ARTICLE https://doi.org/10.1038/s41467-020-20802-z OPEN Excitonic density wave and spin-valley superfluid in bilayer transition metal dichalcogenide ✉ ✉ Zhen Bi 1,2 & Liang Fu1 Artificial moiré superlattices in 2d van der Waals heterostructures are a new venue for realizing and controlling correlated electronic phenomena. Recently, twisted bilayer WSe2 emerged as a new robust moiré system hosting a correlated insulator at moiré half-filling over 1234567890():,; a range of twist angle. In this work, we present a theory of this insulating state as an excitonic density wave due to intervalley electron–hole pairing. We show that exciton condensation is strongly enhanced by a van Hove singularity near the Fermi level. Our theory explains the remarkable sensitivity of the insulating gap to the vertical electric field. In contrast, the gap is weakly reduced by a perpendicular magnetic field, with quadratic dependence at low field. The different responses to electric and magnetic field can be understood in terms of pair- breaking versus non-pair-breaking effects in a BCS analog of the system. We further predict superfluid spin transport in this electrical insulator, which can be detected by optical spin injection and spatial-temporal imaging. 1 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA. 2 Department of Physics, the Pennsylvania State University, ✉ University Park, PA, USA. email: [email protected]; [email protected] NATURE COMMUNICATIONS | (2021) 12:642 | https://doi.org/10.1038/s41467-020-20802-z | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-20802-z new era of band engineering and novel electronic phase of moiré bands in bilayer TMD originating from K and K0 valleys Adesign is promised by the discovery of moiré superlattices are decoupled at the single-particle level and treated separately in 2d van der Waals heterostructures. Recent experiments hereafter. From the outset, it is important to distinguish moiré on graphene-based moiré systems1–13 unfolded a brand new systems generated by slightly twisting AA and AB stacking bilayer world of strongly interacting electron phases, including correlated TMDs—which differ by a 180° rotation of the top layer. The two insulating states, unconventional superconductivity as well as cases have very different moiré band structures due to spin-valley quantum anomalous hall states. Moiré superlattices based on locking in TMD. For the AA case, the K valleys on two layers with transition metal dichalcogenides (TMD)14–19 lately emerge as a the same spin polarization are nearly aligned, so that interlayer new host of novel electronic phases of matter. Unlike graphene, tunneling is allowed and creates layer-hybridized moiré bands, as the semiconducting TMDs possess a large bandgap and large shown in Fig. 1a. In contrast, for AB, the K valley on one layer is effective electron mass, as well as strong spin–orbit coupling. Due nearly aligned with the K0 valley on the other layer with the to these features, narrow moiré bands appear generically in het- opposite spin polarization, so that interlayer tunneling is for- erobilayer15 or twisted homobilayer TMDs14 without fine-tuning, bidden and the resulting moiré bands have additional spin/layer which provides a fertile ground for correlation physics. In addi- degeneracy14,42. Throughout this work, we consider the moiré tion, the spin-valley locking in TMDs20–22 offers unique oppor- system from twisting AA stacking bilayer TMD, where the full tunities to probe and manipulate electron spins with optical filling for the topmost moiré bands corresponds to two electrons methods17,23. per supercell. 18 In a very recent experiment on WSe2 homobilayers with twist We calculate the moiré band structures using the continuum angles θ ∈ [4°, 5.1°], a striking correlated insulating dome was model from ref. 14 with parameters extracted from first principle found at low temperature within a narrow range of the vertical calculations (see the section “Moiré band structure” in “Meth- electric field, when the relevant moiré band is half-filled. The ods”). Remarkably, we find that the moiré band dispersion is activation energy gap of the insulating state is ~3 meV, which is highly tunable by the displacement field, D. For a certain range of remarkably large among most moiré systems1–3,8–12. However, displacement field, our calculation (Fig. 1b, c) shows that the the insulating gap is smaller by an order of magnitude than either Fermi level at half-filling is close to a van Hove singularity (VHS) the miniband width (about 60−100 meV for θ = 4–5°) or the in the density of states, resulting from saddle points near the 2 0 characteristic Coulomb interaction energy e /ϵL (~30 meV for a corners of the mini-Brillouin zone, KM and KM. The proximity to dielectric constant ϵ = 10 at 4°). The smallness of the insulating VHS is supported by the observed sign change of Hall coefficient gap compared to the interaction energy and bandwidth and its near half-filling18. In the relevant parameter regime, the moiré sensitivity to the displacement field speak against the scenario of a band has a nontrivial Chern number. Details of the topological Mott insulator and call for a new theoretical understanding. properties of the moiré band are discussed in the section “Band In this work, we predict that the insulator in twisted TMD at topology and Berry curvature”. While playing an important role half-filling is an excitonic density wave formed by the pairing of in the flat-band scenario in graphene-based moiré systesms43–45, electrons and holes in minibands at different valleys. This the topological property does not appear to play a significant role electron–hole pairing is strongly enhanced by a van Hove sin- in the weak coupling scenario considered here. gularity24–31 (VHS) near the Fermi level. We show that the Since a diverging DOS near the Fermi level enhances detuning of the displacement field has a pair-breaking effect on correlation effects, the detuning of VHS by the displacement the excitonic insulator, while the Zeeman coupling to an out-of- field is expected to affect the metal–insulator transition. plane magnetic field is non-pair-breaking at leading order. We Motivated by this consideration, we now develop a low-energy introduce a low-energy theory for twisted TMD and calculate the theory by expanding the moiré band from valley K (K0) around 0 phase diagram as a function of the displacement field, tempera- KM (KM). Based on the lattice symmetries, the Taylor expansion ture, and magnetic field, finding good agreement with the up to the third order in momentum takes the general form experiment18. ε ðkÞ¼αk2 ξðkÞ; ξðkÞ¼ 3 À 2 ð Þ Remarkably, owning to spin-valley locking in TMD32,33, our ± ± ky 3kykx 1 excitonic insulator is also a spin superfluid and thus enables coherent spin transportation over long distances. The possibility of where ± is the valley index related by time-reversal symmetry T : ck;þ ! cÀk;À; ck;À !ÀcÀk;þ. spin supercurrent was initially theorized in magnetic insulators fi α fi with easy-plane anistropy34–38. Its signatures have been reported The coef cient depends on the displacement eld D (see “Methods” for more details). It is useful to first consider a critical in recent electrical measurements on quantum Hall state in gra- fi α = k = 39–41 displacement eld Dc where 0. Then, 0 becomes a phene and antiferromagnetic insulator Cr2O3 , where spin Hall effect is used to generate and detect a nonequilibrium spin monkey saddle point where three energy contour lines interact29,46, resulting in a high-order van Hove singularities accumulation. A key advantage of 2d TMDs is that the spin 27–29,46 polarization can be easily generated and detected by purely optical (hVHS) with a power-law-divergent density of states means. It has been shown that a circularly polarized light can = ρ ðEÞ1=jEj1 3; ð2Þ efficiently generate spin polarizations in TMDs due to the spin- α¼0 20–22 valley locking and valley-selective coupling to chiral photons . as shown in Fig. 1c. The local spin polarization can be read out by measuring the fl difference in re ectance of right- and left-circularly polarized Intervalley exciton order. At low carrier density, the dominant lights, i.e., the circular dichroism spectroscopy17,23. Therefore, interaction is Coulomb repulsionP R within the same valley and twisted TMDs provide an ideal platform for studying spin trans- 0 0 between two valleys: H ¼ drdr V ; ðr À r Þn ;rn ;r0 ; where port with a fully optical setup. We propose an all-optical setup for int i;j i j i j = ¼ y spin injection and spatial–temporal imaging of spin transport. i, j ± is valley index and ni;r ci;rci;r is density operator of a given valley. The interacting Hamiltonian is reminiscent of bilayer quantum Hall system47, where the layer degree of freedom Results plays the role of the valley. At half-filling of each layer, Coulomb Moiré band structure. We consider the electronic band structure repulsion leads to interlayer exciton condensation. However, of twisted homobilayer WSe2. At a small twist angle, the two sets unlike Landau levels, here the moiré band in twisted bilayer 2 NATURE COMMUNICATIONS | (2021) 12:642 | https://doi.org/10.1038/s41467-020-20802-z | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-020-20802-z ARTICLE Fig. 1 Moiré band structures. a The moiré band structure for one valley of twisted bilayer WSe2. At a critical displacement field Dc ≅ 35 meV, the KM becomes a saddle point of the dispersion. b The density of states (with various D near Dc) at twist angle θ = 4°.