Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice.

Studies of two-dimensional electron systems in a strong magnetic field revealed the quantum Hall effect1, a topological state of matter featuring a finite Chern number C and chiral edge states2,3. Haldane4 later theorized that Chern insulators with integer quantum Hall effects could appear in lattice models with complex hopping parameters even at zero magnetic field. The ABC-trilayer graphene/hexagonal boron nitride (ABC-TLG/hBN) moiré superlattice provides an attractive platform with which to explore Chern insulators because it features nearly flat moiré minibands with a valley-dependent, electrically tunable Chern number5,6. Here we report the experimental observation of a correlated Chern insulator in an ABC-TLG/hBN moiré superlattice. We show that reversing the direction of the applied vertical electric field switches the moiré minibands of ABC-TLG/hBN between zero and finite Chern numbers, as revealed by large changes in magneto-transport behaviour. For topological hole minibands tuned to have a finite Chern number, we focus on quarter filling, corresponding to one hole per moiré unit cell. The Hall resistance is well quantized at h/2e2 (where h is Planck's constant and e is the charge on the electron), which implies C = 2, for a magnetic field exceeding 0.4 tesla. The correlated Chern insulator is ferromagnetic, exhibiting substantial magnetic hysteresis and a large anomalous Hall signal at zero magnetic field. Our discovery of a C = 2 Chern insulator at zero magnetic field should open up opportunities for discovering correlated topological states, possibly with topological excitations7, in nearly flat and topologically nontrivial moiré minibands.

Publisher Correction: Tunable correlated Chern insulator and ferromagnetism in a moiré superlattice.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Seeded growth of single-crystal black phosphorus nanoribbons.

Two-dimensional materials have emerged as an important research frontier for overcoming the challenges in nanoelectronics and for exploring new physics. Among them, black phosphorus, with a combination of a tunable bandgap and high mobility, is one of the most promising systems. In particular, black phosphorus nanoribbons show excellent electrostatic gate control, which can mitigate short-channel effects in nanoscale transistors. Controlled synthesis of black phosphorus nanoribbons, however, has remained an outstanding problem. Here we report large-area growth of black phosphorus nanoribbons directly on insulating substrates. We seed the chemical vapour transport growth with black phosphorus nanoparticles and obtain uniform, single-crystal nanoribbons oriented exclusively along the [100] crystal direction. With comprehensive structural calculations, we discover that self-passivation at the zigzag edges holds the key to the preferential one-dimensional growth. Field-effect transistors based on individual nanoribbons exhibit on/off ratios up to ~104, confirming the good semiconducting behaviour of the nanoribbons. These results demonstrate the potential of black phosphorus nanoribbons for nanoelectronic devices and also provide a platform for investigating the exotic physics in black phosphorus.

High-temperature superconductivity in monolayer Bi2Sr2CaCu2O8+δ.

Although copper oxide high-temperature superconductors constitute a complex and diverse material family, they all share a layered lattice structure. This curious fact prompts the question of whether high-temperature superconductivity can exist in an isolated monolayer of copper oxide, and if so, whether the two-dimensional superconductivity and various related phenomena differ from those of their three-dimensional counterparts. The answers may provide insights into the role of dimensionality in high-temperature superconductivity. Here we develop a fabrication process that obtains intrinsic monolayer crystals of the high-temperature superconductor Bi2Sr2CaCu2O8+δ (Bi-2212; here, a monolayer refers to a half unit cell that contains two CuO2 planes). The highest superconducting transition temperature of the monolayer is as high as that of optimally doped bulk. The lack of dimensionality effect on the transition temperature defies expectations from the Mermin-Wagner theorem, in contrast to the much-reduced transition temperature in conventional two-dimensional superconductors such as NbSe2. The properties of monolayer Bi-2212 become extremely tunable; our survey of superconductivity, the pseudogap, charge order and the Mott state at various doping concentrations reveals that the phases are indistinguishable from those in the bulk. Monolayer Bi-2212 therefore displays all the fundamental physics of high-temperature superconductivity. Our results establish monolayer copper oxides as a platform for studying high-temperature superconductivity and other strongly correlated phenomena in two dimensions.

Signatures of tunable superconductivity in a trilayer graphene moiré superlattice.

Understanding the mechanism of high-transition-temperature (high-Tc) superconductivity is a central problem in condensed matter physics. It is often speculated that high-Tc superconductivity arises in a doped Mott insulator1 as described by the Hubbard model2-4. An exact solution of the Hubbard model, however, is extremely challenging owing to the strong electron-electron correlation in Mott insulators. Therefore, it is highly desirable to study a tunable Hubbard system, in which systematic investigations of the unconventional superconductivity and its evolution with the Hubbard parameters can deepen our understanding of the Hubbard model. Here we report signatures of tunable superconductivity in an ABC-trilayer graphene (TLG) and hexagonal boron nitride (hBN) moiré superlattice. Unlike in 'magic angle' twisted bilayer graphene, theoretical calculations show that under a vertical displacement field, the ABC-TLG/hBN heterostructure features an isolated flat valence miniband associated with a Hubbard model on a triangular superlattice5,6 where the bandwidth can be tuned continuously with the vertical displacement field. Upon applying such a displacement field we find experimentally that the ABC-TLG/hBN superlattice displays Mott insulating states below 20 kelvin at one-quarter and one-half fillings of the states, corresponding to one and two holes per unit cell, respectively. Upon further cooling, signatures of superconductivity ('domes') emerge below 1 kelvin for the electron- and hole-doped sides of the one-quarter-filling Mott state. The electronic behaviour in the ABC-TLG/hBN superlattice is expected to depend sensitively on the interplay between the electron-electron interaction and the miniband bandwidth. By varying the vertical displacement field, we demonstrate transitions from the candidate superconductor to Mott insulator and metallic phases. Our study shows that ABC-TLG/hBN heterostructures offer attractive model systems in which to explore rich correlated behaviour emerging in the tunable triangular Hubbard model.

Dephasing of Strong-Field-Driven Excitonic Autler-Townes Doublets Revealed by Time- and Spectrum-Resolved Quantum-Path Interferometry.

Understanding dephasing mechanisms of strong-field-driven excitons in condensed matter is essential for their applications in quantum-state manipulation and ultrafast optical modulations. However, experimental access to exciton dephasing under strong-field conditions is challenging. In this study, using time- and spectrum-resolved quantum-path interferometry, we investigate the dephasing mechanisms of terahertz-driven excitonic Autler-Townes doublets in MoS_{2}. Our results reveal a dramatic increase in the dephasing rate beyond a threshold field strength, indicating exciton dissociation as the primary dephasing mechanism. Furthermore, we demonstrate nonperturbative high-order sideband generation in a regime where the driving fields are insufficient to dissociate excitons.

Strength Enhancement of Polyurethane Film by Solution Annealing.

Recently, polyurethane elastomer (TPU) has attracted more and more attention depending on its excellent optical, mechanical, and retreatment properties. The high strength of polyurethane has always been pursued, which can enable its application in more fields. In this work, an aliphatic polyurethane elastomer membrane (HRPU6) was successfully synthesized, and its strength was obviously improved by solvent annealing technology. The tensile strength and adhesion strength can reach 64.56 and 2.58 MPa, but 36.55 and 1.57 MPa only before solvent annealing, respectively. The impact strength of laminated glass based on HRPU has also been significantly improved after solvent annealing, confirmed through drop ball impact testing. It has been confirmed that the increase in strength of HRPU6 is attributed to the enhancement of hydrogen bonding and the improvement of the phase separation degree.

Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2.

Materials research has driven the development of modern nano-electronic devices. In particular, research in magnetic thin films has revolutionized the development of spintronic devices1,2 because identifying new magnetic materials is key to better device performance and design. Van der Waals crystals retain their chemical stability and structural integrity down to the monolayer and, being atomically thin, are readily tuned by various kinds of gate modulation3,4. Recent experiments have demonstrated that it is possible to obtain two-dimensional ferromagnetic order in insulating Cr2Ge2Te6 (ref. 5) and CrI3 (ref. 6) at low temperatures. Here we develop a device fabrication technique and isolate monolayers from the layered metallic magnet Fe3GeTe2 to study magnetotransport. We find that the itinerant ferromagnetism persists in Fe3GeTe2 down to the monolayer with an out-of-plane magnetocrystalline anisotropy. The ferromagnetic transition temperature, Tc, is suppressed relative to the bulk Tc of 205 kelvin in pristine Fe3GeTe2 thin flakes. An ionic gate, however, raises Tc to room temperature, much higher than the bulk Tc. The gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2 opens up opportunities for potential voltage-controlled magnetoelectronics7-11 based on atomically thin van der Waals crystals.

Magnetic Field-Stabilized Wigner Crystal States in a Graphene Moiré Superlattice.

ABC-stacked trilayer graphene on boron nitride (ABC-TLG/hBN) moiré superlattices provides a tunable platform for exploring Wigner crystal states in which the electron correlation can be controlled by electric and magnetic fields. Here we report the observation of magnetic field-stabilized Wigner crystal states in a ABC-TLG/hBN. We show that correlated insulating states emerge at multiple fractional and integer fillings corresponding to ν = 1/3, 2/3, 1, 4/3, 5/3, and 2 electrons per moiré lattice site under a magnetic field. These correlated insulating states can be attributed to generalized Mott states for the integer fillings and generalized Wigner crystal states for the fractional fillings. The generalized Wigner crystal states are stabilized by a vertical magnetic field and are strongest at one magnetic flux quantum per three moiré superlattices. The ν = 2 insulating state persists up to 30 T, which can be described by a Mott-Hofstadter transition at a high magnetic field.

Cadmium tolerance and accumulation from the perspective of metal ion absorption and root exudates in broomcorn millet.

Cadmium (Cd) is a persistent heavy metal that poses environmental and public health concerns. This study aimed to identify the potential biomarkers responsible for Cd tolerance and accumulation by investigating the response of the content of essential metal elements, transporter gene expression, and root exudates to Cd stress in broomcorn millet (Panicum miliaceum). A hydroponics experiment was conducted using two broomcorn millet cultivars with distinct Cd tolerance levels and accumulation phenotypes (Cd-tolerant and Cd-sensitive cultivars). Cd stress inhibited lateral root growth, especially in the Cd-sensitive cultivar. Furthermore, Cd accumulation was significantly greater in the Cd-tolerant cultivar than in the Cd-sensitive cultivar. Cd stress significantly inhibited the absorption of essential metal elements and significantly increased the calcium concentration. Differentially expressed genes involved in metal ion transport were identified via transcriptome analysis. Cd stress altered the composition of root exudates, thus increasing lipid species and decreasing alkaloid, lignan, sugar, and alcohol species. Moreover, Cd stress significantly reduced most alkaloid, organic acid, and phenolic acid exudates in the Cd-tolerant cultivar, while it increased most lipid and phenolic acid exudates in the Cd-sensitive cultivar. Some significantly changed root exudates (ferulic acid, O-coumaric acid, and spermine) are involved in the phenylalanine biosynthesis, and arginine and proline metabolic pathways, thus, may be potential biomarkers of Cd stress response. Overall, metal ion absorption and root exudates are critical for Cd tolerance and accumulation in broomcorn millet. These findings provide valuable insights into improving Cd phytoremediation by applying mineral elements or metabolites.

Tunable Orbital Ferromagnetism at Noninteger Filling of a Moiré Superlattice.

The flat bands resulting from moiré superlattices exhibit fascinating correlated electron phenomena such as correlated insulators, ( Nature 2018, 556 (7699), 80-84), ( Nature Physics 2019, 15 (3), 237) superconductivity, ( Nature 2018, 556 (7699), 43-50), ( Nature 2019, 572 (7768), 215-219) and orbital magnetism. ( Science 2019, 365 (6453), 605-608), ( Nature 2020, 579 (7797), 56-61), ( Science 2020, 367 (6480), 900-903) Such magnetism has been observed only at particular integer multiples of n0, the density corresponding to one electron per moiré superlattice unit cell. Here, we report the experimental observation of ferromagnetism at noninteger filling (NIF) of a flat Chern band in a ABC-TLG/hBN moiré superlattice. This state exhibits prominent ferromagnetic hysteresis behavior with large anomalous Hall resistivity in a broad region of densities centered in the valence miniband at n = -2.3n0. We observe that, not only the magnitude of the anomalous Hall signal, but also the sign of the hysteretic ferromagnetic response can be modulated by tuning the carrier density and displacement field. Rotating the sample in a fixed magnetic field demonstrates that the ferromagnetism is highly anisotropic and likely purely orbital in character.

The outcome of sutured wounds compared with tissue adhesive for paediatric wound closure: A meta-analysis.

A meta-analysis investigation was executed to measure the outcome of sutured wounds (SWs) compared with tissue adhesive (TA) for paediatric wound closure (PWC). A comprehensive literature inspection till February 2023 was applied and 2018 interrelated investigations were reviewed. The 18 chosen investigations enclosed 1697 children with PWC in the chosen investigations' starting point, 977 of them were utilising SWs, and 906 were utilising TA. Odds ratio (OR) in addition to 95% confidence intervals (CIs) were used to compute the value of the effect of SWs compared with TA for PWC by the dichotomous approaches and a fixed or random model. SWs had significantly higher wound cosmetic (WC) scores (mean deviation [MD], 1.70; 95% CI, 0.57-2.84, P = .003), lower wound dehiscence (WD) (OR 0.60; 95% CI, 0.06-0.43, P < .001), and lower cost (MD, -10.22; 95% CI, -10.94 to -9.50, P < .001) compared with those with TA in PWC. No significant difference was found between children utilising SWs and TA in wound infection (WI) (OR, 0.45; 95% CI, 0.15-1.30, P = .14) with no heterogeneity (I2 = 0%) in PWC. SWs had significantly higher WC scores, lower WD, and lower cost, yet, no significant difference was found in WI compared with those with TA in PWC. However, care must be exercised when dealing with its values because of the low sample size of some of the nominated investigations and the low number of selected investigations for the meta-analysis.

Dynamic and Comparative Transcriptome Analyses Reveal Key Factors Contributing to Cadmium Tolerance in Broomcorn Millet.

Broomcorn millet (Panicum miliaceum L.) has great potential in Cd phytoextraction, but its mechanisms are largely unknown. Two contrasting broomcorn millet varieties, 'Ningmi6' (Cd-sensitive variety) and '4452' (Cd-tolerant variety), were investigated through morphological, physiological, and transcriptomic analyses to determine the factors responsible for their differential Cd tolerance and translocation. The Cd-tolerant variety can accumulate more Cd, and its cell wall and vacuole component Cd proportions were higher compared with the Cd-sensitive variety. Under Cd stress, the glutathione content and peroxidase activity of the Cd-tolerant variety were significantly higher than those of the Cd-sensitive variety. Additionally, weighted gene co-expression network analysis (WGCNA) revealed hub modules that were associated with Cd stress and/or variety. Notably, genes involved in these hub modules were significantly enriched for roles in glutathione metabolism, phenylpropanoid biosynthesis, ABC transport, and metal ion transport process. These results suggested that regulation of genes associated with cell wall precipitation and vacuole compartmentalization may increase Cd tolerance and reduce Cd translocation in the Cd-tolerant variety, although it can absorb more Cd. This study provides a foundation for exploring molecular mechanisms of Cd tolerance and transport in broomcorn millet and new insights into improving Cd phytoremediation with this crop through genetic engineering.

Root characteristics critical for cadmium tolerance and reduced accumulation in wheat (Triticum aestivum L.).

Root radial transport is important for cadmium (Cd) absorption and root-shoot translocation. However, the relationship between root structural characteristics and radial transport of Cd in wheat is still unclear. Six wheat cultivars with different Cd tolerance and accumulation characteristics were used to investigate the roles of root phenotype, microstructure, and apoplastic and symplastic pathways in Cd uptake and root-shoot transport in pot culture. Longer root length, smaller root diameter, and more numerous root tips were more conducive to Cd absorption, while thicker roots were able to retain more Cd, thus reducing root-shoot transport and improving Cd tolerance of shoots. Cd stress can induce the deposition of apoplastic barriers in wheat roots, and the deposition of the apoplastic barrier increases under greater stress. The formation of apoplastic barriers can reduce Cd absorption and transfer to the shoot, and the presence of passage cells can weaken this effect. The cell wall thickening induced by Cd stress enhanced Cd adsorption capacity in wheat roots, but there was no significant correlation between Cd content and polysaccharide content in the cell wall. The up-regulated expression of TaHMA3 and TaVP1, which encode proteins related to Cd compartmentalization, was associated with increased Cd tolerance in wheat and decreased Cd translocation to aboveground parts. The morphology and anatomy of roots appear to play critical roles in Cd tolerance, uptake, and translocation in wheat. The present study provides useful information for the selection of wheat cultivars with low Cd accumulation.

Coexistence of large conventional and planar spin Hall effect with long spin diffusion length in a low-symmetry semimetal at room temperature.

The spin Hall effect (SHE) is usually observed as a bulk effect in high-symmetry crystals with substantial spin-orbit coupling (SOC), where the symmetric spin-orbit field imposes a widely encountered trade-off between spin Hall angle (θSH) and spin diffusion length (Lsf), and spin polarization, spin current and charge current are constrained to be mutually orthogonal. Here, we report a large θSH of 0.32 accompanied by a long Lsf of 2.2 µm at room temperature in a low-symmetry few-layered semimetal MoTe2, thus identifying it as an excellent candidate for simultaneous spin generation, transport and detection. In addition, we report that longitudinal spin current with out-of-plane polarization can be generated by both transverse and vertical charge current, due to the conventional and a newly observed planar SHE, respectively. Our study suggests that manipulation of crystalline symmetries and strong SOC opens access to new charge-spin interconversion configurations and spin-orbit torques for spintronic applications.

Flat Chern Band from Twisted Bilayer MnBi_{2}Te_{4}.

We construct a continuum model for the moiré superlattice of twisted bilayer MnBi_{2}Te_{4}, and study the band structure of the bilayer in both ferromagnetic (FM) and antiferromagnetic (AFM) phases. We find the system exhibits highly tunable Chern bands with Chern number up to 3. We show that a twist angle of 1° turns the highest valence band into a flat band with Chern number ±1 that is isolated from all other bands in both FM and AFM phases. This result provides a promising platform for realizing time-reversal breaking correlated topological phases, such as fractional Chern insulator and p+ip topological superconductor. In addition, our calculation indicates that the twisted stacking facilitates the emergence of quantum anomalous Hall effect in MnBi_{2}Te_{4}.

Loss of Annexin A5 expression attenuates the lipopolysaccharide-induced inflammatory response of rat alveolar macrophages.

Acute lung injury (ALI) is a common respiratory syndrome accompanied with an inflammation response. Annexin A5 (AnxA5) has anti-thrombotic, anti-apoptotic, and anti-inflammatory properties. The current study aims to explore the potential effect of AnxA5 on lipopolysaccharide (LPS)-induced inflammatory response in alveolar macrophages (AMs). Rat AMs (NR8383) were used in this study, and the cell viabilities at 4, 8, and 16 h after LPS administration with gradient concentrations were determined using cell counting kit-8 assay. Cell apoptosis and expressions of messenger RNAs (mRNAs) and protein were determined by flow cytometry, quantitative real-time polymerase chain reaction (qRT-PCR), and western blot, respectively. We found that LPS suppressed the viability of AMs in a dose-dependent manner, and it elevated the expression of AnxA5 in AMs. Inhibition of AnxA5 improved the cell viability compared with the LPS group and could reduce the apoptosis rate in comparison with LPS treatment. The knockdown of AnxA5 suppressed the expressions of tumor necrosis factor-α (TNF-α), interleukin (IL-1ß), and IL-6 at both protein and mRNA levels and regulated the expressions of apoptosis-related molecules (Bax, Bcl-2, and caspase-3). Moreover, the knockdown of AnxA5 improved the expression levels of inhibitory κB (IκB) and nuclear factor E2-related factor 2 (Nrf2) but inhibited the expression of nuclear transcription factor κB (NF-κB), compared with the LPS group. SN50 and ML385 were used to validate this signaling, and the inhibition of AnxA5 suppressed the LPS-induced inflammation, indicating that AnxA5 may be a potential anti-inflammatory target. In addition, NF-κB/Nrf2 signaling pathway may also be involved in the LPS-induced inflammatory response of rat alveolar macrophages.

Mapping Dynamical Magnetic Responses of Ultrathin Micron-Size Superconducting Films Using Nitrogen-Vacancy Centers in Diamond.

Two-dimensional superconductors have attracted growing interest because of their scientific novelty, structural tunability, and useful properties. Studies of their magnetic responses, however, are often hampered by difficulties to grow large-size samples of high quality and uniformity. We report here an imaging method that employed NV- centers in diamond as a sensor capable of mapping out the microwave magnetic field distribution on an ultrathin superconducting film of micron size. Measurements on a 33 nm thick film and a 125 nm thick bulklike film of Bi2Sr2CaCu2O8+δ revealed that the alternating current (ac) Meissner effect (or repulsion of ac magnetic field) set in at 78 and 91 K, respectively; the latter was the superconducting transition temperature of both films. The unusual ac magnetic response of the thin film presumably was due to thermally excited vortex-antivortex diffusive motion in the film. Spatial resolution of our ac magnetometer was limited by optical diffraction and the noise level was at 14 µT/Hz1/2. The technique could be extended with better detection sensitivity to extract local ac conductivity/susceptibility of ultrathin or monolayer superconducting samples as well as ac magnetic responses of other two-dimensional exotic thin films of limited lateral size.

[Effect of Schizonepetae Herba and Saposhnikoviae Radix on expression of AQP4 and AQP8 in colonic mucosa of rats with ulcerative colitis].

The aim of this paper was to investigate the effect of Schizonepetae Herba and Saposhnikoviae Radix(wind medicine) on the expression of AQP4 and AQP8 in colonic mucosa in rats with ulcerative colitis(UC). A total of 35 healthy SD male rats were randomly divided into normal group(gavaged with normal saline), DSS model group, as well as low, middle, and high dose wind medicine groups(Schizonepeta and Saposhnikovia 1∶1, gavaged at dosages of 6, 12, and 24 g·kg~(-1)·d~(-1)), with 7 in each group. UC rat model was established by free drinking of 3% dextran sulphate sodium(DSS) solution for 10 days. At the end of the 10 th day after the treatment, mice were put to death to collect colonic mucosa. The length of colon was measured; the colonic mucosal injury index(CMDI) and pathological changes of colon were observed. ELISA method was used for measuring the content of serum IL-1, IL-8, and immunohistochemical method was used to measure AQP4, AQP8 protein expressions in colon mucosa. The expressions of AQP4, AQP8 mRNA were measured by Real-time PCR. As compared with the normal group, the length of colon tissue was significantly reduced(P<0.01), CMDI scores and pathological scores were significantly increased(P<0.01), the levels of serum IL-1 and IL-8 were significantly increased(P<0.05) in model group; the immunohistochemical results showed that the protein expressions of AQP4, AQP8 were lower; the color was light yellow or brown; AQP4, AQP8 mRNA expressions in colon mucosa were significantly decreased in model group(P<0.01). CMDI scores, pathological scores, and the levels of serum IL-1, IL-8 in high, middle, low dose wind medicine groups were obvious lower than those in the model group(P<0.01 or P<0.05); the protein expressions of AQP4, AQP8 were higher; the color was chocolate brown or dark brown; the length of colon tissue, and the expressions of AQP4, AQP8 mRNA were obvious higher in wind medicine groups(P<0.01 or P<0.05). Schizonepetae Herba and Saposhnikoviae Radix could significantly improve the symptoms and histopathology of UC model rats and accelerate the intestinal mucosal healing. The mechanism may be related with up-regulating the expression level of AQP4 and AQP8 in colonic mucosa.

Gate-Tunable Topological Flat Bands in Trilayer Graphene Boron-Nitride Moiré Superlattices.

We investigate the electronic structure of the flat bands induced by moiré superlattices and electric fields in nearly aligned ABC trilayer graphene (TLG) boron-nitride (BN) interfaces where Coulomb effects can lead to correlated gapped phases. Our calculations indicate that valley-spin resolved isolated superlattice flat bands that carry a finite Chern number C=3 proportional to the layer number can appear near charge neutrality for appropriate perpendicular electric fields and twist angles. When the degeneracy of the bands is lifted by Coulomb interactions, these topological bands can lead to anomalous quantum Hall phases that embody orbital and spin magnetism. Narrow bandwidths of ∼10 meV achievable for a continuous range of twist angles θ≲0.6° with moderate interlayer potential differences of ∼50 meV make the TLG-BN systems a promising platform for the study of electric-field tunable Coulomb-interaction-driven spontaneous Hall phases.