The valence band maxima of most group-VI transition metal dichalcogenide thin films remain at the $Γ$-point all the way from bulk to bilayer. In this paper we develop a continuum theory of the moirè minibands that are formed in the valence bands of $Γ$-valley homobilayers by a small relative twist. Our effective theory is benchmarked against large-scale ab initio electronic structure calculations that account for lattice relaxation. As a consequence of an emergent $D_6$ symmetry we find that low-energy $Γ$-valley moirè holes differ qualitatively from their K-valley counterparts addressed previously; in energetic order the first three bands realize i) a single-orbital model on a honeycomb lattice, ii) a two-orbital model on a honeycomb lattice, and iii) a single-orbital model on a kagome lattice.
Transition metal dichalcogenide (TMD) bilayers have recently emerged as a robust and tunable moiré system for studying and designing correlated electron physics. In this work, by combining large-scale first principle calculation and continuum model approach, we provide an electronic structure theory that maps long-period heterobilayer TMD superlattices onto diatomic crystals with cations and anions. We find that the interplay between moiré potential and Coulomb interaction leads to filling-dependent charge transfer between MM and MX regions several nanometers apart. We show that the insulating…
Small-twist-angle transition metal dichalcogenide (TMD) heterobilayers develop isolated flat moiré bands that are approximately described by triangular lattice generalized Hubbard models [PhysRevLett.121.026402]. In this article we explore the metallic and insulating states that appear under different control conditions at a density of one-electron per moiré period, and the transitions between them. By combining fully self-consistent Hartree-Fock theory calculations with strong-coupling expansions around the atomic limit, we identify four different magnetic states and one nonmagnetic state nea…
Spontaneous order of layer pseudospins in two-dimensional bilayers is common in quantum Hall systems, where it is responsible for hysteretic responses to gate fields in states with Ising order, and giant drag voltages in states with XY (spontaneous inter-layer phase coherence) order. In this article we predict that layer pseudospin order will also occur in double-moiré strongly correlated two-dimensional electron systems. We comment on similarities and differences in the competition between the two types of order in quantum Hall and double-moiré systems, and relate our findings to previous wor…
Moir{é} superlattice realized in two-dimensional heterostructures offers an exciting platform to access strongly-correlated electronic states. In this work, we study transition metal dichalcogenides (TMD) Moir{é} superlattices with time-reversal-symmetry and nontrivial spin{/valley}-Chern numbers. Utilizing realistic material parameters and the method of exact diagonalization, we find that at a certain twisting angle and fractional filling, gapped fractional topological states, i.e., fractional Chern insulators, are naturally {stabilized} by simply introducing the Coulomb repulsion. In contras…
Chern-Textured Exciton Insulators with Valley Spiral Order in Moiré Materials
Rudolf Peierls Centre for Theoretical Physics（鲁道夫·皮尔斯理论物理中心）；Princeton Center for Theoretical Science, Princeton University（普林斯顿大学理论科学中心）；Department of Physics, University of Zurich（苏黎世大学物理系）；Department of Physics, Ghent University（根特大学物理系）
We show that the "dark condensates" that arise when excitons form a Bose-Einstein condensate in a material with an indirect bandgap are not completely dark to optical emission. Rather, such states are "leaky condensates" in which optical emission is facilitated by many-body interactions. We analyze the properties of these leaky condensates in the context of twisted bilayers of transition metal dichalcogenides, which host strongly interacting excitons and an indirect bandgap. We show that this interaction-driven "leaky" emission dominates photoluminescence at low temperatures, with distinctive …
We present a comprehensive dynamical mean field study of the triangular lattice moiré Hubbard model, which is believed to represent the physics of moiré bilayer transition metal dichalcogenides. In these materials, important aspects of the band structure including the bandwidth and the order and location of van Hove singularities can be tuned by varying the interlayer potential. We present a magnetic and metal-insulator phase diagram and a detailed study of the dependence of the resistivity on temperature, band filling and interlayer potential. We find that transport displays Fermi liquid, str…
Moiré heterobilayer transition metal dichalcogenides (TMDs) emerge as an ideal system for simulating the single-band Hubbard model and interesting correlated phases have been observed in these systems. Nevertheless, the moiré bands in heterobilayer TMDs were believed to be topologically trivial. Recently, it was reported that both a quantum valley Hall insulating state at filling $ν=2$ (two holes per moiré unit cell) and a valley-polarized quantum anomalous Hall state at filling $ν=1$ were observed in AB stacked moiré MoTe$_2$/WSe$_2$ heterobilayers. However, how the topologically nontrivial s…
Twisting moiré heterostructures to the flatband regime allows for the formation of strongly correlated quantum states, since the dramatic reduction of the bandwidth can cause the residual electronic interactions to set the principal energy scale. An effective description for such correlated moiré heterostructures, derived in the strong-coupling limit at integer filling, generically leads to spin-valley Heisenberg models. Here we explore the emergence and stability of spin liquid behavior in an SU(2)$^{\mathrm{spin}}\otimes$SU(2)$^{\mathrm{valley}}$ Heisenberg model upon inclusion of Hund's-ind…
Valley polarization of moiré interlayer exciton complexes driven by many-body interactions
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；National Institute for Materials Science（物质材料研究机构）；Research Center for Electronic and Optical Materials, National Institute for Materials Science（物质材料研究机构电子光学材料研究中心）；Research Center for Materials Nanoarchitectonics, National Institute for Materials Science（物质材料研究机构材料纳米架构研究中心）
Recent experiments have observed correlated insulating and possible superconducting phases in twisted homobilayer transition metal dichalcogenides (TMDs). Besides the spin-valley locked moire bands due to the intrinsic Ising spin-orbit coupling, homobilayer moire TMDs also possess either logarithmic or power-law divergent Van Hove singularities (VHS) near the Fermi surface, controllable by an external displacement field. The former and the latter are dubbed conventional and higher-order VHS, respectively. Here, we perform a perturbative renormalization group (RG) analysis to unbiasedly study t…
Exotic correlated insulating phases emerge in the ABC-stacked trilayer graphene-boron nitride Moire super-lattice at both quarter and half-filling. A single-band minimal model with valley contrasting staggered-flux is proposed to capture the relevant band structure, where the conspiracy of perfect Fermi-surface nesting and van Hove singularity strongly enhance the valley fluctuation, leading to inter-valley spiral (IVS) order at half filling. In this paper, we consider a strong coupling U(1)$_{v}\times $SU(2)$_{s}$ symmetric spin-valley model to obtain the correlated insulating state and the p…
One salient feature of systems with Moiré superlattice is that, Chern number of "minibands" originating from each valley of the original graphene Brillouin zone becomes a well-defined quantized number because the miniband from each valley can be isolated from the rest of the spectrum due to the Moiré potential. Then a Moiré system with a well-defined valley Chern number can become a nonchiral topological insulator with $U(1) \times Z_3 $ symmetry and a $\mathbb{Z}$ classification at the free fermion level. Here we demonstrate that the strongly interacting nature of the Moiré system reduces the…
Twisted van der Waals heterostructures have recently been proposed as a condensed-matter platform for realizing controllable quantum models due to the low-energy moiré bands with specific charge distributions in moiré superlattices. Here, combining angle-resolved photoemission spectroscopy with sub-micron spatial resolution (μ-ARPES) and scanning tunneling microscopy (STM), we performed a systematic investigation on the electronic structure of 5.1° twisted bilayer WSe2 that hosts correlated insulating and zero-resistance states. Interestingly, contrary to one's expectation, moiré bands were ob…
Moiré superlattices formed in two-dimensional semiconductor heterobilayers provide a new realization of Hubbard model physics in which the number of electrons per effective atom can be tuned at will. We report on an exact diagonalization study of the electronic properties of half-filled narrow moiré bands in which correlation strengths are varied by changing twist angles or interaction strengths. We construct a phase diagram for the bilayer, identifying where the metal-insulator phase transition occurs, estimating the sizes of the charge gaps in the insulating phase, and commenting on the natu…
We theoretically consider two-dimensional moiré transition metal dichalcogenide (TMD) bilayers, which are strongly correlated in the sense that the on-site Coulomb interaction is comparable to or larger than the hopping kinetic energy between the moiré lattice sites. The system accommodates many symmetry-broken ground states both in charge and isospin sectors at various commensurate rational fillings such as 1/2, 1/3, 1/4, 2/3, etc. We investigate two complementary important aspects of the dependence of the symmetry breaking on (1) the range of the electron-electron interaction, which can in p…
Moiré superlattice systems such as transition metal dichalcogenide heterobilayers have garnered significant recent interest due to their promising utility as tunable solid state simulators. Recent experiments on a WSe$_2$/WS$_2$ heterobilayer detected incompressible charge ordered states that one can view as generalized Wigner crystals. The tunability of the hetero-TMD Moiré system presents an opportunity to study the rich set of possible phases upon melting these charge-ordered states. Here we use Monte Carlo simulations to study these intermediate phases in between incompressible charge-orde…
We compute the valley/magnetic phase diagram of mono layers of transition metal dichalcogenides in the hole doped region where spin-orbit effects are particularly relevant. Taking into account the moderate to high local electron-electron interactions due to the presence of transition metal atoms, we show that the system is unstable to an itinerant ferromagnetic phase where all charge carriers are spin and valley polarized. This phase shows an anomalous charge Hall and anomalous spin-Hall response, and may thus be detected experimentally.
Twisted bilayer transition metal dichalcogenides have emerged as important model systems for the investigation of correlated electron physics because their interaction strength, carrier concentration, band structure, and inversion symmetry breaking are controllable by device fabrication, twist angle, and most importantly, gate voltage, which can be varied in situ. The low energy physics of some of these materials has been shown to be described by a "moiré Hubbard model" generalized from the usual Hubbard model by the addition of strong, tunable spin orbit coupling and inversion symmetry breaki…
The long wavelength moiré superlattices in twisted 2D structures have emerged as a highly tunable platform for strongly correlated electron physics. We study the moiré bands in twisted transition metal dichalcogenide homobilayers, focusing on WSe$_2$, at small twist angles using a combination of first principles density functional theory, continuum modeling, and Hartree-Fock approximation. We reveal the rich physics at small twist angles $θ<4^\circ$, and identify a particular magic angle at which the top valence moiré band achieves almost perfect flatness. In the vicinity of this magic angle, …
The interplay between strong correlations and non-trivial topology in twisted moiré systems can give rise to a rich landscape of ordered states that intertwine the spin, valley, and charge degrees of freedom. In this paper, we investigate the properties of a system that displays long-range valley-polarized nematic order. Besides breaking the threefold rotational symmetry of the triangular moiré superlattice, this type of order also breaks twofold rotational and time-reversal symmetries, which leads to interesting properties. First, we develop a phenomenological model to describe the onset of t…
The observation of strongly-correlated states in moiré systems has renewed the conceptual interest in magnetic systems with higher SU(4) spin symmetry, e.g. to describe Mott insulators where the local moments are coupled spin-valley degrees of freedom. Here, we discuss a numerical renormalization group scheme to explore the formation of spin-valley ordered and unconventional spin-valley liquid states at zero temperature based on a pseudo-fermion representation. Our generalization of the conventional pseudo-fermion functional renormalization group approach for $\mathfrak{su}$(2) spins is capabl…
We introduce and study a minimum two-orbital Hubbard model on a triangular lattice, which captures the key features of both the trilayer ABC-stacked graphene-boron nitride heterostructure and twisted transition metal dichalcogenides in a broad parameter range. Our model comprises first- and second-nearest neighbor hoppings with valley-contrasting flux that accounts for trigonal warping in the band structure. For the strong-coupling regime with one electron per site, we derive a spin-orbital exchange Hamiltonian and find the semiclassical ground state to be a spin-valley density wave. We show t…
Moiré materials provide fertile ground for the correlated and topological quantum phenomena. Among them, the quantum anomalous Hall (QAH) effect, in which the Hall resistance is quantized even under zero magnetic field, is a direct manifestation of the intrinsic topological properties of a material and an appealing attribute for low-power electronics applications. The QAH effect has been observed in both graphene and transition metal dichalcogenide (TMD) moiré materials. It is thought to arise from the interaction-driven valley polarization of the narrow moiré bands. Here, we show surprisingly…
By enabling control of valley degrees of freedom in transition-metal dichalcogenides, valley-selective circular dichroism has become a key concept in valleytronics. In this manuscript, we show that valley excitons -- bound electron-hole pairs formed at either the K or K' valleys upon absorption of circularly-polarized light -- are chiral quasiparticles characterized by a finite orbital angular momentum (OAM). We further formulate an ab-initio many-body theory of valley-selective circular dichroism and valley excitons based on the Bethe-Salpeter equation. Beside governing the interaction with c…
Experimental demonstrations of tunable correlation effects in magic-angle twisted bilayer graphene have put two-dimensional moiré quantum materials at the forefront of condensed-matter research. Other twisted few-layer graphitic structures, boron-nitride, and homo- or hetero-stacks of transition metal dichalcogenides (TMDs) have further enriched the opportunities for analysis and utilization of correlations in these systems. Recently, within the latter material class, strong spin-orbit coupling or excitonic physics were experimentally explored. The observation of a Mott insulating state and ot…
Motivated by the recent observation of evidences of ferromagnetism in correlated insulating states in systems with Moiré superlattices, we study a two-orbital quantum antiferromagnetic model on the triangular lattice, where the two orbitals physically correspond to the two valleys of the original graphene sheet. For simplicity this model has a SU(2)$^s$$\otimes$SU(2)$^v$ symmetry, where the two SU(2) symmetries correspond to the rotation within the spin and valley space respectively. Through analytical argument, Schwinger boson analysis and also DMRG simulation, we find that even though all th…
Dynamic Moiré Potentials and Robust Wigner Crystallization in Large-Scale Twisted Transition Metal Dichalcogenides
Hefei National Research Center for Physical Sciences at the Microscale, and Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家物理科学（微观）研究中心和合肥国家实验室）；Graduate School of Science and Technology, Keio University（庆应义塾大学理工学部研究生院）；Keio University Sustainable Quantum Artificial Intelligence Center (KSQAIC), Keio University（庆应义塾大学可持续量子人工智能中心）；State Key Laboratory of Precision and Intelligent Chemistry, Department of Chemical Physics, and Anhui Center for Applied Mathematics, University of Science and Technology of China（中国科学技术大学精密与智能化学国家重点实验室、化学物理系和安徽应用数学中心）
We use a variational technique to study the many-body phase diagram of electrons and holes in $n$-doped and $p$-doped monolayer transition metal dichalcogenides (TMDs). We find a total of four different phases. $i$) A fully spin polarized and valley polarized ferromagnetic state. $ii$) A state with no global spin polarization but with spin polarization in each valley separately, i.e. spin-valley locking. $iii$) A state with spin polarization in one of the valleys and little to no spin polarization in the other valley. $iv$) A paramagnetic state with no valley polarization. These phases are sep…
We introduce a new approach for the correlation energy of one- and two-valley two-dimensional electron gas (2DEG) systems. Our approach is based on a random phase approximation at high densities and a classical approach at low densities, with interpolation between the two limits. This approach gives excellent agreement with available Quantum Monte Carlo (QMC) calculations. We employ the two-valley 2DEG model to describe the electron correlations in monolayer transition metal dichalcogenides (TMDs). The zero-temperature transition from a Fermi liquid to a quantum Wigner crystal phase in monolay…
While transition metal dichalcogenide (TMD) based moire materials have been shown to host various correlated electronic phenomena, topological states have not been experimentally observed until now. In this work, using first principles calculations and continuum modeling, we reveal the displacement field induced topological moire bands in AB-stacked TMD heterobilayer MoTe2/WSe2. Valley contrasting Chern bands with non-trivial spin texture are formed from interlayer hybridization between MoTe2 and WSe2 bands of nominally opposite spins. Our study establishes a recipe for creating topological ba…
Tsientang Institute for Advanced Study（紫江高等研究院）；Sichuan Normal University（四川师范大学）；Max Planck Institute for the Structure and Dynamics of Matter（马克斯·普朗克物质结构与动力学研究所）；City University of Hong Kong（香港城市大学）；Songshan Lake Materials Laboratory（松山湖材料实验室）；Center for Computational Quantum Physics (CCQ), The Flatiron Institute（纽约联邦研究所计算量子物理中心）；Donostia International Physics Center (DIPC)（多诺斯蒂亚国际物理中心）；Liaoning University（辽宁大学）
Twisted multilayers of two-dimensional (2D) materials are an increasingly important platform for investigating quantum phases of matter, and in particular, strongly correlated electrons. The moiré pattern introduced by the relative twist between layers creates effective potentials of long-wavelength, leading to electron localization. However, in contrast to the abundance of 2D materials, few twisted heterostructures have been studied until now. Here we develop a first-principle continuum theory to study the electronic bands introduced by moire patterns of twisted Janus transition metal dichalc…
Recent experiment has shown that the ABC-stacked trilayer graphene-boron nitride Moire super-lattice at half-filling is a Mott insulator. Based on symmetry analysis and effective band structure calculation, we propose a valley-contrasting chiral tight-binding model with local Coulomb interaction to describe this Moire super-lattice system. By matching the positions of van Hove points in the low-energy effective bands, the valley-contrasting staggered flux per triangle is determined around $π/2$. When the valence band is half-filled, the Fermi surfaces are found to be perfectly nested between t…
Moire superlattices of transition metal dichalcogenide (TMD) bilayers have been shown to host correlated electronic states, which arises from the interplay of emergent moire potential and long-range Coulomb interactions. Here we theoretically investigate structural relaxation and single-particle electronic properties in moire superlattices of transition metal dichalcogenide homobilayer and study the ground state charge orders in the effective honeycomb lattice of MX and XM region. From the large-scale density functional theory calculation and continuum model with layer degrees of freedom, we f…
Moiré in $Γ$-valley square lattice: Copper- and iron-based superconductor simulation in a single device
International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science（物质纳米结构国际中心（WPI-MANA），国立材料研究所）；Department of Physics, Harvard University（哈佛大学物理系）；Department of Physics, University of Connecticut（康涅狄格大学物理系）；School of Physics, Georgia Institute of Technology（佐治亚理工学院物理学院）
