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We show that an s-wave superconductor may become topological in the presence of a magnetic field which leads to formation of an Abrikosov vortex lattice. Below the upper critical field, a new superconducting state with a nontrivial topological number emerges, which we call Abrikosov-Chern superconducting state. Deeper in the superconducting domain, the topological number changes in steps, eventually reaching zero. Our theory uncovers the nature of evolution from an integer quantum Hall state having a cyclotron gap above the upper critical field to the topologically-trivial s-wave superconductor carrying finite-energy Caroli-de Gennes-Matricon levels at low field. Topological transitions manifest as changes in the number of edge modes, detectable through tunneling microscopy and thermal transport measurements., Comment: 5 pages, 3 figures

A microscopic mechanism for chiral superconductivity from Coulomb repulsion is proposed for spin- and valley-polarized state of rhombohedral multilayer graphene. The superconducting state occurs at low density, has chiral $p$-wave pairing symmetry, and exhibits highest $T_c$ close to a Lifshitz transition from annular to simply-connected Fermi sea. This Lifshitz transition also marks a topological phase transition from a trivial to a topological superconducting phase hosting Majorana fermions. The chirality of the superconducting order parameter is selected by the chirality of the valley-polarized Bloch electrons. Our results are in reasonable agreement with observations in a recent experiment on tetralayer graphene [arXiv:2408.15233], Comment: 7+4 pages, 4+4 figures

Superconductors can form ideal diodes carrying nondissipative supercurrents in the forward direction and dissipative currents in the backward direction. The Josephson diode has proven to be a promising design where the junction between the two superconductors comprises the weakest link and thus provides the dominant mechanism. We here propose a Josephson diode based on a single magnetic material with a conical spin structure. The helical spin rotation produces Rashba-like band splitting inversely proportional to the rotation period. Together with the Zeeman splitting caused by the time-reversal symmetry breaking of the noncoplanar spin texture, this results in a large diode efficiency close to the $0-\pi$ transition of the magnetic Josephson junction., Comment: Main text: 6 pages, 4 figures. Supplemental material: 2 pages, 2 figures

Chiral superconductors are unconventional superconducting states that break time reversal symmetry spontaneously and typically feature Cooper pairing at non-zero angular momentum. Such states may host Majorana fermions and provide an important platform for topological physics research and fault-tolerant quantum computing. Despite of intensive search and prolonged studies of several candidate systems, chiral superconductivity has remained elusive so far. Here we report the discovery of unconventional superconductivity in rhombohedral tetra-layer graphene. We observed two superconducting states in the gate-induced flat conduction bands with Tc up to 300 mK and charge density ne as low as 2.4*1011 cm-2, appearing robustly in three different devices, where electrons reside close to a proximate WSe2 layer, far away from WSe2, and in the absence of WSe2 respectively. Spontaneous time-reversal-symmetry-breaking (TRSB) due to electron's orbital motion is found, and several observations indicate the chiral nature of these superconducting states, including 1. In the superconducting state, Rxx shows fluctuations at zero magnetic field and magnetic hysteresis versus an out-of-plane magnetic field B, which are absent from all other superconductors; 2. one superconducting state develops within a spin- and valley-polarized quarter-metal phase, and is robust against the neighboring spin-valley-polarized quarter-metal state under B; 3. the normal states show anomalous Hall signals at zero magnetic field and magnetic hysteresis. We also observed a critical B > 0.9 Tesla, higher than any graphene superconductivity reported so far and indicates a strong-coupling superconductivity close the BCS-BEC crossover. Our observations establish a pure carbon material for the study of topological superconductivity, and pave the way to explore Majorana modes and topological quantum computing.

We study the entire energy spectrum of an electron droplet in the lowest Landau level. By exact diagonalization calculations, we find highly excited states in the middle of the spectrum that display unexpected density distribution and pair correlation. We show that these exceptional excited states contain tightly bound electron bubbles with local filling $\nu = 1$ that form various ordered structures. Remarkably, these bubble excited states are shown to exist for both the $1/r$ Coulomb interaction and the $1/r^3$ dipole interaction. The experimental realization of bubble excited states in moir\'e materials under a magnetic field is also discussed., Comment: Main: 6 pages, 4 figures. SM: 11 pages, 6 figures

We introduce the notion of nonreciprocal superconductors where inversion and time-reversal symmetries are broken, giving rise to an asymmetric energy dispersion. We demonstrate that nonreciprocal superconductivity can be detected by Andreev reflection. In particular, a transparent junction between a normal metal and a nonreciprocal superconductor generally exhibits an asymmetric current-voltage characteristic, which serves as a defining feature of nonreciprocal superconductivity. Unlike the superconducting diode effects, our detection scheme has the advantage of avoiding large critical currents that turn the superconducting state to normal. Finally, we discuss candidates for nonreciprocal superconductivity, including graphene, UTe2, as well as engineered platforms., Comment: 17 pages, 12 figures

We show that the static structure factor of general many-body systems with $U(1)$ symmetry has a lower bound determined only by the ground state Chern number. Our bound relies only on causality and non-negative energy dissipation, and holds for a wide range of systems. We apply our theory to (fractional) Chern insulators, (fractional) quantum spin Hall insulators, topological superconductors, and chiral spin liquids. Our results uncover a universal feature of topological phases beyond the quantized response., Comment: 5 pages, 2 figures for the main text and 5 pages, 1 figure for Supplemental Materials

Moir\'e materials provide an ideal platform for exploring quantum phases of matter. However, solving the many-electron problem in moir\'e systems is challenging due to strong correlation effects. We introduce a powerful variational representation of quantum states, many-body neural Bloch wavefunction, to solve many-electron problems in moir\'e materials accurately and efficiently. Applying our method to the semiconductor heterobilayer WSe2/WS2 , we obtain a generalized Wigner crystal at filling factor n = 1/3, a Mott insulator n = 1, and a correlated insulator with local magnetic moments and antiferromagnetic spin correlation at n = 2. Our neural network approach improves the simulation accuracy of strongly interacting moir\'e materials and paves the way for discovery of new quantum phases with variational learning principle in a unified framework.

In a PN junction, the separation between positive and negative charges leads to diode transport. In the past few years, the intrinsic diode transport in noncentrosymmetric polar conductors has attracted great interest, because it suggests novel nonlinear applications and provides a symmetry-sensitive probe of Fermi surface. Recently, such studies have been extended to noncentrosymmetric superconductors, realizing the superconducting diode effect. Here, we show that, even in a centrosymmetric crystal without directional charge separation, the spins of an antiferromagnet (AFM) can generate a spatial directionality, leading to an AFM diode effect. We observe large second-harmonic transport in a nonlinear electronic device enabled by the compensated AFM state of even-layered MnBi2Te4. We also report a novel electrical sum-frequency generation (SFG), which has been rarely explored in contrast to the well-known optical SFG in wide-gap insulators. We demonstrate that the AFM enables an in-plane field-effect transistor and harvesting of wireless electromagnetic energy. The electrical SFG establishes a powerful method to study nonlinear electronics built by quantum materials. The AFM diode effect paves the way for potential device concepts including AFM logic circuits, self-powered AFM spintronics, and other applications that potentially bridge nonlinear electronics with AFM spintronics., Comment: 33+8 pages, 14+2 figures

Motivated by a recent experiment reporting the fractional quantum spin Hall effect in twisted ${\rm MoTe}_2$, we investigate microscopically the prospects of realizing exotic topologically ordered states beyond conventional quantum Hall physics. We show that a non-Abelian spin Hall insulator, a state of two copies of the non-Abelian Moore-Read state, can be stabilized at half filling of time-reversal conjugate Chern bands. We elucidate that the existence of this phase relies on the reduction of opposite-spin interactions at short distances to overcome the Ising ferromagnetism. Moreover, we demonstrate that band mixing provides a generic mechanism for this reduction to be achieved. Quite remarkably, we find that a renormalization of opposite-spin interactions at short distances as small as 15 % of the moir\'e period is sufficient for a direct transition to a completely spin unpolarized phase which supports the non-Abelian spin Hall insulator. Furthermore, we show that the non-Abelian spin Hall insulator can either break time-reversal symmetry or preserve it depending on the underlying topological order., Comment: 10 pages, 7 figures

The recent observation of the fractional quantum anomalous Hall effect in moir\'e fractional Chern insulators (FCI) provides opportunities for investigating zero magnetic field anyons. So far, both experimental and theoretical results suggest that filling > 1/3 FCI states in the first Chern band share features with those of the lowest Landau level (LL). To create the possibility of realizing non-Abelian anyons, one route is to engineer higher flat Chern bands that mimic higher LLs. Here, we investigate the interaction, topology, and ferromagnetism of the second moir\'e miniband in twisted MoTe2 bilayer (tMoTe2). Around filling factor v = -3, i.e., half-filling of the second miniband, we uncover spontaneous ferromagnetism and an incipient Chern insulator state. By measuring the anomalous Hall effect as a function of twist angle, we find that the Chern numbers (C) of the top two moir\'e flat bands have opposite sign (C = -+1) at twist angles above 3.1{\deg} but the same sign (C = -1) around 2.6{\deg}. This observation is consistent with the recently predicted twist-angle dependent band topology, resulting from the competition between moir\'e ferroelectricity and piezoelectricity. As we increase the magnetic field, only the small twist-angle device (2.6{\deg}) experiences a topological phase transition with an emergent C = -2 state. This is attributed to a Zeeman field-induced band crossing between opposite valleys, with the determined C = -1 for the top two bands. Our results lay a firm foundation for understanding the higher flat Chern bands, which is essential for the prediction or discovery of non-Abelian FCIs., Comment: 24 pages, 4 figures

We introduce the concept of quantum weight as a fundamental property of quantum many-body systems that is encoded in the ground-state static structure factor and characterizes density fluctuation at long wavelength. We show that quantum weight of three-dimensional electron systems is related by a sum rule to the inverse dielectric function, which describes electron energy loss spectrum. Using this relation, we derive an upper and a lower bound on the quantum weight of real materials in terms of their electron density, static dielectric constant, and plasmon energy. For systems with short-range interactions or Coulomb systems in reduced dimensions, we derive a sum rule relating the quantum weight to the optical conductivity and establish a remarkable connection with the quantum geometry of many-body ground states. Our work highlights quantum weight as a key material parameter that can be experimentally determined., Comment: Supersedes arXiv:2401.13847. 5 pages, 1 figures + Supplemental Materials (3 pages)

The convergence of topology and correlations represents a highly coveted realm in the pursuit of novel quantum states of matter. Introducing electron correlations to a quantum spin Hall (QSH) insulator can lead to the emergence of a fractional topological insulator and other exotic time-reversal-symmetric topological order, not possible in quantum Hall and Chern insulator systems. However, the QSH insulator with quantized edge conductance remains rare, let alone that with significant correlations. In this work, we report a novel dual QSH insulator within the intrinsic monolayer crystal of TaIrTe4, arising from the interplay of its single-particle topology and density-tuned electron correlations. At charge neutrality, monolayer TaIrTe4 demonstrates the QSH insulator that aligns with single-particle band structure calculations, manifesting enhanced nonlocal transport and quantized helical edge conductance. Interestingly, upon introducing electrons from charge neutrality, TaIrTe4 only shows metallic behavior in a small range of charge densities but quickly goes into a new insulating state, entirely unexpected based on TaIrTe4's single-particle band structure. This insulating state could arise from a strong electronic instability near the van Hove singularities (VHS), likely leading to a charge density wave (CDW). Remarkably, within this correlated insulating gap, we observe a resurgence of the QSH state, marked by the revival of nonlocal transport and quantized helical edge conduction. Our observation of helical edge conduction in a CDW gap could bridge spin physics and charge orders. The discovery of a dual QSH insulator introduces a new method for creating topological flat minibands via CDW superlattices, which offer a promising platform for exploring time-reversal-symmetric fractional phases and electromagnetism., Comment: 23 pages, 15 figures, submitted version

We uncover a new superconducting state with partial spin polarization induced by a magnetic field. This state, which we call "magnonic superconductor", lacks a conventional pairing order parameter, but is characterized instead by a composite order parameter that represents the binding of electron pairs and magnons. We rigorously demonstrate the existence of magnonic superconductivity with high transition temperature in one- and two-dimensional Hubbard models with repulsive interaction. We further show that magnonic Cooper pairs can attract to form higher-charge bound states, which can give rise to charge-$6e$ superconductivity., Comment: 17 pages, 13 figures In the updated version we added new results

We explore the possibility of quantum states of matter that are compressible but have vanishing DC conductivity in the absence of disorder. We show that the composite Fermi liquid emerging from strong interaction in a generic Chern band has zero Drude weight, in stark contrast to normal Fermi liquids. Our work establishes the absence of Drude weight as the defining property of the composite Fermi liquid phase, which distinguishes it from the Fermi liquid or other types of non-Fermi liquids. Our findings point to a possibly wide class of gapless quantum phases with unexpected transport and optical properties., Comment: 5 + 6

Motivated by the recent discovery of fractional quantum anomalous Hall states in moir\'e systems, we consider the possibility of realizing non-Abelian phases in topological minibands. We study a family of moir\'e systems, skyrmion Chern band (SCB) models, which can be realized in two-dimensional semiconductor/magnetic skyrmion heterostructures and also capture the essence of twisted transition metal dichalcogenide (TMD) homobilayers. We show using many-body exact diagonalization that, in spite of strong Berry curvature variations in momentum space, the non-Abelian Moore-Read state can be realized at half filling of the second miniband. These results demonstrate the feasibility of non-Abelian fractionalization in moir\'e systems without Landau levels and shed light on the desirable conditions for their realization. In particular, we highlight the prospect of realizing the Moore-Read state in twisted semiconductor bilayers., Comment: V1: 5 pages, 4 figures. V2: 5 pages, 5 figures. Added exact diagonalization results for twisted MoTe2. V3: 5 pages, 5 figures (main) + 13 pages, 9 figures (SM). Added results on Pfaffian vs anti-Pfaffian

We predict the emergence a state of matter with intertwined ferromagnetism, charge order and topology in fractionally filled moir\'e superlattice bands. Remarkably, these quantum anomalous Hall crystals exhibit a quantized integer Hall conductance that is different than expected from the filling and Chern number of the band. Microscopic calculations show that this phase is robustly favored at half-filling ($\nu=1/2$) at larger twist angles of the twisted semiconductor bilayer $t$MoTe$_2$, Comment: 5 + 5 pages, PRL Editors' Suggestion

The pyrochlore lattice, a three-dimensional network of corner-sharing tetrahedra, is a promising material playground for correlated topological phases arising from the interplay between spin-orbit coupling (SOC) and electron-electron interactions. Due to its geometrically frustrated lattice structure, exotic correlated states on the pyrochlore lattice have been extensively studied using various spin Hamiltonians in the localized limit. On the other hand, the topological properties of the electronic structure in the pyrochlore lattice have rarely been explored, due to the scarcity of pyrochlore materials in the itinerant paramagnetic limit. Here, we explore the topological electronic band structure of pyrochlore superconductor RbBi$_{2}$ using angle-resolved photoemission spectroscopy. Thanks to the strong SOC of the Bi pyrochlore network, we experimentally confirm the existence of three-dimensional (3D) massless Dirac fermions enforced by nonsymmorphic symmetry, as well as a 3D quadratic band crossing protected by cubic crystalline symmetry. Furthermore, we identify an additional 3D linear Dirac dispersion associated with band inversion protected by threefold rotation symmetry. These observations reveal the rich non-trivial band topology of itinerant pyrochlore lattice systems in the strong SOC regime. Through manipulation of electron correlations and SOC of the frustrated pyrochlore lattices, this material platform is a natural host for exotic phases of matter, including the fractionalized quantum spin Hall effect in the topological Mott insulator phase, as well as axion electrodynamics in the axion insulator phase., Comment: 18 pages, 4 figures

We show that topological flat minibands can be engineered in a class of narrow gap semiconductor films using only an external electrostatic superlattice potential. We demonstrate that, for realistic material parameters, these bands are capable of hosting correlated topological phases such as integer and fractional quantum anomalous Hall states and composite Fermi liquid phases at zero magnetic field. Our results provide a path towards the realization of fractionalized topological states in a broad range of materials.

We establish a universal relation between the energy gap and the static dielectric constant for all insulating states. This relation yields an upper bound on the energy gap, which only depends on the electron density and electronic dielectric constant. We identify two types of energy gaps associated with transverse and longitudinal excitations at long wavelength, which correspond to the optical gap and the plasmon energy respectively. Their upper bounds are set by the dielectric constant and its inverse respectively. The transverse gap bound is calculated for a wide range of materials and compared with the measured optical gap. A remarkable case is cubic boron nitride, in which the direct gap reaches \SI{72}{\percent} of the bound. Our results are derived from the Kramers-Kronig relation and the $f$-sum rule, and therefore rest on general physical principles., Comment: 6 pages + references + Supplemental materials (4 pages). 5 figure, 1 table. Significantly expanded in v2

Twisted homobilayer transition metal dichalcogenide (TMD) attracts an expanding experimental interest recently for exhibiting a variety of topological and magnetic states even at zero magnetic field. Most of the studies right now focus on hole filling nu_h < 1, while a rich phase diagram at higher hole filling calls for more investigation. We perform a thorough survey of possible interaction-driven phases at higher integer hole fillings. We first construct the continuum model from a first-principles calculation, and then perform a self-consistent Hartree-Fock study of the interacting ground states. We identify various valley polarized (VP) states at odd integer fillings and intervalley coherent (IVC) states at even integer fillings and discuss the energetics competition among them. We also discuss the origin and the experimental implications of the curious Chern insulator at nu_h = 2., Comment: 6 pages, 3 figures

Semiconductor moir\'e superlattices provide a versatile platform to engineer new quantum solids composed of artificial atoms on moir\'e sites. Previous studies have mostly focused on the simplest correlated quantum solid - the Fermi-Hubbard model - where intra-atom interactions are simplified to a single onsite repulsion energy U. These studies have revealed novel quantum phases ranging from Mott insulators to quantum anomalous Hall insulators at a filling of one electron per moir\'e unit cell. New types of quantum solids should arise at even higher filling factors where the multi-electron configuration of moir\'e artificial atoms provides new degrees of freedom. Here we report the experimental observation of Wigner molecular crystals emerging from multi-electron artificial atoms in twisted bilayer WS2 moir\'e superlattices. Moir\'e artificial atoms, unlike natural atoms, can host qualitatively different electron states due to the interplay between quantized energy levels and Coulomb interactions. Using scanning tunneling microscopy (STM), we demonstrate that Wigner molecules appear in multi-electron artificial atoms when Coulomb interactions dominate. Three-electron Wigner molecules, for example, are seen to exhibit a characteristic trimer pattern. The array of Wigner molecules observed in a moir\'e superlattice comprises a new crystalline phase of electrons: the Wigner molecular crystal. We show that these Wigner molecular crystals are highly tunable through mechanical strain, moir\'e period, and carrier charge type. Our study presents new opportunities for exploring quantum phenomena in moir\'e quantum solids composed of multi-electron artificial atoms.

The combination of repulsive and attractive Coulomb interactions in a quantum electron(e)-hole(h) fluid can give rise to novel correlated phases of multiparticle charge complexes such as excitons, trions and biexcitons. Here we report the first experimental realization of an electrically controlled interlayer trion fluid in two-dimensional van der Waals heterostructures. We demonstrate that in the strong coupling regime of electron-hole bilayers, electrons and holes in separate layers can spontaneously form three-particle trion bound states that resemble positronium ions in high energy physics. The interlayer trions can assume 1e-2h and 2e-1h configurations, where electrons and holes are confined in different transition metal dichalcogenide layers. We show that the two correlated holes in 1e-2h trions form a spin-singlet state with a spin gap of ~1meV. By electrostatic gating, the equilibrium state of our system can be continuously tuned into an exciton fluid, a trion fluid, an exciton-trion mixture, a trion-charge mixture or an electron-hole plasma. Upon optical excitation, the system can host novel high-order multiparticle charge complexes including interlayer four-particle complex (tetrons) and five-particle complex (pentons). Our work demonstrates a unique platform to study novel correlated phases of tunable Bose-Fermi mixtures and opens up new opportunities to realize artificial ions/molecules in electronic devices.

Moir\'e materials provide a highly tunable platform in which novel electronic phenomena can emerge. We study strained moir\'e materials in a uniform magnetic field and predict highly anisotropic electrical conductivity which switches easy-axis as magnetic field or strain is varied. The dramatic anisotropy reflects one-dimensional localization (dimensional reduction) of the electron wavefunctions along a crystal axis due to quantum interference effects. This can be understood in an effective one-dimensional quasiperiodic Aubry-Andr\'e-like models, or in a complementary semiclassical picture. This phenomenon should be observable in strained moir\'e materials at realistic fields and low strain disorder, as well as unstrained systems with anisotropic Fermi surfaces., Comment: 6+6 pages, 4 figures

Large-scale moir\'e systems are extraordinarily sensitive, with even minute atomic shifts leading to significant changes in electronic structures. Here, we investigate the lattice relaxation effect on moir\'e band structures in twisted bilayer MoTe$_2$ with two approaches: (a) large-scale plane-wave basis first principle calculation down to $2.88^{\circ}$, (b) transfer learning structure relaxation + local-basis first principles calculation down to $1.1^{\circ}$. We use two types of van der Waals corrections: the D2 method of Grimme and the density-dependent energy correction, and find that the density-dependent energy correction yields a continuous evolution of bandwidth with twist angles. Based on the above results. we develop a more complete continuum model with a single set of parameters for a wide range of twist angles, and perform many-body simulations at $\nu=-1,-2/3, -1/3$., Comment: 6+15 pages, 5+16 figs, updated continuum model fitting, dDsC correction in maintext, D2 correction in the SM

We develop a pairing-based graph neural network for simulating quantum many-body systems. Our architecture augments a BCS-type geminal wavefunction with a generalized pair amplitude parameterized by a graph neural network. Variational Monte Carlo with our neural network simultaneously provides an accurate, flexible, and scalable method for simulating many-electron systems. We apply this method to two-dimensional semiconductor electron-hole bilayers and obtain accurate results on a variety of interaction-induced phases, including the exciton Bose-Einstein condensate, electron-hole superconductor, and bilayer Wigner crystal. Our study demonstrates the potential of physically-motivated neural network wavefunctions for quantum materials simulations.

The quantum anomalous Hall effect (QAHE) is a robust topological phenomenon featuring quantized Hall resistance at zero magnetic field. We report the QAHE in a rhombohedral pentalayer graphene/monolayer WS2 heterostructure. Distinct from other experimentally confirmed QAHE systems, this system has neither magnetic element nor moir\'e superlattice effect. The QAH states emerge at charge neutrality and feature Chern numbers C = +-5 at temperatures up to about 1.5 K. This large QAHE arises from the synergy of the electron correlation in intrinsic flat bands of pentalayer graphene, the gate-tuning effect, and the proximity-induced Ising spin-orbit-coupling. Our experiment demonstrates the potential of crystalline two-dimensional materials for intertwined electron correlation and band topology physics, and may enable a route for engineering chiral Majorana edge states., Comment: to be published in Science

Quantum geometry of electronic wavefunctions results in fascinating topological phenomena. A prominent example is the intrinsic anomalous Hall effect (AHE) in which a Hall voltage arises in the absence of an applied magnetic field. The AHE requires a coexistence of Berry curvature and spontaneous time-reversal symmetry breaking. These conditions can be realized in two-dimensional moir\'e systems with broken $xy$-inversion symmetry ($C_{2z}$) that host flat electronic bands. Here, we explore helical trilayer graphene (HTG), three graphene layers twisted sequentially by the same angle forming two misoriented moir\'e patterns. Although HTG is globally $C_{2z}$-symmetric, surprisingly we observe clear signatures of topological bands. At a magic angle $\theta_\mathrm{m}\approx 1.8^\circ$, we uncover a robust phase diagram of correlated and magnetic states using magnetotransport measurements. Lattice relaxation leads to large periodic domains in which $C_{2z}$ is broken on the moir\'e scale. Each domain harbors flat topological bands with valley-contrasting Chern numbers $\pm(1,-2)$. We find correlated states at integer electron fillings per moir\'e unit cell $\nu=1,2,3$ and fractional fillings $2/3,7/2$ with the AHE arising at $\nu=1,3$ and $2/3,7/2$. At $\nu=1$, a time-reversal symmetric phase appears beyond a critical electric displacement field, indicating a topological phase transition. Finally, hysteresis upon sweeping $\nu$ points to first-order phase transitions across a spatial mosaic of Chern domains separated by a network of topological gapless edge states. We establish HTG as an important platform that realizes ideal conditions for exploring strongly interacting topological phases and, due to its emergent moir\'e-scale symmetries, demonstrates a novel way to engineer topology.

Motivated by the recent experimental breakthrough on the observation of the fractional quantum anomalous Hall (FQAH) effects in semiconductor and graphene moir\'{e} materials, we explore the rich physics associated with the coexistence of FQAH effect and the charge density wave (CDW) order that spontaneously breaks the translation symmetry. We refer to a state with both properties as "FQAH-crystal". We show that the interplay between FQAH effect and CDW can lead to a rich phase diagram including multiple topological phases and topological quantum phase transitions at the same moir\'e filling. In particular, we demonstrate the possibility of direct quantum phase transitions from a FQAH-crystal with Hall conductivity $\sigma_H = - 2/3$ to a trivial CDW insulator with $\sigma_H = 0$, and more interestingly, to a QAH-crystal with $\sigma_H= -1$., Comment: 9 pages

The fractional quantum anomalous Hall effect (FQAHE), the analog of the fractional quantum Hall effect1 at zero magnetic field, is predicted to exist in topological flat bands under spontaneous time-reversal-symmetry breaking. The demonstration of FQAHE could lead to non-Abelian anyons which form the basis of topological quantum computation. So far, FQAHE has been observed only in twisted MoTe2 (t-MoTe2) at moire filling factor v > 1/2. Graphene-based moire superlattices are believed to host FQAHE with the potential advantage of superior material quality and higher electron mobility. Here we report the observation of integer and fractional QAH effects in a rhombohedral pentalayer graphene/hBN moire superlattice. At zero magnetic field, we observed plateaus of quantized Hall resistance Rxy = h/(ve^2) at filling factors v = 1, 2/3, 3/5, 4/7, 4/9, 3/7 and 2/5 of the moire superlattice respectively. These features are accompanied by clear dips in the longitudinal resistance Rxx. In addition, at zero magnetic field, Rxy equals 2h/e^2 at v = 1/2 and varies linearly with the filling factor-similar to the composite Fermi liquid (CFL) in the half-filled lowest Landau level at high magnetic fields. By tuning the gate displacement field D and v, we observed phase transitions from CFL and FQAH states to other correlated electron states. Our graphene system provides an ideal platform for exploring charge fractionalization and (non-Abelian) anyonic braiding at zero magnetic field, especially considering a lateral junction between FQAHE and superconducting regions in the same device., Comment: Nature, in press

Remarkable recent experiments have observed fractional quantum anomalous Hall (FQAH) effects at zero field and unusually high temperatures in twisted semiconductor bilayer $t$MoTe$_2$. Intriguing observations in these experiments such as the absence of integer Hall effects at twist angles where a fractional Hall effect is observed, do however remain unexplained. The experimental phase diagram as a function of twist angle remains to be established. By comprehensive numerical study, we show that band mixing has large qualitative and quantitative effects on the energetics of competing states and their energy gaps throughout the twist angle range $\theta\leq 4^\circ$. This lays the ground for the detailed realistic study of a rich variety of strongly correlated twisted semiconductor multilayers and an understanding of the phase diagram of these fascinating systems., Comment: 5 + 2 pages, 7 Figures

Recent experiments on the twisted semiconductor bilayer system $t$MoTe$_2$ have observed integer and fractional quantum anomalous Hall effects, which occur in topological moir\'e bands at zero magnetic field. Here, we present a global phase diagram of $t$MoTe$_2$ throughout the filling range $0< n\leq 1$ substantiated by exact diagonalization calculations. At a magic angle, we find that the system resembles the lowest Landau level (LLL) to a remarkable degree, exhibiting an abundance of incompressible fractional quantum anomalous Hall states and compressible anomalous composite Fermi liquid states. Away from the magic angle, particle-hole symmetry is strongly broken. Some LLL-like features remain robust near half-filling, while others are replaced, predominantly by charge density waves near $n=0$ and anomalous Hall Fermi liquids near $n=1$. Among LLL-like phases, we find the anomalous composite Fermi liquid at $n=\frac{1}{2}$ to be most robust against deviations from the magic angle. Within the band-projected model, we show that strong particle-hole asymmetry above the magic angle results from interaction-enhanced quasiparticle dispersion near $n=1$. Our work sets the stage for future exploration of LLL-like and beyond-LLL phases in fractional quantum anomalous Hall systems., Comment: New numerical results and supplementary material added. Accepted to Phys. Rev. B. 8 + 8 pages, 7 + 9 figures

Topological materials are at the forefront of quantum materials research, offering tremendous potential for next-generation energy and information devices. However, current investigation of these materials remains largely focused on performance and often neglects the crucial aspect of sustainability. Recognizing the pivotal role of sustainability in addressing global pollution, carbon emissions, resource conservation, and ethical labor practices, we present a comprehensive evaluation of topological materials based on their sustainability and environmental impact. Our approach involves a hierarchical analysis encompassing cost, toxicity, energy demands, environmental impact, social implications, and resilience to imports. By applying this framework to over 16,000 topological materials, we establish a sustainable topological materials database. Our endeavor unveils environmental-friendly topological materials candidates which have been previously overlooked, providing insights into their environmental ramifications and feasibility for industrial scalability. The work represents a critical step toward industrial adoption of topological materials, offering the potential for significant technological advancements and broader societal benefits.

Ferroic orders describe spontaneous polarization of spin, charge, and lattice degrees of freedom in materials. Materials featuring multiple ferroic orders, known as multiferroics, play important roles in multi-functional electrical and magnetic device applications. 2D materials with honeycomb lattices offer exciting opportunities to engineer unconventional multiferroicity, where the ferroic orders are driven purely by the orbital degrees of freedom but not electron spin. These include ferro-valleytricity corresponding to the electron valley and ferro-orbital-magnetism supported by quantum geometric effects. Such orbital multiferroics could offer strong valley-magnetic couplings and large responses to external fields-enabling device applications such as multiple-state memory elements, and electric control of valley and magnetic states. Here we report orbital multiferroicity in pentalayer rhombohedral graphene using low temperature magneto-transport measurements. We observed anomalous Hall signals Rxy with an exceptionally large Hall angle (tan{\Theta}H > 0.6) and orbital magnetic hysteresis at hole doping. There are four such states with different valley polarizations and orbital magnetizations, forming a valley-magnetic quartet. By sweeping the gate electric field E we observed a butterfly-shaped hysteresis of Rxy connecting the quartet. This hysteresis indicates a ferro-valleytronic order that couples to the composite field E\cdot B, but not the individual fields. Tuning E would switch each ferroic order independently, and achieve non-volatile switching of them together. Our observations demonstrate a new type of multiferroics and point to electrically tunable ultra-low power valleytronic and magnetic devices.

The integer quantum anomalous Hall (QAH) effect is a lattice analog of the quantum Hall effect at zero magnetic field. This striking transport phenomenon occurs in electronic systems with topologically nontrivial bands and spontaneous time-reversal symmetry breaking. Discovery of its putative fractional counterpart in the presence of strong electron correlations, i.e., the fractional quantum anomalous Hall (FQAH) effect, 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 MoTe$_2$. At zero magnetic field, near filling factor $\nu = -1$ (one hole per moir\'e unit cell) we see an extended integer QAH plateau in the Hall resistance $R_\text{xy}$ that is quantized to $h/e^2 \pm 0.1 \%$ while the longitudinal resistance $R_\text{xx}$ vanishes. Remarkably, at $\nu=-2/3$ and $-3/5$ we see plateau features in $R_\text{xy}$ at $3h/2e^2 \pm 1\%$ and $5h/3e^2 \pm 3\%$, respectively, while $R_\text{xx}$ remains small. All these features shift linearly in an 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. In addition, at zero magnetic field, $R_\text{xy}$ is approximately $2h/e^2$ near half filling ($\nu = -1/2$) and varies linearly as $\nu$ is tuned. This behavior resembles that of the composite Fermi liquid in the half-filled lowest Landau level of a two-dimensional electron gas at high magnetic field. Direct observation of the FQAH and associated effects paves the way for researching charge fractionalization and anyonic statistics at zero magnetic field., Comment: 15 pages, 4 figures for main text. 8 extended data figures

We introduce density imbalanced electron-hole bilayers at a commensurate 2 : 1 density ratio as a platform for realizing novel phases involving electrons, excitons and trions. Three length scales are identified which characterize the interplay between kinetic energy, intralayer repulsion, and interlayer attraction. By a combination of theoretical analysis and numerical calculation, we find a variety of strong-coupling phases in different parameter regions, including quantum crystals of electrons, excitons, and trions. We also propose an "excitonic supersolid" phase that features electron crystallization and exciton superfluidity simultaneously. The material realization and experimental signature of these phases are discussed in the context of semiconductor transition metal dichalcogenide bilayers., Comment: Published in PRL. Revised main text and supplement. Main: 6 pages, 4 figures. SM: 8 pages, 6 figures

Quantum geometry - the geometry of electron Bloch wavefunctions - is central to modern condensed matter physics. Due to the quantum nature, quantum geometry has two parts, the real part quantum metric and the imaginary part Berry curvature. The studies of Berry curvature have led to countless breakthroughs, ranging from the quantum Hall effect in 2DEGs to the anomalous Hall effect (AHE) in ferromagnets. However, in contrast to Berry curvature, the quantum metric has rarely been explored. Here, we report a new nonlinear Hall effect induced by quantum metric by interfacing even-layered MnBi2Te4 (a PT-symmetric antiferromagnet (AFM)) with black phosphorus. This novel nonlinear Hall effect switches direction upon reversing the AFM spins and exhibits distinct scaling that suggests a non-dissipative nature. Like the AHE brought Berry curvature under the spotlight, our results open the door to discovering quantum metric responses. Moreover, we demonstrate that the AFM can harvest wireless electromagnetic energy via the new nonlinear Hall effect, therefore enabling intriguing applications that bridges nonlinear electronics with AFM spintronics., Comment: 19 pages, 4 figures and a Supplementary Materials with 66 pages, 4 figures and 3 tables. Originally submitted to Science on Oct. 5, 2022

Electronic ferroelectricity represents a new paradigm where spontaneous symmetry breaking driven by electronic correlations, in contrast to traditional lattice-driven ferroelectricity, leads to the formation of electric dipoles. Despite the potential application advantages arising from its electronic nature, switchable electronic ferroelectricity remains exceedingly rare. Here, we report the discovery of an electronic ratchet effect that manifests itself as switchable electronic ferroelectricity in a layer-contrasting graphene-boron nitride moir\'e heterostructure. Our engineered layer-asymmetric moir\'e potential landscapes result in layer-polarized localized and itinerant electronic subsystems. At particular fillings of the localized subsystem, we find a ratcheting injection of itinerant carriers in a non-volatile manner, leading to a highly unusual ferroelectric response. Strikingly, the remnant polarization can be stabilized at multiple (quasi-continuous) states with behavior markedly distinct from known ferroelectrics. Our experimental observations, simulations, and theoretical analysis suggest that dipolar excitons are the driving force and elementary ferroelectric units in our system. This signifies a new type of electronic ferroelectricity where the formation of dipolar excitons with aligned moments generates a macroscopic polarization and leads to an electronically-driven ferroelectric response, which we term excitonic ferroelectricity. Such new ferroelectrics, driven by quantum objects like dipolar excitons, could pave the way to innovative quantum analog memory and synaptic devices.

Recent experiments have produced evidence for fractional quantum anomalous Hall (FQAH) states at zero magnetic field in the semiconductor moir\'e superlattice system $t$MoTe$_2$. Here we argue that a composite fermion description, already a unifying framework for the phenomenology of 2d electron gases at high magnetic fields, provides a similarly powerful perspective in this new context. To this end, we present exact diagonalization evidence for composite Fermi liquid states at zero magnetic field in $t$MoTe$_2$ at fillings $n=\frac{1}{2}$ and $n=\frac{3}{4}$. We dub these non-Fermi liquid metals anomalous composite Fermi liquids (ACFLs), and we argue that they play a central organizing role in the FQAH phase diagram. We proceed to develop a long wavelength theory for this ACFL state that offers concrete experimental predictions upon doping the composite Fermi sea, including a Jain sequence of FQAH states and a new type of commensurability oscillations originating from the superlattice potential intrinsic to the system., Comment: Phys. Rev. Lett. journal version. Editors' Suggestion and Featured in Physics. main: 5 pages, 3 figures. SM: 5 pages, 3 figures

Twisted homobilayer transition metal dichalcogenide (TMD) offers a versatile platform for exploring band topology, interaction-driven phases, and magnetic orders. We study the interaction-driven phases in twisted TMD homobilayers and their low-energy collective excitations, focusing on the effect of band topology on magnetism and its thermal stability. From Hartree-Fock theory of the continuum model, we identify several magnetic and topological phases. By tuning the displacement field, we find two phase transitions involving a change in topology and magnetism respectively. We analyze the magnon spectrum, revealing the crucial role of band topology in stabilizing 2D ferromagnetism by amplifying easy-axis magnetic anisotropy, resulting in a large magnon gap of up to 7meV. As the magnon gap is directly tied to the stability of the magnetic phase to thermal fluctuations, our findings have several important experimental implications., Comment: 5 pages, 5 figures. (v2) Added SM. (v3) Improved convergence, added NLSM analysis

We provide a universal tight bound on the energy gap of topological insulators by exploring relationships between topology, quantum geometry, and optical absorption. Applications of our theory to infrared absorption near topological band inversion, magnetic circular dichorism in Chern insulators, and topological gap in moir\'e materials are demonstrated., Comment: 1+10 pages, 1 figure

We propose helical trilayer graphene (HTG), a helical structure featuring identical rotation angles $\theta\approx 1.5^\circ$ 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 supermoir\'e (or moir\'e-of-moir\'e) structure, we show that HTG locally relaxes into large regions of a periodic single-moir\'e structure in which $C_{2z}$ is broken, giving rise to flat topological bands carrying valley-Chern numbers $C=\pm(1,-2)$. These bands feature near-ideal quantum geometry and are isolated from remote bands by a large gap $E_{\mathrm{gap}}\sim 100$ meV, making HTG a promising platform for experimental realization of correlated topological states such as integer and fractional quantum anomalous Hall states in $C=1$ and $2$ bands.