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Flat bands and nontrivial topological physics are two important topics of condensed matter physics. With a unique stacking configuration analogous to the Su-Schrieffer-Heeger (SSH) model, rhombohedral graphite (RG) is a potential candidate for realizing both flat bands and nontrivial topological physics. Here we report experimental evidence of topological flat bands (TFBs) on the surface of bulk RG, which are topologically protected by bulk helical Dirac nodal lines via the bulk-boundary correspondence. Moreover, upon {\it in situ} electron doping, the surface TFBs show a splitting with exotic doping evolution, with an order-of-magnitude increase in the bandwidth of the lower split band, and pinning of the upper band near the Fermi level. These experimental observations together with Hartree-Fock calculations suggest that correlation effects are important in this system. Our results demonstrate RG as a new platform for investigating the rich interplay between nontrivial band topology, correlation effects, and interaction-driven symmetry-broken states., Comment: 15 pages, 5 figures

Twisted bilayer graphene (tBLG) provides a fascinating platform for engineering flat bands and inducing correlated phenomena. By designing the stacking architecture of graphene layers, twisted multilayer graphene can exhibit different symmetries with rich tunability. For example, in twisted monolayer-bilayer graphene (tMBG) which breaks the C2z symmetry, transport measurements reveal an asymmetric phase diagram under an out-of-plane electric field, exhibiting correlated insulating state and ferromagnetic state respectively when reversing the field direction. Revealing how the electronic structure evolves with electric field is critical for providing a better understanding of such asymmetric field-tunable properties. Here we report the experimental observation of field-tunable dichotomic electronic structure of tMBG by nanospot angle-resolved photoemission spectroscopy (NanoARPES) with operando gating. Interestingly, selective enhancement of the relative spectral weight contributions from monolayer and bilayer graphene is observed when switching the polarity of the bias voltage. Combining experimental results with theoretical calculations, the origin of such field-tunable electronic structure, resembling either tBLG or twisted double-bilayer graphene (tDBG), is attributed to the selectively enhanced contribution from different stacking graphene layers with a strong electron-hole asymmetry. Our work provides electronic structure insights for understanding the rich field-tunable physics of tMBG., Comment: 4 figures

Magic-angle twisted bilayer graphene (MATBG) exhibits correlated phenomena such as superconductivity and Mott insulating state related to the weakly dispersing flat band near the Fermi energy. Beyond its moir\'e period, such flat band is expected to be sensitive to lattice relaxations. Thus, clarifying the evolution of the electronic structure with twist angle is critical for understanding the physics of MATBG. Here, we combine nanospot angle-resolved photoemission spectroscopy and atomic force microscopy to resolve the fine electronic structure of the flat band and remote bands, and their evolution with twist angles from 1.07$^\circ$ to 2.60$^\circ$. Near the magic angle, dispersion is characterized by a flat band near the Fermi energy with a strongly reduced bandwidth. Moreover, near 1.07$^\circ$, we observe a spectral weight transfer between remote bands at higher binding energy and extract the modulated interlayer spacing near the magic angle. Our work provides direct spectroscopic information on flat band physics and highlights the role of lattice relaxations., Comment: 22 pages, 5 figures, Nature Materials, in press

The diverse and intriguing phenomena observed in twisted bilayer systems, such as graphene and transition metal dichalcogenides, prompted new questions about the emergent effects that they may host. However, the practical challenge of realizing these structures on a scale large enough for spectroscopic investigation, remains a significant hurdle, resulting in a scarcity of direct measurements of the electronic band structure of twisted transition metal dichalcogenide bilayers. Here we present a systematic nanoscale angle-resolved photoemission spectroscopy investigation of bulk, single layer, and twisted bilayer WS2 with a small twist angle of 4.4{\deg}. The experimental results are compared with theoretical calculations based on density functional theory along the high-symmetry directions {\Gamma}-K and {\Gamma}-M. Surprisingly, the electronic band structure measurements suggest a structural relaxation occurring at 4.4{\deg} twist angle, and formation of large, untwisted bilayer regions., Comment: 23 pages, 11 figures

We present a common framework to study varieties in great generality from a categorical point of view. The main application of this study is in the setting of algebraic categories, where we introduce Birkhoff varieties which are essentially subvarieties of algebraic categories, and we get a generalization of Birkhoff's variety theorem. In particular, we show that Birkhoff varieties are coreflexive equalizers. The key of this generalization is to give a more general concept of equation for subvarieties of algebraic categories. In order to get our characterization of Birkhoff varieties, we study inserters over algebraic categories, where we generalize some well-known results of algebras for finitary endofunctors over $Set$. By duality, we obtain a characterization of cosubvarieties of coalgebraic categories. Surprisingly, these cosubvarieties turn to be varieties according to our theory of varieties., Comment: 17 pages, comments are welcome

Monolayer transition-metal dichalcogenide (TMD) semiconductors exhibit strong excitonic effects and hold promise for optical and optoelectronic applications. Yet, electron doping of TMDs leads to the conversion of neutral excitons into negative trions, which recombine predominantly non-radiatively at room temperature. As a result, the photoluminescence (PL) intensity is quenched. Here we study the optical and electronic properties of a MoS2/WSe2 heterostructure as a function of chemical doping by Cs atoms performed under ultra-high vacuum conditions. By PL measurements we identify two interlayer excitons and assign them to the momentum-indirect Q-Gamma and K-Gamma transitions. The energies of these excitons are in a very good agreement with ab initio calculations. We find that the Q-Gamma interlayer exciton is robust to the electron doping and is present at room temperature even at a high charge carrier concentration. Submicrometer angle-resolved photoemission spectroscopy (micro-ARPES) reveals charge transfer from deposited Cs adatoms to both the upper MoS2 and the lower WSe2 monolayer without changing the band alignment. This leads to a small (10 meV) energy shift of interlayer excitons. Robustness of the momentum-indirect interlayer exciton to charge doping opens up an opportunity of using TMD heterostructures in light-emitting devices that can work at room temperature at high densities of charge carriers.

The electrical control of a material's conductivity is at the heart of modern electronics. Conventionally, this control is achieved by tuning the density of mobile charge carriers. A completely different approach is possible in Mott insulators such as Ca$_2$RuO$_4$, where an insulator-to-metal transition (IMT) can be induced by a weak electric field or current. This phenomenon has numerous potential applications in, e.g., neuromorphic computing. While the driving force of the IMT is poorly understood, it has been thought to be a breakdown of the Mott state. Using in operando angle-resolved photoemission spectroscopy, we show that this is not the case: The current-driven conductive phase arises with only a minor reorganisation of the Mott state. This can be explained by the co-existence of structurally different domains that emerge during the IMT. Electronic structure calculations show that the boundaries between domains of slightly different structure lead to a drastic reduction of the overall gap. This permits an increased conductivity, despite the persistent presence of the Mott state. This mechanism represents a paradigm shift in the understanding of IMTs, because it does not rely on the simultaneous presence of a metallic and an insulating phase, but rather on the combined effect of structurally inhomogeneous Mott phases., Comment: 12 pages, 14 figures

For each finite primitive subgroup $G$ of $\operatorname{PGL}_4(\mathbb{C})$, we find all the smooth $G$-invariant quartic surfaces. We also find all the faithful representations in $\operatorname{PGL}_4(\mathbb{C})$ of the smooth quartic $G$-invariant surfaces by the groups: $\mathfrak{A}_5,\mathfrak{S}_5, \operatorname{PSL_2(\mathbb{F}_7)},\mathfrak{A}_6,\mathbb{Z}_2^4\rtimes\mathbb{Z}_5$ and $\mathbb{Z}_2^4\rtimes D_{10}$. The primitive representation of these groups are precisely the subgroups of $\operatorname{PGL}_4(\mathbb{C})$ for which $\mathbb{P}^3$ is not $G$-super rigid. As a byproduct, we show that the smooth quartic surface with the biggest group of projective automorphism is given by $\{ x_0^4 + x_1^4 + x_2^4 + x_3^4 + 12 x_0 x_1 x_2 x_3= 0\}$ (unique up to projective equivalence)., Comment: 22 pages, comments are welcome

Exfoliated magnetic 2D materials enable versatile tuning of magnetization, e.g., by gating or providing proximity-induced exchange interaction. However, their electronic band structure after exfoliation has not been probed, most likely due to their photochemical sensitivity. Here, we provide micron-scale angle-resolved photoelectron spectroscopy of the exfoliated intralayer antiferromagnet MnPS$_3$ above and below the N\'{e}el temperature down to one monolayer. The favorable comparison with density functional theory calculations enables to identify the orbital character of the observed bands. Consistently, we find pronounced changes across the N\'{e}el temperature for bands that consist of Mn 3d and 3p levels of adjacent S atoms. The deduced orbital mixture indicates that the superexchange is relevant for the magnetic interaction. There are only minor changes between monolayer and thicker films demonstrating the predominant 2D character of MnPS$_3$. The novel access is transferable to other MPX$_3$ materials (M: transition metal, P: phosphorus, X: chalcogenide) providing a multitude of antiferromagnetic arrangements., Comment: 26 pages, 17 figures

The recent observation of correlated phases in transition metal dichalcogenide moir\'e systems at integer and fractional filling promises new insight into metal-insulator transitions and the unusual states of matter that can emerge near such transitions. Here, we combine real- and momentum-space mapping techniques to study moir\'e superlattice effects in 57.4$^{\circ}$ twisted WSe$_2$ (tWSe$_2$). Our data reveal a split-off flat band that derives from the monolayer $\Gamma$ states. Using advanced data analysis, we directly quantify the moir\'e potential from our data. We further demonstrate that the global valence band maximum in tWSe$_2$ is close in energy to this flat band but derives from the monolayer K-states which show weaker superlattice effects. These results constrain theoretical models and open the perspective that $\Gamma$-valley flat bands might be involved in the correlated physics of twisted WSe$_2$.

Moir\'e superlattices in twisted two-dimensional materials have generated tremendous excitement as a platform for achieving quantum properties on demand. However, the moir\'e pattern is highly sensitive to the interlayer atomic registry, and current assembly techniques suffer from imprecise control of the average twist angle, spatial inhomogeneity in the local twist angle, and distortions due to random strain. Here, we demonstrate a new way to manipulate the moir\'e patterns in hetero- and homo-bilayers through in-plane bending of monolayer ribbons, using the tip of an atomic force microscope. This technique achieves continuous variation of twist angles with improved twist-angle homogeneity and reduced random strain, resulting in moir\'e patterns with highly tunable wavelength and ultra-low disorder. Our results pave the way for detailed studies of ultra-low disorder moir\'e systems and the realization of precise strain-engineered devices.

Twisted van der Waals heterostructures have recently been proposed as a condensed-matter platform for realizing controllable quantum models due to the low-energy moir\'e bands with specific charge distributions in moir\'e superlattices. Here, combining angle-resolved photoemission spectroscopy with sub-micron spatial resolution ({\mu}-ARPES) and scanning tunneling microscopy (STM), we performed a systematic investigation on the electronic structure of 5.1{\deg} twisted bilayer WSe2 that hosts correlated insulating and zero-resistance states. Interestingly, contrary to one's expectation, moir\'e bands were observed only at {\Gamma}-valley but not K-valley in {\mu}-ARPES measurements; and correspondingly, our STM measurements clearly identified the real-space honeycomb- and Kagome-shaped charge distributions at the moir\'e length scale associated with the {\Gamma}-valley moir\'e bands. These results not only reveal the unsual valley dependent moir\'e-modified electronic structure in twisted transition metal dichalcogenides, but also highlight the {\Gamma}-valley moir\'e bands as a promising platform for exploring strongly correlated physics in emergent honeycomb and Kagome lattices at different energy scales., Comment: 29 pages, accepted by PRX

Nearly localized moire flat bands in momentum space, arising at particular twist angles, are the key to achieve correlated effects in transition-metal dichalcogenides. Here, we use angle-resolved photoemission spectroscopy (ARPES) to visualize the presence of a flat band near the Fermi level of van der Waals (vdW) WSe2/MoSe2 heterobilayer grown by molecular beam epitaxy. This flat band is localized near the K point of the Brillouin zone and has a width of several hundred meVs. By combining ARPES measurements with density functional theory (DFT) calculations, we confirm the coexistence of different domains, namely the reference 2H stacking without layer misorientation and regions with arbitrary twist angles. For the 2H-stacked heterobilayer, our ARPES results show strong interlayer hybridization effects, further confirmed by complementary micro- Raman spectroscopy measurements. The spin-splitting of the valence band at K is determined to be 470 meV. The valence band maximum (VBM) position of the heterobilayer is located at the Gamma point. The energy difference between the VBM at Gamma and the K point is of -60 meV, which is a stark difference compared to individual 1L WSe2 and 1L WSe2, showing both a VBM at K.

Phase separation in the nanometer- to micrometer-scale is characteristic for correlated materials, for example, high temperature superconductors, colossal magnetoresistance manganites, Mott insulators, etc. Resolving the electronic structure with spatially-resolved information is critical for revealing the fundamental physics of such inhomogeneous systems yet this is challenging experimentally. Here by using nanometer- and micrometer-spot angle-resolved photoemission spectroscopies (NanoARPES and MicroARPES), we reveal the spatially-resolved electronic structure in the stripe phase of IrTe$_2$. Each separated domain shows two-fold symmetric electronic structure with the mirror axis aligned along 3 equivalent directions, and 6$\times$1 replicas are clearly identified. Moreover, such electronic structure inhomogeneity disappears across the stripe phase transition, suggesting that electronic phase with broken symmetry induced by the 6$\times$1 modulation is directly related to the stripe phase transition of IrTe$_2$. Our work demonstrates the capability of NanoARPES and MicroARPES in elucidating the fundamental physics of phase-separated materials.

We study the valence band structure of ReSe$_{2}$ crystals with varying thickness down to a single layer using nanoscale angle-resolved photoemission spectroscopy and density functional theory. The width of the top valence band in the direction perpendicular to the rhenium chains decreases with decreasing number of layers, from 280 meV for the bulk to 61 meV for monolayer. This demonstrates increase of in-plane anisotropy induced by changes in the interlayer coupling and suggests progressively more one-dimensional character of electronic states in few-layer rhenium dichalcogenides.

In atomically thin transition metal dichalcogenide semiconductors, there is a crossover from indirect to direct bandgap as the thickness drops to one monolayer, which comes with a fast increase of the photoluminescence signal. Here, we show that for different alloy compositions of WS2(1-x)Se2x this trend may be significantly affected by the alloy content and we demonstrate that the sample with the highest Se ratio presents a strongly reduced effect. The highest micro-PL intensity is found for bilayer WS2(1-x)Se2x (x = 0.8) with a decrease of its maximum value by only a factor of 2 when passing from mono- to bi-layer. To better understand this factor and explore the layer-dependent band structure evolution of WS2(1-x)Se2x, we performed a nano-angle resolved photoemission spectroscopy study coupled with first-principles calculations. We find that the high micro-PL value for bilayer WS2(1-x)Se2x (x = 0.8) is due to the overlay of direct and indirect optical transitions. This peculiar high PL intensity in WS2(1-x)Se2x opens the way for spectrally tunable light-emitting devices.

Interfacing bulk conducting topological Bi$_2$Se$_3$ films with s-wave superconductors initiates strong superconducting order in the nontrivial surface states. However, bulk insulating topological (Bi$_{1-x}$Sb$_{x})_2$Te$_3$ films on bulk Nb instead exhibit a giant attenuation of surface superconductivity, even for films only two-layers thick. This massive suppression of proximity pairing is evidenced by ultrahigh-resolution band mappings and by contrasting quantified superconducting gaps with those of heavily n-doped topological Bi$_2$Se$_3$/Nb. The results underscore the limitations of using superconducting proximity effects to realize topological superconductivity in nearly intrinsic systems., Comment: Accepted at Physical Review Letters: https://journals.aps.org/prl/accepted/b0073Y70Ddc12c71a13a3ac871102f6c7e1a1bd7b

We have performed scanning angle-resolved photoemission spectroscopy with a nanometer-sized beam spot (nano-ARPES) on the cleaved surface of Pb5Bi24Se41, which is a member of the (PbSe)5(Bi2Se3)3m homologous series (PSBS) with m = 4 consisting of alternate stacking of the topologically-trivial insulator PbSe bilayer and four quintuple layers (QLs) of the topological insulator Bi2Se3. This allows us to visualize a mosaic of topological Dirac states at a nanometer scale coming from the variable thickness of the Bi2Se3 nano-islands (1-3 QLs) that remain on top of the PbSe layer after cleaving the PSBS crystal, because the local band structure of topological origin changes drastically with the thickness of the Bi2Se3 nano-islands. A comparison of the local band structure with that in ultrathin Bi2Se3 films on Si(111) gives us further insights into the nature of the observed topological states. This result demonstrates that nano-ARPES is a very useful tool for characterizing topological heterostructures., Comment: 6 pages, 3 figures

Hexagonal boron nitride (hBN) is the supporting substrate of choice for two-dimensional material devices because it is atomically flat and chemically inert. However, due to the small size of mechanically exfoliated hBN flakes, electronic structure studies of 2D materials supported by hBN using angle-resolved photoemission spectroscopy (ARPES) are challenging. Here we investigate the electronic band structure of a Bernal-stacked bilayer graphene sheet on a hexagonal boron nitride (BLG/hBN) flake using nanospot ARPES (nanoARPES). By fitting high-resolution energy vs. momentum electronic band spectra, we extract the tight-binding parameters for BLG on hBN. In addition, we reveal spatial variations of the alignment angle between BLG and hBN lattices via inhomogeneity of the electronic bands near the Fermi level. We confirmed these findings by scanning tunneling microscopy measurements obtained on the same device. Our results from spatially resolved nanoARPES measurements of BLG/hBN heterostructures are instrumental for understanding experiments that utilize spatially averaging techniques such as electronic transport and optical spectroscopy.

Electrostatic gating is pervasive in materials science, yet its effects on the electronic band structure of materials has never been revealed directly by angle-resolved photoemission spectroscopy (ARPES), the technique of choice to non-invasively probe the electronic band structure of a material. By means of a state-of-the-art ARPES setup with sub-micron spatial resolution, we have investigated a heterostructure composed of Bernal-stacked bilayer graphene (BLG) on hexagonal boron nitride and deposited on a graphite flake. By voltage biasing the latter, the electric field effect is directly visualized on the valence band as well as on the carbon 1s core level of BLG. The band gap opening of BLG submitted to a transverse electric field is discussed and the importance of intralayer screening is put forward. Our results pave the way for new studies that will use momentum-resolved electronic structure information to gain insight on the physics of materials submitted to the electric field effect.