Dirac quantum criticality in twisted double bilayer transition metal dichalcogenides

TL;DR

Sets the stage for Dirac quantum criticality in ABBA-stacked twisted double bilayer TMDs at $ν=2$, showing a continuous DSM-to-AFM transition in the $(2+1)$-dimensional Gross-Neveu-Heisenberg universality class with emergent Lorentz invariance; a strong-coupling and self-consistent Hartree-Fock analysis maps angle- and pressure-tuned phase diagrams and reveals competition with ferromagnetism at small twist angles. The framework incorporates a realistic Γ-valley continuum model, long-range Coulomb interactions, uniaxial pressure, and residual heterostrain, providing quantitative predictions for critical twists, gaps, and scaling exponents. Finite heterostrain introduces a noninteracting gap and leads to a low-temperature Heisenberg universality crossover, outlining a rich quantum critical regime accessible in experiments on twisted double bilayer WSe$_2$. The work offers experimentally testable signatures, such as moiré-scale Néel order and specific scaling relations in magnetization, gaps, and dynamic spin structure factors, enabling direct exploration of fermionic quantum criticality in moiré materials.

Abstract

We investigate the phase diagram of moiré double bilayer transition metal dichalcogenides with ABBA stacking as a function of twist angle and applied pressure. At hole filling $ν= 2$ per moiré unit cell, the noninteracting system hosts a Dirac semimetal with graphene-like low-energy bands in the moiré Brillouin zone. At small twist angles, the Fermi velocity is reduced and interactions dominate the low-temperature behavior. A strong-coupling analysis identifies insulating ferromagnetic and antiferromagnetic ground-state candidates, characterized by spin-density modulations set by the moiré scale. Using a realistic continuum model with long-range Coulomb interactions, we perform self-consistent Hartree-Fock calculations to study the competition between these states. Varying the twist angle or pressure drives a transition from a Dirac semimetal to an antiferromagnetic insulator, which breaks SU(2) spin rotation and two-fold lattice rotation symmetries. Within a renormalization group analysis of the most general symmetry-allowed low-energy field theory, we show that this semimetal-to-insulator transition is continuous and belongs to the (2+1)D relativistic Gross-Neveu-Heisenberg universality class with $N = 2$ four-component Dirac fermions. Finite heterostrain, relevant in realistic samples, induces a crossover from Gross-Neveu-Heisenberg universality at intermediate temperatures to conventional (2+1)D Heisenberg criticality at the lowest temperatures. Further decreasing the twist angle can cause a level crossing from the antiferromagnetic insulator into a ferromagnetic insulator with spin-split bands. Our results provide a comprehensive theoretical framework that complements and elucidates recent experiments in twisted double bilayer WSe$_2$.

Figures (11)

• Figure 1: (a) Real-space view of the two inner layers of ABBA-stacked twisted double bilayer TMDs, highlighting the moiré unit cell (dashed hexagon). Locally, regions of approximate AA, AB, and BA stacking can be identified. The inset illustrates the out-of-plane structure: an AB-stacked bilayer atop a BA-stacked bilayer, with a small twist angle $\theta$ between them. (b) The Brillouin zones of the top (blue hexagon) and bottom (red hexagon) bilayers are rotated relative to each other by a twist angle $\theta$, producing a moiré Brillouin zone (small dashed hexagon) with high-symmetry points $\boldsymbol{\kappa}$ and $\boldsymbol{\kappa}'$ at its corners. Neighboring moiré Brillouin zones are connected by the moiré reciprocal lattice vectors $\mathbf{G}_j$.
• Figure 2: (a) Hartree-Fock energy $E_\mathrm{HF}$ of the antiferromagnetic (blue) and ferromagnetic (red) states relative to the noninteracting ground-state energy $E_0$ in the full model [Eq. \ref{['eq:full-model']}] for an effective permittivity $\epsilon_\mathrm{eff} = 110$, representative of realistic values for twisted double bilayer WSe$_2$. Calculations were performed on an $18 \times 18$ momentum-space grid, keeping two bands per spin species, with initial states preserving the respective symmetries of each candidate phase. At small twist angles, the two states are nearly degenerate, reflecting the approximate chiral symmetry, yet the antiferromagnetic state is ultimately favored across the entire range. (b) Same as (a) but for $\epsilon_\mathrm{eff} = 40$, enhancing the relative weight of the antichiral interaction compared with the kinetic contribution. In this case, the ferromagnetic state is favored at sufficiently small twist angles.
• Figure 3: (a) Charge density $n(\mathbf R)$ (grayscale) and spin density $\langle \hat{s}_z \rangle(\mathbf R)$ (overlaid color scale, opacity indicating magnitude) in the Dirac semimetal (DSM) ground state of the full Hamiltonian [Eq. \ref{['eq:full-model']}] at twist angle $\theta = 4^\circ$ and effective permittivity $\epsilon_\text{eff} = 110$, representative of realistic values for twisted double bilayer WSe$_2$. Light gray lines indicate the microscopic lattices of the inner TMD layers. The charge density forms an emergent honeycomb pattern on the moiré scale. Densities are normalized to the value $n_{\mathrm A}$ at the A site of the emergent honeycomb lattice. In the Dirac semimetal phase, the spin density vanishes, rendering the overlaid color scale transparent. (b) Same as (a), but for twist angle $\theta = 2^\circ$. The ground state is antiferromagnetic (AFM), with antiparallel spins on the two sites of the emergent honeycomb lattice. The moiré scale is enlarged compared to (a). (c) Same as (a), but for twist angle $\theta = 1^\circ$ and reduced effective permittivity $\epsilon_\text{eff} = 40$. The ground state is ferromagnetic (FM), with parallel spins on the two sites of the emergent honeycomb lattice. (d) Electronic spectrum at twist angle $\theta = 4^\circ$ and effective permittivity $\epsilon_\text{eff} = 110$, representative of realistic values for twisted double bilayer WSe$_2$. Dirac points at the moiré Brillouin zone corners $\boldsymbol{\kappa}$ and $\boldsymbol{\kappa}’$ remain gapless and spin degenerate. Dashed gray lines show the noninteracting bands for comparison. Energies are referenced to the Fermi level $\varepsilon_\mathrm{F}$ at filling $\nu = 2$ holes per moiré unit cell. (e) Same as (d), but for twist angle $\theta = 2^\circ$. In contrast to the Dirac semimetal, the spectrum is fully gapped while remaining spin-degenerate, realizing an antiferromagnetic insulator. (f) Same as (d), but for twist angle $\theta = 1^\circ$ and reduced effective permittivity $\epsilon_\text{eff} = 40$. The spectrum is fully gapped and spin-split, realizing a ferromagnetic insulator. The inset indicates the path through the moiré Brillouin zone along which the energy bands are shown.
• Figure 4: (a) Total and staggered magnetization densities $m_\mathrm{FM}$ and $m_\mathrm{AFM}$ at the centers of the AB and BA regions of the moiré lattice as functions of twist angle $\theta$ for fixed effective permittivity $\epsilon_\text{eff} = 110$, representative of realistic values for twisted double bilayer WSe$_2$. Gray points show $m_\mathrm{AFM}$ at finite system sizes $L = 12, 15, 18$, while blue points indicate the extrapolation to the thermodynamic limit. The results reveal a continuous quantum phase transition from an antiferromagnetic insulator for $\theta < \theta_\mathrm{c}$ to a symmetric Dirac semimetal for $\theta > \theta_\mathrm{c}$, with critical twist angle $\theta_\mathrm{c} \simeq 2.7^\circ$. (b) Same as (a), but for reduced effective permittivity $\epsilon_\text{eff} = 40$. At small twist angles, antiferromagnetic and ferromagnetic orders compete, leading to a strong first-order transition through a level crossing near $\theta \simeq 1.1^\circ$. Below this angle, the staggered magnetization vanishes and the ground state develops a finite net magnetization. (c) Quantum phase diagram as a function of twist angle $\theta$ and effective permittivity $\epsilon_\text{eff}$. Regions of antiferromagnetic (AFM) order, ferromagnetic (FM) order, and the symmetric Dirac semimetal (DSM) phase are identified. Insets indicate the low-energy band structure. The antiferromagnetic-to-ferromagnet transition is first-order, while the Dirac-semimetal-to-antiferromagnetic-insulator transition is continuous and belongs to the relativistic Gross-Neveu-Heisenberg universality class. Dashed gray lines indicate parameter cuts shown in (a) and (b).
• Figure 5: Staggered magnetization density $m_\mathrm{AFM}$ (blue) and interaction-induced gap $\Delta$ (orange) as functions of applied uniaxial pressure $p$ along the out-of-plane direction, for fixed twist angle $\theta = 2.75^\circ$ and effective permittivity $\epsilon_\mathrm{eff} = 110$. At low pressures, the ground state is a fully symmetric Dirac semimetal. Increasing pressure drives the system into an antiferromagnetic insulator. The continuous transition occurs at a critical pressure $p_\mathrm{c} \simeq 0.2\,\mathrm{GPa}$, with its precise value strongly dependent on the sample's twist angle.