Wigner molecular crystals from multielectron moiré artificial atoms.

Semiconductor moiré superlattices provide a versatile platform to engineer quantum solids composed of artificial atoms on moiré sites. Previous studies have mostly focused on the simplest correlated quantum solid-the Fermi-Hubbard model-in which intra-atom interactions are simplified to a single onsite repulsion energy U. Here we report the experimental observation of Wigner molecular crystals emerging from multielectron artificial atoms in twisted bilayer tungsten disulfide moiré superlattices. Using scanning tunneling microscopy, we demonstrate that Wigner molecules appear in multielectron artificial atoms when Coulomb interactions dominate. The array of Wigner molecules observed in a moiré superlattice comprises a crystalline phase of electrons: the Wigner molecular crystal, which is shown to be highly tunable through mechanical strain, moiré period, and carrier charge type.

Mapping twist-tuned multiband topology in bilayer WSe2.

Semiconductor moiré superlattices have been shown to host a wide array of interaction-driven ground states. However, twisted homobilayers have been difficult to study in the limit of large moiré wavelengths, where interactions are most dominant. In this study, we conducted local electronic compressibility measurements of twisted bilayer WSe2 (tWSe2) at small twist angles. We demonstrated multiple topological bands that host a series of Chern insulators at zero magnetic field near a "magic angle" around 1.23°. Using a locally applied electric field, we induced a topological quantum-phase transition at one hole per moiré unit cell. Our work establishes the topological phase diagram of a generalized Kane-Mele-Hubbard model in tWSe2, demonstrating a tunable platform for strongly correlated topological phases.

Magic-angle helical trilayer graphene.

We propose magic-angle helical trilayer graphene (HTG), a helical structure featuring identical rotation angles between three consecutive layers of graphene, as a unique and experimentally accessible platform for realizing exotic correlated topological states of matter. While nominally forming a supermoiré (or moiré-of-moiré) structure, we show that HTG locally relaxes into large regions of a periodic single-moiré structure realizing flat topological bands carrying nontrivial valley Chern number. These bands feature near-ideal quantum geometry and are isolated from remote bands by a very large energy gap, making HTG a promising platform for experimental realization of correlated topological states such as integer and fractional quantum anomalous Hall states.

Superconductivity and strong interactions in a tunable moiré quasicrystal.

Electronic states in quasicrystals generally preclude a Bloch description1, rendering them fascinating and enigmatic. Owing to their complexity and scarcity, quasicrystals are underexplored relative to periodic and amorphous structures. Here we introduce a new type of highly tunable quasicrystal easily assembled from periodic components. By twisting three layers of graphene with two different twist angles, we form two mutually incommensurate moiré patterns. In contrast to many common atomic-scale quasicrystals2,3, the quasiperiodicity in our system is defined on moiré length scales of several nanometres. This 'moiré quasicrystal' allows us to tune the chemical potential and thus the electronic system between a periodic-like regime at low energies and a strongly quasiperiodic regime at higher energies, the latter hosting a large density of weakly dispersing states. Notably, in the quasiperiodic regime, we observe superconductivity near a flavour-symmetry-breaking phase transition4,5, the latter indicative of the important role that electronic interactions play in that regime. The prevalence of interacting phenomena in future systems with in situ tunability is not only useful for the study of quasiperiodic systems but may also provide insights into electronic ordering in related periodic moiré crystals6-12. We anticipate that extending this platform to engineer quasicrystals by varying the number of layers and twist angles, and by using different two-dimensional components, will lead to a new family of quantum materials to investigate the properties of strongly interacting quasicrystals.

Tunable spin and valley excitations of correlated insulators in Γ-valley moiré bands.

Moiré superlattices formed from transition metal dichalcogenides support a variety of quantum electronic phases that are highly tunable using applied electromagnetic fields. While the valley degree of freedom affects optoelectronic properties in the constituent transition metal dichalcogenides, it has yet to be fully explored in moiré systems. Here we establish twisted double-bilayer WSe2 as an experimental platform to study electronic correlations within Γ-valley moiré bands. Through local and global electronic compressibility measurements, we identify charge-ordered phases at multiple integer and fractional moiré fillings. By measuring the magnetic field dependence of their energy gaps and the chemical potential upon doping, we reveal spin-polarized ground states with spin-polaron quasiparticle excitations. In addition, an applied displacement field induces a metal-insulator transition driven by tuning between Γ- and K-valley moiré bands. Our results demonstrate control over the spin and valley character of the correlated ground and excited states in this system.

Artificial Atoms, Wigner Molecules, and an Emergent Kagome Lattice in Semiconductor Moiré Superlattices.

Semiconductor moiré superlattices comprise an array of artificial atoms and provide a highly tunable platform for exploring novel electronic phases. We introduce a theoretical framework for studying moiré quantum matter that treats intra-moiré-atom interactions exactly and is controlled in the limit of large moiré period. We reveal an abundance of new physics arising from strong electron interactions when there are multiple electrons within a moiré unit cell. In particular, at filling factor n=3, the Coulomb interaction within each three-electron moiré atom leads to a three-lobed "Wigner molecule." When their size is comparable to the moiré period, the Wigner molecules form an emergent Kagome lattice. Our Letter identifies two universal length scales characterizing the kinetic and interaction energies in moiré materials and demonstrates a rich phase diagram due to their interplay.

Moiré Landau Fans and Magic Zeros.

We study the energy spectrum of moiré systems under a uniform magnetic field. The superlattice potential generally broadens Landau levels into Chern bands with finite bandwidth. However, we find that these Chern bands become flat at a discrete set of magnetic fields which we dub "magic zeros." The flat band subspace is generally different from the Landau level subspace in the absence of the moiré superlattice. By developing a semiclassical quantization method and taking account of superlattice induced Bragg reflection, we prove that magic zeros arise from the simultaneous quantization of two distinct k-space orbits. For instance, we show the chiral model of TBG has flat bands at special fields for any twist angle in the nth Landau level for |n|>1. The flat bands at magic zeros provide a new setting for exploring crystalline fractional quantum Hall physics.

One-dimensional Luttinger liquids in a two-dimensional moiré lattice.

The Luttinger liquid (LL) model of one-dimensional (1D) electronic systems provides a powerful tool for understanding strongly correlated physics, including phenomena such as spin-charge separation1. Substantial theoretical efforts have attempted to extend the LL phenomenology to two dimensions, especially in models of closely packed arrays of 1D quantum wires2-13, each being described as a LL. Such coupled-wire models have been successfully used to construct two-dimensional (2D) anisotropic non-Fermi liquids2-6, quantum Hall states7-9, topological phases10,11 and quantum spin liquids12,13. However, an experimental demonstration of high-quality arrays of 1D LLs suitable for realizing these models remains absent. Here we report the experimental realization of 2D arrays of 1D LLs with crystalline quality in a moiré superlattice made of twisted bilayer tungsten ditelluride (tWTe2). Originating from the anisotropic lattice of the monolayer, the moiré pattern of tWTe2 hosts identical, parallel 1D electronic channels, separated by a fixed nanoscale distance, which is tuneable by the interlayer twist angle. At a twist angle of approximately 5 degrees, we find that hole-doped tWTe2 exhibits exceptionally large transport anisotropy with a resistance ratio of around 1,000 between two orthogonal in-plane directions. The across-wire conductance exhibits power-law scaling behaviours, consistent with the formation of a 2D anisotropic phase that resembles an array of LLs. Our results open the door for realizing a variety of correlated and topological quantum phases based on coupled-wire models and LL physics.

Quantum anomalous Hall effect from intertwined moiré bands.

Electron correlation and topology are two central threads of modern condensed matter physics. Semiconductor moiré materials provide a highly tuneable platform for studies of electron correlation1-12. Correlation-driven phenomena, including the Mott insulator2-5, generalized Wigner crystals2,6,9, stripe phases10 and continuous Mott transition11,12, have been demonstrated. However, non-trivial band topology has remained unclear. Here we report the observation of a quantum anomalous Hall effect in AB-stacked MoTe2 /WSe2 moiré heterobilayers. Unlike in the AA-stacked heterobilayers11, an out-of-plane electric field not only controls the bandwidth but also the band topology by intertwining moiré bands centred at different layers. At half band filling, corresponding to one particle per moiré unit cell, we observe quantized Hall resistance, h/e2 (with h and e denoting the Planck's constant and electron charge, respectively), and vanishing longitudinal resistance at zero magnetic field. The electric-field-induced topological phase transition from a Mott insulator to a quantum anomalous Hall insulator precedes an insulator-to-metal transition. Contrary to most known topological phase transitions13, it is not accompanied by a bulk charge gap closure. Our study paves the way for discovery of emergent phenomena arising from the combined influence of strong correlation and topology in semiconductor moiré materials.

Magic in twisted transition metal dichalcogenide bilayers.

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 WSe2, 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∘, and identify a particular magic angle at which the top valence moiré band achieves almost perfect flatness. In the vicinity of this magic angle, we predict the realization of a generalized Kane-Mele model with a topological flat band, interaction-driven Haldane insulator, and Mott insulators at the filling of one hole per moiré unit cell. The combination of flat dispersion and uniformity of Berry curvature near the magic angle holds promise for realizing fractional quantum anomalous Hall effect at fractional filling. We also identify twist angles favorable for quantum spin Hall insulators and interaction-induced quantum anomalous Hall insulators at other integer fillings.

Spin-textured Chern bands in AB-stacked transition metal dichalcogenide bilayers.

While transition-metal dichalcogenide (TMD)-based moiré materials have been shown to host various correlated electronic phenomena, topological states have not been experimentally observed until now [T. Li et al., Quantum anomalous Hall effect from intertwined moiré bands. arXiv [Preprint] (2021). https://arxiv.org/abs/2107.01796 (Accessed 5 July 2021)]. In this work, using first-principle calculations and continuum modeling, we reveal the displacement field-induced topological moiré bands in AB-stacked TMD heterobilayer [Formula: see text]/[Formula: see text] Valley-contrasting Chern bands with nontrivial spin texture are formed from interlayer hybridization between [Formula: see text] and [Formula: see text] bands of nominally opposite spins. Our study establishes a recipe for creating topological bands in AB-stacked TMD bilayers in general, which provides a highly tunable platform for realizing quantum-spin Hall and interaction-induced quantum anomalous Hall effects.

Obtaining highly excited eigenstates of the localized XX chain via DMRG-X.

We benchmark a variant of the recently introduced density matrix renormalization group (DMRG)-X algorithm against exact results for the localized random field XX chain. We find that the eigenstates obtained via DMRG-X exhibit a highly accurate l-bit description for system sizes much bigger than the direct, many-body, exact diagonalization in the spin variables is able to access. We take advantage of the underlying free fermion description of the XX model to accurately test the strengths and limitations of this algorithm for large system sizes. We discuss the theoretical constraints on the performance of the algorithm from the entanglement properties of the eigenstates, and its actual performance at different values of disorder. A small but significant improvement to the algorithm is also presented, which helps significantly with convergence. We find that, at high entanglement, DMRG-X shows a bias towards eigenstates with low entanglement, but can be improved with increased bond dimension. This result suggests that one must be careful when applying the algorithm for interacting many-body localized spin models near a transition.This article is part of the themed issue 'Breakdown of ergodicity in quantum systems: from solids to synthetic matter'.

Nonzero-temperature entanglement negativity of quantum spin models: Area law, linked cluster expansions, and sudden death.

We show that the bipartite logarithmic entanglement negativity (EN) of quantum spin models obeys an area law at all nonzero temperatures. We develop numerical linked cluster (NLC) expansions for the "area-law" logarithmic entanglement negativity as a function of temperature and other parameters. For one-dimensional models the results of NLC are compared with exact diagonalization on finite systems and are found to agree very well. The NLC results are also obtained for two dimensional XXZ and transverse field Ising models. In all cases, we find a sudden onset (or sudden death) of negativity at a finite temperature above which the negativity is zero. We use perturbation theory to develop a physical picture for this sudden onset (or sudden death). The onset of EN or its magnitude are insensitive to classical finite-temperature phase transitions, supporting the argument for absence of any role of quantum mechanics at such transitions. On approach to a quantum critical point at T=0, negativity shows critical scaling in size and temperature.

Early Breakdown of Area-Law Entanglement at the Many-Body Delocalization Transition.

We introduce the numerical linked cluster expansion as a controlled numerical tool for the study of the many-body localization transition in a disordered system with continuous nonperturbative disorder. Our approach works directly in the thermodynamic limit, in any spatial dimension, and does not rely on any finite size scaling procedure. We study the onset of many-body delocalization through the breakdown of area-law entanglement in a generic many-body eigenstate. By looking for initial signs of an instability of the localized phase, we obtain a value for the critical disorder, which we believe should be a lower bound for the true value, that is higher than current best estimates from finite size studies. This implies that most current methods tend to overestimate the extent of the localized phase due to finite size effects making the localized phase appear stable at small length scales. We also study the mobility edge in these systems as a function of energy density, and we find that our conclusion is the same at all examined energies.