The recent observation of the fractional quantum anomalous Hall effect in moir & eacute; fractional Chern insulators provides an opportunity to investigate zero magnetic field anyons. One approach for potentially realizing non-abelian anyons is to engineer higher flat Chern bands that mimic higher Landau levels. We investigate the interaction, topology and ferromagnetism of the second moir & eacute; miniband in twisted MoTe2 bilayers. At half-filling of the second miniband, we observed spontaneous ferromagnetism and an incipient Chern insulator state. The Chern numbers of the top two moir & eacute; flat bands exhibited opposite signs for twist angles above 3.1 degrees but had the same sign near 2.6 degrees, consistent with theoretical predictions. In the 2.6 degrees device, increasing the magnetic field induced a topological phase transition due to band-crossing between opposite valleys, resulting in an emergent state with Chern number C = -2. Additionally, an insulating state at half-filling of the second valley-polarized band indicates that a charge-ordered state is favoured over the fractional Chern insulator state. These findings lay a foundation for understanding the higher flat Chern bands, which are crucial for the discovery of non-abelian fractional Chern insulators.

The half-filled lowest Landau level is a fascinating platform for researching interacting topological phases. A celebrated example is the composite Fermi liquid, a non-Fermi liquid formed by composite fermions in strong magnetic fields1-10. Its zero-field counterpart is predicted in a twisted MoTe2 bilayer (tMoTe2)11,12-a recently discovered fractional Chern insulator exhibiting the fractional quantum anomalous Hall effect13-16. Although transport measurements at ν = -1/2 show signatures consistent with a zero-field composite Fermi liquid14, new probes are crucial to investigate the state and its elementary excitations. Here, by using the unique valley properties of tMoTe2, we report optical signatures of a zero-field composite Fermi liquid. We measured the degree of circular polarization (ρ) of trion photoluminescence versus hole doping and electric field. We found that, within the phase space showing robust ferromagnetism, ρ is near unity for Fermi liquid states. However, ρ is quenched at both integer and fractional Chern insulators, and in a hole doping range near ν = -1/2. Temperature, optical excitation power and electric-field-dependence measurements demonstrate that the quenching of ρ is a direct consequence of an energy gap (pseudogap) for electronic excitations of the Chern insulators (composite Fermi liquid): because the local spin-polarized excitations necessary to form trions are strongly suppressed, trion formation at the corresponding filling factors relies on optically generated unpolarized itinerant holes. Our work highlights a new excitonic probe of zero-field fractional Chern insulator physics, unique to tMoTe2.

In the absence of time reversal symmetry, orbital magnetization provides a sensitive probe of topology and interactions, with particularly rich phenomenology in Chern insulators where topological edge states carry large equilibrium currents. Here, we use a nanoscale superconducting sensor to map the magnetic fringe fields in twisted bilayers of MoTe$_2$, where transport and optical sensing experiments have revealed the formation of fractional Chern insulator (FCI) states at zero magnetic field. At a temperature of 1.6K, we observe oscillations in the local magnetic field associated with fillings $\nu=-1,-2/3,-3/5,-4/7$ and $-5/9$ of the first moiré hole band, consistent with the formation of FCIs at these fillings. By quantitatively reconstructing the magnetization, we determine the local thermodynamic gaps of the most robust FCI state at $\nu=-2/3$, finding $^{-2/3}\Delta$ as large as 7 meV. Spatial mapping of the charge density- and displacement field-tuned magnetic phase diagram further allows us to characterize sample disorder, which we find to be dominated by both inhomogeneity in the effective unit cell area as well as inhomogeneity in the band edge offset and bound dipole moment. Our results highlight both the challenges posed by structural disorder in the study of twisted homobilayer moiré systems and the opportunities afforded by the remarkably robust nature of the underlying correlated topological states.

In moiré materials with flat electronic bands and suitable quantum geometry, strong correlations can give rise to novel topological states of matter. The nontrivial band topology of twisted molybdenum ditelluride (tMoTe_2) – responsible for its fractional quantum anomalous Hall (FQAH) states – is predicted to arise from a layer-pseudospin skyrmion lattice. Tracing the layer polarization of wavefunctions within the moiré unit cell can thus offer crucial insights into the band topology. Here, we use scanning tunneling microscopy and spectroscopy (STM/S) to probe the layer-pseudospin skyrmion textures of tMoTe_2. We do this by simultaneously visualizing the moiré lattice structure and the spatial localization of its electronic states. We find that the wavefunctions associated with the topological flat bands exhibit a spatially-dependent layer polarization within the moiré unit cell. This is in excellent agreement with our theoretical modeling, thereby revealing a direct microscopic connection between the structural properties of tMoTe_2 and its band topology. Our work enables new pathways for engineering FQAH states with strain, as well as future STM studies of the intertwined correlated and topological states arising in gate-tunable devices.

The ability to control the underlying lattice geometry of a system may enable transitions between emergent quantum ground states. We report in situ gate switching between honeycomb and triangular lattice geometries of an electron many-body Hamiltonian in rhombohedral (R)-stacked molybdenum ditelluride (MoTe2) moiré bilayers, resulting in switchable magnetic exchange interactions. At zero electric field, we observed a correlated ferromagnetic insulator near one hole per moiré unit cell with a widely tunable Curie temperature up to 14 K. Applying an electric field switched the system into a half-filled triangular lattice with antiferromagnetic interactions; further doping this layer-polarized superlattice tuned the antiferromagnetic exchange interaction back to ferromagnetic. Our work demonstrates R-stacked MoTe2 moirés to be a laboratory for engineering correlated states with nontrivial topology.

The integer quantum anomalous Hall (QAH) effect is a lattice analogue of the quantum Hall effect at zero magnetic field(1-3). This phenomenon occurs in systems with topologically non-trivial bands and spontaneous time-reversal symmetry breaking. Discovery of its fractional counterpart in the presence of strong electron correlations, that is, the fractional QAH effect(4-7), would open a new chapter in condensed matter physics. Here we report the direct observation of both integer and fractional QAH effects in electrical measurements on twisted bilayer MoTe2. At zero magnetic field, near filling factor nu = -1 (one hole per moire unit cell), we see an integer QAH plateau in the Hall resistance R-xy quantized to h/e(2) +/- 0.1%, whereas the longitudinal resistance R-xx vanishes. Remarkably, at nu = -2/3 and -3/5, we see plateau features in R-xy at 3/2h/e(2) +/- 1% and 5/3h/e(2) +/- 3%, respectively, whereas R-xx remains small. All features shift linearly versus applied magnetic field with slopes matching the corresponding Chern numbers -1, -2/3 and -3/5, precisely as expected for integer and fractional QAH states. Additionally, at zero magnetic field, R-xy is approximately 2h/e(2) near half-filling (nu = -1/2) and varies linearly as nu is tuned. This behaviour resembles that of the composite Fermi liquid in the half-filled lowest Landau level of a two-dimensional electron gas at high magnetic field(8-14). Direct observation of the fractional QAH and associated effects enables research in charge fractionalization and anyonic statistics at zero magnetic field.

The interplay between spontaneous symmetry breaking and topology can result in exotic quantum states of matter. A celebrated example is the quantum anomalous Hall (QAH) state, which exhibits an integer quantum Hall effect at zero magnetic field thanks to its intrinsic ferromagnetism. In the presence of strong electron-electron interactions, exotic fractional-QAH (FQAH) states at zero magnetic field can emerge. These states could host fractional excitations, including non-Abelian anyons - crucial building blocks for topological quantum computation. Flat Chern bands are widely considered as a desirable venue to realize the FQAH state. For this purpose, twisted transition metal dichalcogenide homobilayers in rhombohedral stacking have recently been predicted to be a promising material platform. Here, we report experimental signatures of FQAH states in 3.7-degree twisted MoTe2 bilayer. Magnetic circular dichroism measurements reveal robust ferromagnetic states at fractionally hole filled moiré minibands. Using trion photoluminescence as a sensor, we obtain a Landau fan diagram which shows linear shifts in carrier densities corresponding to the v=-2/3 and -3/5 ferromagnetic states with applied magnetic field. These shifts match the Streda formula dispersion of FQAH states with fractionally quantized Hall conductance of -2/3$e^2/h$ and -3/5$e^2/h$, respectively. Moreover, the v=-1 state exhibits a dispersion corresponding to Chern number -1, consistent with the predicted QAH state. In comparison, several non-ferromagnetic states on the electron doping side do not disperse, i.e., are trivial correlated insulators. The observed topological states can be further electrically driven into topologically trivial states. Our findings provide clear evidence of the long-sought FQAH states, putting forward MoTe2 moiré superlattices as a fascinating platform for exploring fractional excitations.

Mechanical deformation of a crystal can have a profound effect on its physical properties. Notably, even small modifications of bond geometry can completely change the size and sign of magnetic exchange interactions and thus the magnetic ground state. Here we report the strain tuning of the magnetic properties of the A-type layered antiferromagnetic semiconductor CrSBr achieved by designing a strain device that can apply continuous, in situ uniaxial tensile strain to two-dimensional materials, reaching several percent at cryogenic temperatures. Using this apparatus, we realize a reversible strain-induced antiferromagnetic-to-ferromagnetic phase transition at zero magnetic field and strain control of the out-of-plane spin-canting process. First-principles calculations reveal that the tuning of the in-plane lattice constant strongly modifies the interlayer magnetic exchange interaction, which changes sign at the critical strain. Our work creates new opportunities for harnessing the strain control of magnetism and other electronic states in low-dimensional materials and heterostructures.

The emergence of atomically thin van der Waals magnets provides a new platform for the studies of two-dimensional magnetism and its applications. However, the widely used measurement methods in recent studies cannot provide quantitative information of the magnetization nor achieve nanoscale spatial resolution. These capabilities are essential to explore the rich properties of magnetic domains and spin textures. Here, we employ cryogenic scanning magnetometry using a single-electron spin of a nitrogen-vacancy center in a diamond probe to unambiguously prove the existence of magnetic domains and study their dynamics in atomically thin CrBr 3 . By controlling the magnetic domain evolution as a function of magnetic field, we find that the pinning effect is a dominant coercivity mechanism and determine the magnetization of a CrBr 3 bilayer to be about 26 Bohr magnetons per square nanometer. The high spatial resolution of this technique enables imaging of magnetic domains and allows to locate the sites of defects that pin the domain walls and nucleate the reverse domains. Our work highlights scanning nitrogen-vacancy center magnetometry as a quantitative probe to explore nanoscale features in two-dimensional magnets.

Moiré superlattices of twisted nonmagnetic two-dimensional (2D) materials are highly controllable platforms for the engineering of exotic correlated and topological states. Here, we report emerging magnetic textures in small-angle twisted 2D magnet chromium triiodide (CrI3). Using single-spin quantum magnetometry, we directly visualized nanoscale magnetic domains and periodic patterns, a signature of moiré magnetism, and measured domain size and magnetization. In twisted bilayer CrI3, we observed the coexistence of antiferromagnetic (AFM) and ferromagnetic (FM) domains with disorder-like spatial patterns. In twisted double-trilayer CrI3, AFM and FM domains with periodic patterns appear, which is in good agreement with the calculated spatial magnetic structures that arise from the local stacking-dependent interlayer exchange interactions in CrI3 moiré superlattices. Our results highlight magnetic moiré superlattices as a platform for exploring nanomagnetism.

The development of van der Waals (vdW) crystals and their heterostructures has created a fascinating platform for exploring optoelectronic properties in the two-dimensional (2D) limit. With the recent discovery of 2D magnets, the control of the spin degree of freedom can be integrated to realize 2D spin-optoelectronics. Here, we report spin photovoltaic effects in vdW heterostructures of 2D magnet chromium triiodide (CrI3) sandwiched by graphene contacts. The photocurrent displays a distinct dependence on light helicity, which can be tuned by varying the magnetic states and photon energy. Circular polarization–resolved absorption measurements reveal that these observations originate from magnetic order–coupled and, thus, helicity-dependent charge-transfer excitons. The photocurrent displays multiple plateaus as the magnetic field is swept, associated with different CrI3 spin configurations. Giant photo-magnetocurrent is observed, which tends to infinity for a small applied bias. Our results pave the way to explore emergent photospintronics by engineering magnetic vdW heterostructures.

Magnetic-domain structure and dynamics play an important role in understanding and controlling the magnetic properties of two-dimensional magnets, which are of interest to both fundamental studies and applications. However, the probe methods based on the spin-dependent optical permeability and electrical conductivity can neither provide quantitative information of the magnetization nor achieve nanoscale spatial resolution. These capabilities are essential to image and understand the rich properties of magnetic domains. Here, we employ cryogenic scanning magnetometry using a single-electron spin of a nitrogen-vacancy center in a diamond probe to unambiguously prove the existence of magnetic domains and study their dynamics in atomically thin CrBr3. The high spatial resolution of this technique enables imaging of magnetic domains and allows to resolve domain walls pinned by defects. By controlling the magnetic domain evolution as a function of magnetic field, we find that the pinning effect is a dominant coercivity mechanism with a saturation magnetization of about 26μB/nm2 for bilayer CrBr3. The magnetic-domain structure and pinning-effect dominated domain reversal process are verified by micromagnetic simulation. Our work highlights scanning nitrogen-vacancy center magnetometry as a quantitative probe to explore two-dimensional magnetism at the nanoscale.

William Dembski’s and Michael Behe’s recent article, Irreducible Complexity Revisited, provides an important update on the irreducible complexity argument since the publication some 8 years ago of Behe’s book, Darwin’s Black Box. Their article, however, exhibits some confusion, or at least a lack of explicit clarification, regarding the interplay among specified complexity, cumulative complexity and irreducible complexity. In the present article, I analyze the relationship of these concepts and show that the argument from irreducible complexity cannot be divorced from the broader argument of specified complexity. While this has been previously acknowledged in a broad sense, I make explicit irreducible complexity’s dependence on specified complexity, including specified complexity as applied to cumulative complexity, and further demonstrate why this dependence causes the irreducible complexity argument to break down in the evolutionist’s mind. Indeed, this dependence is directly responsible for evolutionists’ ability to acknowledge the existence of irreducibly complex biological features while still rejecting the irreducible complexity argument. Finally, the present article demonstrates how the arguments from complexity can be better understood within the context of an integrated approach to the complexity space. With a better understanding of the complexity space, the concepts of irreducible complexity and specified complexity can be fortified and focused in order to bring the full weight of these arguments to bear on evolutionary claims. In his recent essay, Irreducible Complexity Revisited, 2 William Dembski provides an update on the irreducible complexity argument, and a brief response to criticisms raised, since the publication some 8 years ago of Michael Behe’s book, Darwin’s Black Box. Dembski (and Behe, who is acknowledged as a co-author of the essay) presents a number of arguments previously advanced, as well as a probability equation for assessing the likelihood of a Darwinian pathway creating an irreducibly complex system. An update on irreducible complexity is timely, given the responses to Behe’s book raised by Darwinists and the continuing assertion – popular within evolutionary circles – that Behe’s arguments have somehow been refuted. Dembski shows that this common refrain