{M}}_{z}$ symmetry is not accidental. In the AA-stacked case, it can be shown to hold exactly for arbitrary moiré harmonics within the local-stacking approximation⁴⁶. In the limit of vanishing twist angle (θ → 0), the moiré Hamiltonian can be constrained by the exact symmetries of the untwisted bilayer configuration. The inversion symmetry of the untwisted AA-stacked bilayer gives rise to the ${\tilde{M}}_{z}$ symmetry of the moiré Hamiltonian, as shown in Supplementary Information Sections VB and VI. In the AB-stacked case, the true in-plane mirror symmetry of the untwisted bilayer leads to an effective inversion symmetry $\tilde{I}$ of the corresponding moiré Hamiltonian, which also acts non-symmorphically in momentum space. In the simplified AB-stacked model, the approximate C_2z symmetry, combined with the $\tilde{I}$ symmetry, leads to an ${\tilde{M}}_{z} = C_{2 z} \tilde{I}$ symmetry of the system.

Projective fermion representations that realize momentum-space non-symmorphic symmetries have previously been proposed in magnetic systems⁴² or systems subjected to a large magnetic field (on the order of thousands of Tesla)^40,41,47. M-point moiré materials provide the first experimentally viable realization of these symmetries in any (that is, magnetic or non-magnetic) system. To better understand the origin of the momentum-space non-symmorphic action of the ${\tilde{M}}_{z}$ symmetry, we construct a simple one-dimensional tight-binding model that incorporates it. The resulting ladder model, shown in Fig. 5a, mimics the dispersion of an atomic band in the M-point moiré Hamiltonian for valley η = 0 along the $\hat{x}$ direction (see Supplementary Information Section VI). Each unit cell is threaded by a uniform perpendicular magnetic field, enclosing a π-flux. Because π-flux and (−π)-flux are equivalent, the model also respects time-reversal and ${\tilde{M}}_{z}$ symmetry. In the Fourier-transformed basis ${\hat{b}}_{k, l}^{†} = \frac{1}{\sqrt{N}} \sum_{n} {\hat{b}}_{n, l}^{†} e^{i k n}$ , the ${\tilde{M}}_{z}$ symmetry acts non-symmorphically as ${\tilde{M}}_{z} {\hat{b}}_{k, l}^{†} {\tilde{M}}_{z}^{- 1} = {\hat{b}}_{k + π, - l}^{†}$ , ensuring that the spectra of the Hamiltonian at k and k + π are identical, as shown in Fig. 5b.

**Fig. 5: Momentum-space non-symmorphic symmetries.**

Hubbard and Luttinger simulators

Within each valley, the first two sets of spinful bands in SnSe₂ and ZrS₂ bilayers are individually Wannierizable, with their bandwidths tunable by adjusting the twist angle. Given the excellent SU(2) symmetry, these M-point moiré systems become effective simulators of the Hubbard model when Coulomb interactions are included (H. Hu et al., to be published). However, owing to the extra valley degree of freedom, these systems go beyond the single-band U(2) Hubbard model, instead realizing a six-flavour U(2) × U(2) × U(2) Hubbard model.

Another key distinction from the standard Hubbard model can arise from the ${\tilde{M}}_{z}$ symmetry. In real space, ${\tilde{M}}_{z}$ does not change the position along the moiré heterostructure. As a result, the continuum moiré Hamiltonian can be made diagonal in the ${\tilde{M}}_{z}$ basis. Because ${(T_{a_{M_{1}}}^{'})}^{2} {(T_{a_{M_{2}}}^{'})}^{- 1}$ and $T_{a_{M_{2}}}^{'}$ both commute with ${\tilde{M}}_{z}$ , each mirror sector of valley η = 0 will feature reduced translation symmetry specified by the rectangular lattice vectors $2 a_{M_{1}} - a_{M_{2}}$ and $a_{M_{2}}$ . The $T_{a_{M_{1}}}^{'}$ operator anticommutes with ${\tilde{M}}_{z}$ , exchanging the two mirror sectors. The Wannier orbitals of any atomic band—such as the first conduction band of AA-stacked SnSe₂ from Fig. 5c—can therefore be split by their ${\tilde{M}}_{z}$ eigenvalues: the orbitals of each mirror sector are displaced by $a_{M_{1}}$ and form two interpenetrating rectangular lattices shown in Fig. 5d. Within each mirror sector and in valley η = 0, the interorbital separation is larger by a factor of $\sqrt{3}$ along the $\hat{x}$ direction compared with the $\hat{y}$ one. Provided that the Wannier orbital spread is approximately isotropic (as it happens for the first band of AA-stacked SnSe₂ but not in the first band of twisted ZrS₂), this will lead to reduced hopping along $\hat{x}$ compared with $\hat{y}$ (see Supplementary Information Section VI). As the tunnelling between ${\tilde{M}}_{z}$ sectors is forbidden, the system in each valley will behave quasi-one-dimensionally, with flatter dispersion along the $C_{3 z}^{η} \hat{x}$ direction, effectively emulating a Luttinger model. In the three-valley system, this quasi-one-dimensional behaviour causes the U(2) × U(2) × U(2) local moments to couple differently along three C_3z-related directions, similar (but not identical) to the couplings of the Kitaev model⁴⁵.

We note, however, that quasi-one-dimensionality along the $C_{3 z}^{η} \hat{y}$ direction (that is, flatter dispersion along the $C_{3 z}^{η} \hat{x}$ direction) is not an inherent or universal feature of M-point moiré materials. Instead, it is the presence of the effective ${\tilde{M}}_{z}$ symmetry, not previously identified, that plays a more general role. Together with approximately isotropic Wannier orbitals for the bands, the effective ${\tilde{M}}_{z}$ symmetry can enforce one-dimensional behaviour in the single-particle valley-projected moiré Hamiltonian. However, this symmetry is also compatible with two-dimensional physics in general (see Methods). For instance, because of the elongated Wannier orbitals, twisted ZrS₂ exhibits excellent effective ${\tilde{M}}_{z}$ symmetry, but its first set of conduction bands is not quasi-one-dimensional along the $C_{3 z}^{η} \hat{y}$ direction.

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