🔬 condensed matter

Quantum spin Hall crystals at fractional filling of twisted MoTe $_2$

This paper predicts and classifies interaction-driven quantum spin Hall crystals (QSHCs) emerging at fractional fillings in twisted MoTe $_2$ , providing specific evidence for a 9-fold quasi-degenerate $\sqrt{3}\times\sqrt{3}$ charge-ordered state at $\nu = -8/3$ near a $5^\circ$ twist that exhibits distinct topological phases protected by time-reversal or valley symmetry.

Original authors: Xiaoyang Shen, Raul Perea-Causin, Christopher Ekman, Jiong-Hao Wang, Hui Liu, Emil J. Bergholtz

Published 2026-08-27

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CC BY 4.0

Original authors: Xiaoyang Shen, Raul Perea-Causin, Christopher Ekman, Jiong-Hao Wang, Hui Liu, Emil J. Bergholtz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern physics, scientists are constantly searching for materials that can conduct electricity without losing energy, a property known as superconductivity, or materials that can guide electric currents along their edges while blocking them in the center. One of the most promising platforms for discovering these exotic states is a class of materials called twisted bilayers. Imagine taking two sheets of a single-atom-thin crystal and stacking them on top of each other, but rotating one slightly so that the atoms do not line up perfectly. This misalignment creates a new, larger pattern across the surface, much like the interference pattern seen when two window screens are overlaid at a slight angle. This pattern, known as a moiré superlattice, acts as a new kind of trap for electrons, forcing them to interact with one another in ways that are impossible in ordinary materials. These interactions can give rise to strange new phases of matter, including states where the material behaves like an insulator in its interior but conducts electricity along its edges, a phenomenon known as the quantum spin Hall effect. For years, researchers have focused on how these materials behave when they are filled with a specific number of electrons, but a new question has emerged: what happens when the material is filled with a fraction of that number, and can the electrons arrange themselves into a crystal-like pattern while still maintaining these special conducting properties?

A team of researchers has now predicted the existence of a new state of matter that answers this question, describing a "quantum spin Hall crystal" that emerges in a specific material called twisted bilayer MoTe2. Using advanced computer simulations, the team found that at a particular fractional filling of electrons, specifically when the material is filled to minus eight-thirds of its capacity, the electrons spontaneously organize themselves into a repeating pattern. This is not just a simple arrangement; it is a complex crystal where the electrons break the natural symmetry of the underlying atomic lattice, creating a new, larger grid. Remarkably, despite this crystalline order, the material retains the ability to conduct electricity along its edges in a way that is protected by fundamental symmetries of nature. The researchers identified that this state is driven by the interactions between the electrons themselves, rather than by external magnetic fields or other forces. The simulations show that as the electrons settle into this new crystal structure, they form distinct groups based on their spin and valley properties, creating a state that is robust and stable under specific conditions.

The study reveals that this new crystal state is not a single, uniform phase but rather a family of closely related states that are nearly identical in energy. The researchers discovered that these states can be organized into a structure where some versions preserve a fundamental symmetry known as time-reversal symmetry, while others break it. In the versions that break this symmetry, the electrons spontaneously choose a direction, creating a magnetic-like order, yet they still maintain a special topological property that protects the edge currents. The team found that the most stable version of this crystal, which they calculated to be slightly lower in energy than its symmetric counterpart, involves a specific shift in the electron pattern between the two different layers of the material. This shift causes the electron density to stagger, reducing the repulsive force between the electrons and making the crystal more stable. The researchers also showed that this new state competes with other known phases, such as a different type of crystal where electrons in the two layers lock together in a coherent way, but their calculations suggest that the quantum spin Hall crystal becomes the preferred state when the interactions between electrons are tuned to a specific strength.

To reach these conclusions, the researchers built a detailed computer model of the twisted bilayer MoTe2, simulating the behavior of electrons as they moved through the complex landscape created by the twisted layers. They adjusted various parameters in their model, such as the angle of the twist and the strength of the electrical forces between the electrons, to see how the system responded. They found that at a twist angle of approximately five degrees, the electrons naturally formed a repeating pattern that expands the unit cell by a factor of √3 in both directions. This pattern, known as a √3 × √3 crystal, was found to host the unique quantum spin Hall properties. The team carefully analyzed the energy of different possible arrangements and found that the state they predicted was energetically favorable, meaning it would likely be the state the material naturally settles into. They also confirmed that this state is distinct from other known phases, such as those where the material simply becomes a metal or a standard insulator, by checking for specific signatures like the presence of edge currents and the absence of certain magnetic orders.

The implications of this work are significant for the ongoing search for new quantum materials. The researchers suggest that their predicted state could be detected in experiments by looking for specific signatures, such as the quantized flow of spin-polarized currents along the edges of the material, combined with the presence of the repeating charge pattern. They propose that techniques like scanning tunneling microscopy, which can image the arrangement of atoms and electrons on a surface, could be used to visualize the crystal structure, while transport measurements could reveal the unique conducting properties. The study also highlights the delicate balance between different types of order in these materials, showing how the competition between electron interactions can lead to the emergence of entirely new phases. By mapping out the conditions under which these states appear, the researchers provide a roadmap for experimentalists who are working to create and observe these exotic states in the laboratory.

While the findings are based on theoretical simulations rather than direct experimental observation, the confidence in the results is bolstered by the robustness of the calculations. The researchers tested their predictions against different system sizes and parameters to ensure that the results were not just a numerical artifact but a genuine physical phenomenon. They also ruled out other potential explanations, such as the idea that the material might simply be a collection of two independent layers or that the observed effects were due to a different type of magnetic ordering. The simulations consistently showed that the quantum spin Hall crystal is a distinct and stable phase that arises from the interplay of topology and electron correlations. The team acknowledges that future work will be needed to fully understand the collective behavior of these states and to explore whether similar phenomena occur at other fractional fillings or in different materials. However, the current study provides a clear and detailed picture of a new class of quantum matter, offering a fresh perspective on how electrons can organize themselves to create states that are both crystalline and topologically protected.

The discovery of this state adds a new chapter to the story of twisted bilayer materials, demonstrating that the rich physics of these systems extends far beyond the integer fillings that have been the focus of much recent research. It suggests that the landscape of quantum phases in these materials is even more diverse than previously thought, with new possibilities emerging at fractional fillings where the electrons are forced to share the available space in complex ways. The ability to predict and classify these states using theoretical models gives scientists a powerful tool for guiding experimental efforts, potentially leading to the discovery of materials with unprecedented properties. As the field of moiré materials continues to evolve, the insights gained from this work will likely play a crucial role in shaping our understanding of how topology and correlations can combine to create the next generation of quantum technologies. The researchers' work stands as a testament to the power of theoretical physics to illuminate the hidden possibilities within the quantum world, inviting further exploration into the intricate dance of electrons in these twisted crystals.

Technical Summary: Quantum Spin Hall Crystals at Fractional Filling of Twisted MoTe $_2$

Problem Statement
Moiré superlattices, particularly twisted bilayer MoTe $_2$ , serve as a versatile platform for correlated topological quantum matter. While the fractional quantum anomalous Hall effect (FQAHE) has been observed at small twist angles, these states typically involve spontaneous breaking of time-reversal symmetry (TRS) via valley polarization. Conversely, TRS-protected topological phases, such as the quantum spin Hall insulator (QSHI) at integer filling ( $\nu = -2$ ), have received less attention in the context of fractional fillings. A central open question is whether interaction-driven charge-ordered states at fractional fillings can support the quantum spin Hall effect (QSHE) while preserving or interplaying with TRS and discrete translation symmetries. Specifically, can a crystalline order emerge that gaps the Fermi surface to produce minibands with opposite topological characters in spin-valley sectors, thereby realizing a "Quantum Spin Hall Crystal" (QSHC)?

Methodology
The authors employ a combination of continuum modeling and self-consistent Hartree-Fock (SCHF) calculations to investigate twisted bilayer MoTe $_2$ at a fractional filling of $\nu = -8/3$ and twist angles near $5^\circ$ .

Modeling: The study starts from a non-interacting continuum model of twisted MoTe $_2$ . To analyze commensurate crystallization, the band structure is folded from the moiré Brillouin zone (mBZ) into the Brillouin zone of a $\sqrt{3} \times \sqrt{3}$ superlattice (crystal Brillouin zone, cBZ).
Interaction Treatment: The authors perform momentum-space SCHF calculations within the cBZ, retaining both $K$ and $K'$ valleys. They project onto active crystal minibands and tune parameters including twist angle ( $\theta$ ), dielectric constant ( $\epsilon$ ), and displacement field ( $V_z$ ).
Order Parameters: Crystalline orders are distinguished via order parameters $\Phi_\mu(Q)$ in valley space, distinguishing intravalley charge/spin channels ( $\Phi_0, \Phi_z$ ) and intervalley coherent (IVC) channels ( $\Phi_{x,y}$ ).
Topology and Symmetry Analysis: The topological nature of the phases is determined using the Fukui–Hatsugai–Suzuki method for Chern numbers and Wilson loop eigenphases. The classification of the QSHC family relies on analyzing the interplay between TRS, valley $U(1)_v$ symmetry, and the relative translation between $K$ and $K'$ valley crystals.
Validation: The study includes exact diagonalization (ED) on a one-band-per-valley model to verify that the ground state in the large-twist-angle regime is not valley-polarized, justifying the focus on TRS-related competing orders. Finite-size and band-cutoff convergence tests are performed to ensure the robustness of the energy splitting between candidate states.

Key Contributions and Results

Prediction of QSHCs: The authors predict the existence of interaction-driven Quantum Spin Hall Crystals (QSHCs) at $\nu = -8/3$ . These states are characterized by a $\sqrt{3} \times \sqrt{3}$ charge order that gaps the non-interacting Dirac cones at the cBZ corners ( $\kappa, \kappa'$ ).
Phase Diagram: In the large twist angle regime ( $\theta \in [4^\circ, 6^\circ]$ ), the authors identify four distinct phases:
1. QSHC: Dominated by the intravalley charge channel ( $\Phi_0$ ). It exhibits valley Chern numbers $C_K = -C_{K'} = 1$ , resulting in a spin-Chern number $C_s = 1$ . The Berry curvature concentrates at $\kappa$ and $\kappa'$ .
2. AFM $_z$ Crystal: A trivial crystal with $C_K = C_{K'} = 0$ and strong out-of-plane Néel-like spin order ( $\Phi_z$ ).
3. IVC Crystal: A topologically trivial intervalley-coherent state characterized by finite $\Phi_{x,y}$ and a finite-momentum in-plane spin texture.
4. Gapless Metal: A regime where the interaction-driven gap closes.
  The QSHC occupies an intermediate window ( $\theta \simeq 5\text{--}5.5^\circ$ ). The transition between the IVC crystal and QSHC is driven by the competition between intervalley and intravalley channels; reducing the effective interaction strength (increasing $\epsilon$ ) suppresses the intervalley channel, stabilizing the QSHC.
Z $_3$ Classification of QSHC Families: A major theoretical contribution is the classification of QSHC states based on the relative displacement between the $K$ and $K'$ valley crystals.
- The $\sqrt{3} \times \sqrt{3}$ order generates 9 quasi-degenerate states ( $3 \times 3$ ).
- These states are organized into a $Z_3$ structure labeled by an index $\delta \in \{0, \pm 1\}$ , representing the relative shift of the $K'$ crystal with respect to the $K$ crystal.
- $\delta = 0$ (Singlet): Preserves TRS. Characterized by a non-trivial $Z_2$ topological invariant.
- $\delta = \pm 1$ (Doublet): Spontaneously breaks TRS but preserves valley $U(1)_v$ . Characterized by an integer spin-Chern number $C_s = 1$ . These two states form a degenerate doublet exchanged by TRS.
- The intervalley Hartree interaction lifts the degeneracy between these sectors. The $\delta = \pm 1$ states are slightly energetically favored over $\delta = 0$ because the relative translation staggers the charge density modulations, reducing intervalley Coulomb repulsion. The energy difference is small ( $\sim 10^{-3}$ meV) but robust.
General Classification Framework: The paper extends this $Z_3$ structure to a general classification of QSHC families for arbitrary commensurate superlattices. The relative-shift sectors are labeled by the group quotient $D = L/L_s$ . The symmetry protecting the topology (TRS, $U(1)_v$ , or magnetic TRS) depends on whether the sector is self-conjugate or part of a TRS-related pair. Table I in the paper summarizes these families for various crystal patterns (e.g., $\sqrt{3} \times \sqrt{3}$ , period-2 stripes, $2 \times 2$ ).

Significance and Claims
The paper claims to establish a new class of topological matter: interaction-driven Quantum Spin Hall Crystals. Unlike the QSHI at integer filling, the QSHC arises from the interplay of correlations and spontaneous symmetry breaking at fractional filling. The work provides a unified framework for understanding how crystal order and topology intertwine, specifically showing that QSHCs cannot be described as a trivial stack of two independent Quantum Anomalous Hall Crystals (QAHCs) but are bound by intervalley interactions into a single spin-Hall crystal with a rich $Z_3$ internal structure.

The authors assert that their findings offer a unified framework for crystal phases with quantum spin Hall character. They propose that the QSHC phase at $\nu = -8/3$ is distinguishable experimentally by the combination of a quantized spin Hall response and $\sqrt{3} \times \sqrt{3}$ charge order, detectable via transport or scanning probe microscopy. The paper modestly notes that while SCHF captures the crystal order, it cannot capture fractionalized topological orders; however, the stability of the QSHC against competing valley-polarized states is supported by exact diagonalization results showing a non-polarized ground state in the relevant regime. The near-degeneracy of the $Z_3$ family invites future study of collective excitations and potential transitions between sectors.

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