We report local conductivity imaging of edge states in monolayer graphene by millikelvin microwave impedance microscopy. At the charge-neutrality point, as the magnetic field increases, the local conductivity at the edge drops to zero more slowly than in the bulk. This behavior is consistent with the calculated spatial profile of the charge gap in the canted antiferromagnetic phase. For comparison, we also perform microwave imaging of integer quantum Hall states away from neutrality, which host dissipationless chiral edge channels. The evolution of the edge signal as a function of the bulk gap is fundamentally different between the Landau level filling factor ν=0 and |ν|≥1 integer quantum Hall states, which can be qualitatively explained by numerical simulations and theoretical analysis. Our results provide a comprehensive microscopic picture of the edge and bulk states as the Fermi level moves across the unique Landau-level spectrum of graphene.

Macroscopic quantum coherence emerges when bosons condense into a Bose-Einstein condensate (BEC). First observed as a single-component superfluid in helium, BECs later emerged in ultracold atomic gases at nanokelvin temperatures as weakly interacting quantum fluids, which can also host multicomponent spinor condensates with rich internal degrees of freedom. Excitons provide a promising solid-state platform for BECs that can combine strong interactions, electrical tunability, high transition temperatures, and multicomponent order. Yet, conclusive evidence for condensation has remained elusive. Here, we report evidence of two-component exciton BECs in MoSe2/hBN/WSe2 electron-hole bilayers by directly probing the spin susceptibility of constituent electrons and holes. This heterostructure hosts equilibrium exciton fluids with four spin-valley flavors. Using magneto-optical spectroscopy in a dilution refrigerator, we reveal three exciton condensate phases with distinct flavor polarizations. At zero magnetic field, the many-body ground state is a coherent superposition of two simultaneously condensed intravalley exciton flavors. Under a magnetic field, the intravalley exciton condensate first switches to a two-component intervalley exciton condensate via a first-order quantum phase transition at a weak critical field, and then turns into a fully-polarized single-component condensate at high fields. The two-component condensates persist up to ~1.8 K. Our results establish van der Waals electron-hole bilayers as a versatile platform for exploring strongly interacting, multicomponent exciton BECs.

We analyze interaction-driven charge-density-wave (CDW) states in the spin-valley polarized first valence miniband of twisted MoTe_2 (tMoTe_2) using an adiabatic mapping from the continuum model to an effective Landau-level (LL) problem. When projected to the lowest LL, the leading spatial harmonic of the moiré-periodic potential changes sign at a magic twist angle θ_c where the band reaches its minimum bandwidth. By solving self-consistent Hartree-Fock equations in a multi-LL Hilbert space, we find that triangular-lattice CDW states with density maxima on MX (or XM) sites or on MM sites are favored on opposite sides of the magic angle at most filling factors and that stripe order appears near ν_h=1/2. We show that CDW states at ν_h >1/2 can carry a nonzero total Chern number, providing a natural route to reentrant integer quantum Hall effects and discuss the energy competition between fractional Chern insulator and CDW states.

Two-dimensional (2D) van-der-Waals materials are a promising platform for exciton state engi- neering. In this paper, we study the properties of excitons in 2D group VI transition-metal dichalco- genide (TMD) semiconductors that are modified by a periodic electrostatic potential through the quadratic Stark effect. Using a model that retains only center-of-mass and valley degrees-of-freedom, we find that electrostatic potentials can drive optical valley splitting up to 10meVs and induce valley selective exciton dispersion. We explain why both properties are sensitive to the rotational symmetry of the electrostatic trapping potential using a combination of numerical results and an- alytical approximations. An important consequence of valley-splitting is that the lowest exciton band is non-degenerate and has a linear dispersion around gamma that is expected to suppress thermal excitations, allowing true Bose condensation and superfluidity of excitons in two space dimensions.

We investigate how many-electron excited states emerge in twisted MoSe_{2} homobilayers when the lattice reconstructions evolve. Notably, we identify a new trion resonance that arises in the transition regime of lattice reconstruction, where gradual changes in atomic alignment between the layers occur. Magnetic field-dependent measurements, supported by first-principles calculations, indicate that the exciton forms at the K valley while the doped hole resides in the Γ valley. First-principles calculations further indicate that two nearly degenerate exciton resonances can arise, localized at different sites within the moiré supercell. We propose that the new trion resonance is a "charge-transfer" trion, in which the electron-hole pair is spatially separated from the doped hole. The emergence of these complex excited states stems from the distinct moiré potentials acting on holes and excitons, resulting in their different spatial distribution within the superlattice.

Moir & eacute; materials with opposite nonzero minibands Chern numbers in time-reversal-partner valleys are twodimensional topological insulators at band filling v = 2. We explore the possibility that in this class of moir & eacute; materials intervalley coherence can sometimes be present in interaction-induced insulators at band filling v = 1, using Landau levels with opposite signs of the magnetic field as a convenient generic model. In the absence of intravalley interactions the mean-field ground state at filling factor v = 1 is a gapless intervalley coherent state that maps under a particle-hole transformation of one valley to a strong-field superconducting vortex-lattice state that has been studied previously. When the ratio A of intravalley to intervalley exchange interactions is increased, gapped states appear, one with broken time-reversal symmetry and a quantized Hall effect but no valley polarization and one with broken parity symmetry and zero Hall conductivity. We discuss the possibility that the latter state could be related to the fractional quantum spin Hall effect recently observed at an odd filling factor in a moir & eacute; topological insulator and comment on related systems in which correlations between electrons in bands with opposite Chern numbers might play a key role.

Structural twirls form spontaneously in the domain wall networks of some moiré materials. We show that in heterobilayers, neighboring twirl chiralities tend to antialign, forming staggered patterns that are well described by antiferromagnetic lattice ϕ^{4} theories. In moiré systems with triangular domains, this leads to frustration in the chirality configuration of the twirls and to hysteresis with respect to variation of the average twist angle and possibly other control parameters. We expect that in typical materials, the ordering temperature of twirls is about 10^{3}  K, and that thermal fluctuations in individual twirl chiralities freeze below room temperature.

The layer-dependent Hamiltonians of parallel-stacked MoTe_2 and WSe_2 homobilayer moiré materials are topologically non-trivial, both in real space and in momentum space, and have been shown to support integer and fractional quantum anomalous Hall states, as well as antiferromagnetic and superconducting states. Here, we address the interplay between the antiferromagnetic and superconducting states observed in tWSe_2 when the Fermi level is close to its M-point van Hove singularity and the displacement field is small. We combine DFT with path-integrals to construct a minimal moiré band model that accounts for lattice relaxation along the c-axis and perform Hartree-Fock calculations to identify competing charge and spin ordered states. For tWSe_2 at θ=2.7^∘ and θ=3.65^∘, we find that a layer antiferromagnet (AFM), a stripe spin-density-wave (SDW), and the ferromagnetic Chern insulator (FM) are the primary candidates for the ground state at zero displacement field, and argue that antiferromagnetic spin interactions on the next neighbor bond J_2 can induce a time-reversal symmetry breaking chiral superconducting state.

The recent realization of Hofstadter spectra and fractional Chern insulators in moiré materials has introduced a new ingredient, a periodic lattice potential, to the study of quantum Hall phases. While the fractionalized states in moiré systems are expected to be in the same universality class as their counterparts in Landau levels, the periodic potential can have qualitative and quantitative effects on physical observables. Here, we examine how the magnetoroton collective modes of fractional quantum Hall (FQH) states are altered by external periodic potentials. Employing a single-mode approximation, we derive an effective Hamiltonian for the low-energy neutral excitations expressed in terms of three-point density correlation functions, which are computed using Monte Carlo. Our analysis is applicable to FQH states in graphene with a hexagonal boron nitride (hBN) substrate and also to fractional Chern insulator (FCI) states in twisted MoTe_{2} bilayers. We predict experimentally testable trends in the THz absorption characteristics of FCI and FQH states and estimate the external potential strength at which a soft-mode phase transition occurs between FQH and charge density wave states.

Equilibrium interlayer exciton condensation is common in bilayer quantum Hall systems and is characterized by spontaneous phase coherence between isolated layers. It has been predicted that similar physics can occur in the absence of a magnetic field in some two-dimensional semiconductor bilayers. In this work we consider the case of two transition metal dichalcogenide (TMD) monolayers separated by a twisted hexagonal boron nitride (hBN) bilayer or multilayer. The hBN layers suppress tunneling between the TMD layers so that phase coherence is spontaneous when it is present. When twisted, the hBN layers also form a ferroelectric moiré pattern that applies opposite triangular-lattice modulation potentials to the two TMD layers. We show via mean-field theory that at total hole filling per moiré unit cell ν=1, this geometry can favor a chiral p-wave exciton condensate state in which the quantum anomalous Hall effect coexists with counter-flow superfluidity. We present a mean-field phase diagram for TMD hole bilayers modulated by twisted hBN, discuss the conditions needed for the realization of the p-wave condensate state, and propose experiments that could confirm its presence.

The Pauli exclusion principle combined with interactions between fermions is a basic mechanism across condensed-matter systems giving rise to a spontaneous breaking of the spin-space rotation symmetry of spin-ordered phases. Ferromagnetism is a conventional manifestation of spin ordering which leads to numerous applications, e.g., in spintronic information technologies. Altermagnetism, whose recent discovery was largely motivated by spintronics, stands apart from conventional magnetism in the sense that it spontaneously breaks not only spin-space but also real-space rotation symmetries, while it preserves a symmetry combining spin-space and real-space rotations. This is realized on crystals by a collinear compensated ordering of spins with a characteristic d, g or i-wave symmetry. Our Perspective goes beyond the theory of spin arrangements on crystals by connecting altermagnetism to basic notions in condensed matter physics. Specifically, we reflect on the analogies and distinctions of altermagnetism as compared to superfluid 3He and theories of spin ordering in the momentum space generated by other higher-partial-wave instabilities of a Fermi-liquid. On one hand, all these physical systems have in common the extraordinary combination of spontaneous breaking of spin-space and real-space rotation symmetries. On the other hand, we point out that there are key differences, both at the symmetry level and, particularly, at the level of microscopic mechanisms of ordering. These explain the comparatively large abundance, robustness and utility of altermagnetism, as predicted by the symmetry-classification of spin arrangements on crystals and ab initio calculations, and supported by initial experiments.

At a magic relative twist angle, magic angle twisted bilayer graphene (MATBG) has an octet of flat bands that can host strong correlation physics when partially filled. A key theoretical discovery in MATBG is the existence of ferromagnetic Slater determinants as exact ground states of the corresponding flat band interacting (FBI) Hamiltonian. The FBI Hamiltonian describes the behavior of electrons that interact with each other in a high-dimensional space, and is constructed from the band structure of the non-interacting Bistritzer–MacDonald model at the chiral limit. A key property of the FBI Hamiltonian for MATBG is that it is frustration free and can be written as a sum of non-commuting terms. In this work, we provide a complete characterization of the ground state manifold of the FBI Hamiltonian, proving that it is precisely the linear span of such ferromagnetic Slater determinants.

Spontaneous intervalley coherence is suspected in several different graphene multilayer systems, but is difficult to confirm because of a paucity of convenient experimental signatures. Here we suggest that magneto-conductance features associated with quantum corrections to Drude conductivity can serve as a smoking gun for intervalley coherence that does not break time-reversal symmetry. In this class of ordered multilayer quantum transport corrections can produce weak localization or weak antilocalization, depending on whether the valley order belongs to the orthogonal or symplectic symmetry class. Our analysis motivates low-temperature weak-field magnetoresistance measurements in graphene multilayers in which time-reversal invariant intervalley coherent order is conjectured.

The question of whether or not passive sub-wavelength cavities can alter the properties of quantum materials is currently attracting a great deal of attention. In this Article we show that the Fermi liquid parameters of a two-dimensional metal are modified by cavity polariton modes, and that these changes can be monitored by measuring a paradigmatic magneto-transport phenomenon, Shubnikov-de Haas oscillations in a weak perpendicular magnetic field. This effect is intrinsic, and totally unrelated to disorder. As an illustrative example, we carry out explicit calculations of the quasiparticle velocity of graphene in a planar van der Waals cavity formed by natural hyperbolic crystals and metal gates. The largest effects of the cavity occur when the phonon polariton modes of the former match energetically the graphene plasmon. For typical graphene carrier densities this occurs in the Terahertz spectral range.

Non-Abelian fractionalized quasiparticles in half-integer quantum Hall states have been proposed as a platform for the realization of topologically protected fault-tolerant qubits. We observe half-integer quantum Hall states out to remarkably large Landau level filling factors larger than 30 in dual-gated ∼3-10  nm-thick graphite devices. We identify most as single-component states stabilized by fractional correlations in exterior bilayers. Multicomponent (3,3,1) states with strong correlations between opposite-valley states on opposite sides of the graphite interior also appear. The facile integration of graphite with top and bottom surface gates makes thin film graphite an attractive platform to explore the physics of non-Abelian fractionalized quasiparticles.

The growth of a hexagonal FeTe (α-FeTe) thin layer on three-dimensional topological insulator Bi2Te3 has been systematically investigated by combining scanning tunneling microscopy/spectroscopy (STM/STS) with density-functional theory (DFT) calculations. Through post-annealing at 560 K after room-temperature deposition of Fe onto Bi2Te3, α-FeTe has been fabricated with an atomic lattice constant of 4.0 ± 0.1 Å, which is much smaller than the 4.4 ± 0.1 Å of the Te-terminated Bi2Te3 surface. The lattice mismatch and a sufficiently large FeTe-Bi2Te3 distance give rise to a moiré pattern with a periodicity of 6.0 ± 0.1 nm, corresponding to a p(15 × 15) superlattice. On the other hand, an enhanced projected density of states (PDOS) is found at about -0.8 eV and +1.4 eV of α-FeTe/Bi2Te3 in the tunneling spectra. These enhancements, as interpreted from the DFT calculation, arise from the contribution of the 3d out-of-plane orbitals in Fe. DFT calculations also reveal nonzero net magnetization and indicate ferromagnetic (FM) order in the system - contrasting the well-studied bicollinear antiferromagnetism in β-FeTe/Bi2Te3. To summarize, our successful fabrication of α-FeTe/Bi2Te3 provides a distinct platform from β-FeTe/Bi2Te3, enabling an exploration of the interplay between magnetism and interface-induced superconductivity.

Excitonic insulators represent a unique quantum phase of matter that enables the study of exotic quantum bosonic states. Strongly coupled electron-hole bilayers, which host stable dipolar exciton fluids with an exciton density that can be adjusted electrostatically, offer an ideal platform to investigate correlated excitonic insulators. On the basis of electron-hole bilayers made of MoSe2/hexagonal boron nitride/WSe2 heterostructures, here we study the behaviour of excitonic insulators in a perpendicular magnetic field. We report the observation of excitonic quantum oscillations in both Coulomb drag signals and electrical resistance at low to medium magnetic fields. Under a strong magnetic field, we identify multiple quantum phase transitions between the excitonic insulator phase and the bilayer quantum Hall insulator phase. These findings underscore the interplay between the electron-hole interactions and Landau-level quantization, and enable further exploration of quantum phenomena in composite bosonic insulators.

Quantum oscillations in magnetization or resistivity are a defining feature of metals in a magnetic field. The phenomenon is generally not expected in insulators without a Fermi surface. Its observation in Kondo and other correlated insulators provided counterexamples and remains poorly understood. Here we report the observation of resistivity oscillations in a gate-controlled excitonic insulator realized in Coulomb-coupled electron-hole double layers. When the electron or hole cyclotron energy is tuned to exceed the exciton binding energy, recurring transitions arise between the excitonic insulator and layer-decoupled quantum Hall states. Compressibility measurements show an oscillatory exciton binding energy as a function of the magnetic field and electron-hole pair density. Coulomb drag measurements further reveal the signature of finite-angular-momentum excitonic correlations. These findings are qualitatively captured by mean-field calculations. Our study establishes a highly tunable platform based on electron-hole double layers for studying quantum oscillations in correlated insulators.

The lowest Landau level of bilayer graphene has an octet of internal degrees of freedom, composed from spin, valley, and orbital two-level systems. Dominance of n=0 orbitals over n=1 orbitals in low energy quantum fluctuations leads to distinct fractional quantum Hall characteristics compared dominance of n=1 over n=0. The competition between n=0 and n=1 orbitals depends sensitively on particle-hole asymmetry in the single-particle Hamiltonian and on Lamb shifts due to exchange interactions with the negative energy sea, which must be accounted for simultaneously in assessing the orbital competition. We identify the circumstances under which n=1, which supports strong even-denominator fractional quantum Hall states with non-Abelian quasiparticles, emerges robustly as the low-energy Landau level.

The electronic properties of two-dimensional (2D) metals are altered by changes in their three-dimensional dielectric environment. In this Letter we propose that superconductivity can be induced in a 2D metal by resonant coupling between its plasmonic collective modes and optical phonons in a nearby polar dielectric. Specifically, we predict that relatively high-temperature superconductivity can be induced in bilayer graphene twisted to an angle somewhat larger than the magic value by surrounding it with a THz polar dielectric. Our conclusions are based on numerical solutions of Eliashberg equations for massless Dirac fermions with tunable Fermi velocities and Fermi energies, and can be understood qualitatively in terms of a generalized McMillan formula.