Recently, the Hofstadter spectrum of a twisted $\mathrm{WSe_2/MoSe_2}$ heterobilayer has been observed in experiment [C. R. Kometter, et al. Nat.Phys.19, 1861 (2023)], but the origin of Hofstadter states remains unclear. Here, we present a comprehensive theoretical interpretation of the observed Hofstadter states by calculating its accurate Hofstadter spectrum. We point out that the valley Zeeman effect, a unique feature of the transition metal dichalcogenide (TMD) materials, plays a crucial role in determining the shape of the Hofstadter spectrum, due to the narrow bandwidth of the moiré bands. This is distinct from the graphene-based moiré systems. We further predict that the Hofstadter spectrum of the moiré flat band, which was not observed in experiment, can be observed in the same system with a larger twist angle $2^\circ\lesssimθ\lesssim 3^\circ$. Our theory paves the way for further studies of the interplay between the Hofstadter states and correlated insulting states in such moiré lattice systems.
We investigate the optical absorption spectrum and the selection rule for the Hofstadter butterfly in twisted bilayer graphene under magnetic fields. We demonstrate that the absorption spectrum exhibits a self-similar recursive pattern reflecting the fractal nature of the energy spectrum. We find that the optical selection rule has a nested self-similar structure as well, and it is governed by the conservation of the total angular momentum summed over different hierarchies.
Electrons moving through a spatially periodic lattice potential develop a quantized energy spectrum consisting of discrete Bloch bands. In two dimensions, electrons moving through a magnetic field also develop a quantized energy spectrum, consisting of highly degenerate Landau energy levels. In 1976 Douglas Hofstadter theoretically considered the intersection of these two problems and discovered that 2D electrons subjected to both a magnetic field and a periodic electrostatic potential exhibit a self-similar recursive energy spectrum. Known as Hofstadter's butterfly, this complex spectrum resu…
Moiré superlattices in two-dimensional (2D) van der Waals (vdW) heterostructures provide 20 an efficient way to engineer electron band properties. The recent discovery of exotic quantum phases and their interplay in twisted bilayer graphene (tBLG) has built this moiré system one of the most renowned condensed matter platforms (1-10). So far the studies of tBLG has been mostly focused on the lowest two flat moiré bands at the first magic angle θm1 ~ 1.1°, leaving high-order moiré bands and magic angles largely unexplored. Here we report 25 an observation of multiple well-isolated flat moiré ban…
We theoretically explore 2d moiré heterostructures in lattice-commensurate magnetic fields as platforms for quantum simulation of a paradigmatic model of non-Fermi liquid physics: a Fermi-surface coupled to a fluctuating gauge field. In these moiré-Hofstadter (MH) systems, long-wavelength acoustic phonons exhibit singular interactions with electrons analogous to those of electrons with 2d gauge fields. This leads to a breakdown of Fermi-liquid theory at low temperatures. We show that a combination of large moiré-unit cell size, tunable Fermi-surface topology, and enhanced coupling to interlaye…
High-field Josephson effect enabled by a moiré Hofstadter spectrum
Ludwig-Maximilians-Universität（路德维希-马克西米利安大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Nanomaterials and Nanotechnology Research Center (CINN-CSIC), Universidad de Oviedo (UO)（阿斯图里亚斯大学纳米材料与纳米技术研究中心（CINN-CSIC））；Yale University（耶鲁大学）；Aalto University（阿尔托大学）；National Institute for Materials Science（国立材料研究所）
Moiré superlattices engineer band properties and enable observation of fractal energy spectra of Hofstadter butterfly. Recently, correlated-electron physics hosted by flat bands in small-angle moiré systems has been at the foreground. However, the implications of moiré band topology within the single-particle framework are little explored experimentally. An outstanding problem is understanding the effect of band topology on Hofstadter physics, which does not require electron correlations. Our work experimentally studies Chern state switching in the Hofstadter regime using twisted double bilaye…
We propose Landau levels as a probe for the topological character of electronic bands in two-dimensional moiré superlattices. We consider two configurations of twisted double bilayer graphene (TDBG) that have very similar band structures, but show different valley Chern numbers of the flat bands. These differences between the AB-AB and AB-BA configurations of TDBG clearly manifest as different Landau level sequences in the Hofstadter butterfly spectra calculated using the tight-binding model. The Landau level sequences are explained from the point of view of the distribution of orbital magneti…
We investigate the magnetic minibands of a heterostructure consisting of bilayer graphene (BLG) and hexagonal boron nitride (hBN) by numerically diagonalizing a two-band Hamiltonian that describes electrons in BLG in the presence of a moire potential. Due to inversion-symmetry breaking characteristic for the moire potential, the valley symmetry of the spectrum is broken, but despite this, the zero-energy Landau level in BLG survives, albeit with reduced degeneracy. In addition, we derive effective models for the low-energy features in the magnetic minibands and demonstrate the appearance of se…
The Hofstadter problem is the lattice analog of the quantum Hall effect and is the paradigmatic example of topology induced by an applied magnetic field. Conventionally, the Hofstadter problem involves adding $\sim 10^4$ T magnetic fields to a trivial band structure. In this work, we show that when a magnetic field is added to an initially topological band structure, a wealth of remarkable possible phases emerges. Remarkably, we find topological phases which cannot be realized in any crystalline insulators. We prove that threading magnetic flux through a Hamiltonian with nonzero Chern number e…
In magic angle twisted bilayer graphene (MATBG), the moiré superlattice potential gives rise to narrow electronic bands1 which support a multitude of many-body quantum phases. Further richness arises in the presence of a perpendicular magnetic field, where the interplay between moiré and magnetic length scales leads to fractal Hofstadter subbands. In this strongly correlated Hofstadter platform, multiple experiments have identified gapped topological and correlated states, but little is known about the phase transitions between them in the intervening compressible regimes. Here, using a scanni…
The fractal spectrum of magnetic minibands (Hofstadter butterfly), induced by the moiré super- lattice of graphene on an hexagonal crystal substrate, is known to exhibit gapped Dirac cones. We show that the gap can be closed by slightly misaligning the substrate, producing a hierarchy of conical singularities (Dirac points) in the band structure at rational values Phi = (p/q)(h/e) of the magnetic flux per supercell. Each Dirac point signals a switch of the topological quantum number in the connected component of the quantum Hall phase diagram. Model calculations reveal the scale invariant cond…
We calculate the energy bands for graphene monolayers when electrons move through a periodic electrostatic potential in the presence of a uniform perpendicular magnetic field. We clearly demonstrate the quantum fractal nature of the energy bands at reasonably low magnetic fields. We present results for the energy bands as functions of both wave number and magnetic flux through the unit cells of the resulting moire superlattice. The effects due to pseudo-spin coupling and Landau orbit mixing by a strong scattering potential have been exhibited. At low magnetic fields when the Landau orbits are …
In long-wavelength moiré patterns of homobilayer semiconductors, the layer pseudospin of electrons is subject to a sizable Zeeman field that is spatially modulated from the interlayer coupling in moiré. By interference of this spatial modulation with a homogeneous but dynamically tunable component from out-of-plane electric field, we show that the spatial-temporal profile of the overall Zeeman field therefore features a topological texture that can be controlled in an ultrafast timescale by a terahertz field or an interlayer bias. Such dynamical modulation leads to the emergence of an in-plane…
In twisted homobilayer transition metal dichalcogenides, intra- and inter-layer valley excitons hybridize with the layer configurations spatially varying in the moiré. The ground state valley excitons are trapped at two high-symmetry points with opposite electric dipoles in a moiré supercell, forming a honeycomb superlattice of nearest-neighbor dipolar attraction. We find that the spatial texture of layer configuration results in a luminescence anomaly of the moiré trapped excitons, where a tiny displacement by interactions dramatically increases the brightness and changes polarization from ci…
Van der Waals heterostructures comprise a new class of artificial materials formed by stacking atomically-thin planar crystals. Here, we demonstrate band structure engineering of a van der Waals heterostructure composed of a monolayer graphene flake coupled to a rotationally-aligned hexagonal boron nitride substrate. The spatially-varying interlayer atomic registry results both in a local breaking of the carbon sublattice symmetry and a long-range moiré superlattice potential in the graphene. This interplay between short- and long-wavelength effects results in a band structure described by iso…
Emergent Gauge Flux and Spin Ordering in Magnetized Triangular Spin Liquids: Applications to Hofstadter-Hubbard Model
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料中心）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学北京量子器件重点实验室）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）
We study the Hofstadter butterfly and Landau levels of the twisted bilayer graphene (TBG). We show that the nontrivial fragile topology of the lowest two bands near the charge neutral point makes their Hofstadter butterfly generically connected with higher bands, closing the gap between the first and second conduction (valence) bands at a certain magnetic flux per unit cell. We also develop a momentum space method for calculating the TBG Hofstadter butterfly, from which we identify three phases where the Hofstadter butterflies of the lowest two bands and the higher bands are connected in diffe…
Moiré superlattices provide a powerful tool to engineer novel quantum phenomena in two-dimensional (2D) heterostructures, where the interactions between the atomically thin layers qualitatively change the electronic band structure of the superlattice. For example, mini-Dirac points, tunable Mott insulator states, and the Hofstadter butterfly can emerge in different types of graphene/boron nitride moiré superlattices, while correlated insulating states and superconductivity have been reported in twisted bilayer graphene moiré superlattices. In addition to their dramatic effects on the single pa…
In bilayer graphene rotationally faulted to theta=1.1 degrees, interlayer tunneling and rotational misalignment conspire to create a pair of low energy flat band that have been found to host various correlated phenomena at partial filling. Most work to date has focused on the zero magnetic field phase diagram, with magnetic field (B) used as a probe of the B=0 band structure. Here, we show that twisted bilayer graphene (tBLG) in a B as low as 2T hosts a cascade of ferromagnetic Chern insulators with Chern number |C|=1,2 and 3. We argue that the emergence of the Chern insulators is driven by th…
Moiré superlattices of two-dimensional van der Waals materials have emerged as a powerful platform for designing electronic band structures and discovering emergent physical phenomena. A key concept involves the creation of long-wavelength periodic potential and moiré bands in a crystal through interlayer hybridization when two materials are overlaid. Here we demonstrate a new approach based on spatially periodic dielectric screening to create moiré bands in a monolayer semiconductor. It relies on reduced dielectric screening of the Coulomb interactions in monolayer semiconductors and their en…
We develop a new, efficient, and general method to determine the Hofstadter spectrum of isolated narrow bands. The method works for topological as well as for trivial narrow bands by projecting the zero $\mathbf{B}$-field hybrid Wannier states -- which are localized in one direction and Bloch extended in another direction -- onto a representation of the magnetic translation group in the Landau gauge. We then apply this method to find, for the first time, the Hofstadter spectrum for the exact single particle charged excitations in the strong coupling limit of the magic angle twisted bilayer gra…
Moiré excitons are emergent optical excitations in 2D semiconductors with deep moiré superlattice potentials. While these excitations have been realized in several platforms, a system with dynamically tunable moiré potential to tailor the moiré exciton properties is yet to be realized. Here, we present a continuously tunable moiré potential in a monolayer WSe2 that is enabled by its proximity to twisted bilayer graphene (TBG) near the magic-angle. Due to its flat electronic bands, charge distribution is highly localized and forms a triangular lattice in TBG. Tuning the local charge density via…
Moiré superlattices formed in van der Waals bilayers have enabled the creation and manipulation of new quantum states, as is exemplified by the discovery of superconducting and correlated insulating states in twisted bilayer graphene near the magic angle. Twisted bilayer semiconductors may lead to tunable exciton lattices and topological states, yet signatures of moiré excitons have been reported only in closely angularly-aligned bilayers. Here we report tuning of moiré lattice in WS$_{2}$ /MoSe$_{2}$ bilayers over a wide range of twist angles, leading to the continuous tuning of moiré lattice…
Van der Waals moiré materials have emerged as a highly controllable platform to study the electronic correlation phenomena. In particular, robust correlated insulating states have recently been discovered at both integer and fractional filling factors of the semiconductor moiré systems. Here, we reveal the thermodynamic properties of these states by measuring the gate capacitance of MoSe2/WS2 moiré superlattices. We observe a series of incompressible states for filling factor 0 - 8 and anomalously large capacitance (nearly 60% above the device's geometrical capacitance) in the intervening comp…
Semiconductor moiré heterostructures exhibit rich correlation-induced many-body phenomena with signatures of emergent magnetism, Mott insulating states or generalized Wigner crystals observed in optical spectroscopy by probing intralayer excitons. However, as the staggered band alignment in the underlying WSe$_2$/WS$_2$ heterobilayer system separates photoexcited electrons and holes to form lowest-energy interlayer excitons, direct access to interactions between correlated charge states and ground state moiré excitons remained elusive. Here, we use MoSe$_2$/WS$_2$ heterostacks with type-I band…
The optical properties of weakly-doped two-dimensional materials are dominated by strong exciton and trion absorption and luminescence features. In this article we examine the influence of moiré patterns in semiconductor heterobilayers on exciton and trion states in the limit of strong moiré modulation potentials, commenting on similarities and differences compared to the case of excitons and trions in semiconductor quantum dots. We discuss strategies for using optical properties as quantitative probes of moiré materials, and the prospects for exploiting moiré materials to design unique light …
A Catalogue of Topological Moiré Bands in Twisted Semiconductors
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所与凝聚态物理北京国家实验室）；University of Chinese Academy of Sciences（中国科学院大学）；Condensed Matter Physics Data Center of Chinese Academy of Sciences（中国科学院凝聚态物理数据中心）
Unlike conventional two-dimensional (2D) semiconductor superlattices, moiré patterns in 2D materials are flexible and their electronic, magnetic, optical, and mechanical properties depend on their topography. Within a continuous+atomistic theory treating 2D materials as crystalline elastic membranes, we abandon the flat-membrane scenario usually assumed for these materials and address out-of-plane deformations. We confront our predictions to experimental analyses on model systems, epitaxial graphene, and MoS$_2$ on metals and reveal that compression/expansion and bending energies stored in the…
Recent experiments in moiré transition metal dichalcogenide materials have reported the observation of a continuous bandwidth-tuned transition from a metal to a paramagnetic Mott insulator at a fixed filling of one electron per moiré unit cell. The electrical transport measurements reveal a number of puzzling features that are seemingly at odds with the theoretical expectations of an interaction induced, but disorder-free, bandwidth-tuned metal-insulator transition. In this work, we include the effects of long-wavelength inhomogeneities, building on the results for a continuous metal-insulator…
