Energy Gap of the Even-Denominator Fractional Quantum Hall State in Bilayer Graphene.

Bernal bilayer graphene hosts even-denominator fractional quantum Hall states thought to be described by a Pfaffian wave function with non-Abelian quasiparticle excitations. Here, we report the quantitative determination of fractional quantum Hall energy gaps in bilayer graphene using both thermally activated transport and by direct measurement of the chemical potential. We find a transport activation gap of 5.1 K at B=12 T for a half filled N=1 Landau level, consistent with density matrix renormalization group calculations for the Pfaffian state. However, the measured thermodynamic gap of 11.6 K is smaller than theoretical expectations for the clean limit by approximately a factor of 2. We analyze the chemical potential data near fractional filling within a simplified model of a Wigner crystal of fractional quasiparticles with long-wavelength disorder, explaining this discrepancy. Our results quantitatively establish bilayer graphene as a robust platform for probing the non-Abelian anyons expected to arise as the elementary excitations of the even-denominator state.

Cascade of Multielectron Bubble Phases in Monolayer Graphene at High Landau Level Filling.

The phase diagram of an interacting two-dimensional electron system in a high magnetic field is enriched by the varying form of the effective Coulomb interaction, which depends strongly on the Landau level index. While the fractional quantum Hall states that dominate in the lower-energy Landau levels have been explored experimentally in a variety of two-dimensional systems, much less work has been done to explore electron solids owing to their subtle transport signatures and extreme sensitivity to disorder. Here, we use chemical potential measurements to map the phase diagram of electron solid states in N=2, N=3, and N=4 Landau levels in monolayer graphene. Direct comparison between our data and theoretical calculations reveals a cascade of density-tuned phase transitions between electron bubble phases up to two, three, or four electrons per bubble in the N=2, 3, and 4 Landau levels, respectively. Finite-temperature measurements are consistent with melting of the solids for T≈1 K.

Imaging inter-valley coherent order in magic-angle twisted trilayer graphene.

Magic-angle twisted trilayer graphene (MATTG) exhibits a range of strongly correlated electronic phases that spontaneously break its underlying symmetries1,2. Here we investigate the correlated phases of MATTG using scanning tunnelling microscopy and identify marked signatures of interaction-driven spatial symmetry breaking. In low-strain samples, over a filling range of about two to three electrons or holes per moiré unit cell, we observe atomic-scale reconstruction of the graphene lattice that accompanies a correlated gap in the tunnelling spectrum. This short-scale restructuring appears as a Kekulé supercell-implying spontaneous inter-valley coherence between electrons-and persists in a wide range of magnetic fields and temperatures that coincide with the development of the gap. Large-scale maps covering several moiré unit cells further reveal a slow evolution of the Kekulé pattern, indicating that atomic-scale reconstruction coexists with translation symmetry breaking at a much longer moiré scale. We use auto-correlation and Fourier analyses to extract the intrinsic periodicity of these phases and find that they are consistent with the theoretically proposed incommensurate Kekulé spiral order3,4. Moreover, we find that the wavelength characterizing moiré-scale modulations monotonically decreases with hole doping away from half-filling of the bands and depends weakly on the magnetic field. Our results provide essential insights into the nature of the correlated phases of MATTG in the presence of strain and indicate that superconductivity can emerge from an inter-valley coherent parent state.

Universal chiral Luttinger liquid behavior in a graphene fractional quantum Hall point contact.

One-dimensional conductors are described by Luttinger liquid theory, which predicts a power-law suppression of the single-electron tunneling density of states at low voltages. The scaling exponent is predicted to be quantized when tunneling into a single isolated chiral edge state of the fractional quantum Hall effect. We report conductance measurements across a point contact linking integer and fractional quantum Hall edge states (at fillings 1 and [Formula: see text], respectively). At weak coupling, we observe the predicted universal quadratic scaling with temperature and voltage. At strong coupling, we demonstrate perfect Andreev reflection of fractionalized quasiparticles at the point contact. We use the strong coupling physics to realize a nearly dissipationless direct current voltage step-up transformer, whose gain arises directly from topological fractionalization of electrical charge.

Gate-Defined Topological Josephson Junctions in Bernal Bilayer Graphene.

Recent experiments on Bernal bilayer graphene (BLG) deposited on monolayer WSe_{2} revealed robust, ultraclean superconductivity coexisting with sizable induced spin-orbit coupling. Here, we propose BLG/WSe_{2} as a platform to engineer gate-defined planar topological Josephson junctions, where the normal and superconducting regions descend from a common material. More precisely, we show that if superconductivity in BLG/WSe_{2} is gapped and emerges from a parent state with intervalley coherence, then Majorana zero-energy modes can form in the barrier region upon applying weak in-plane magnetic fields. Our results spotlight a potential pathway for "internally engineered" topological superconductivity that minimizes detrimental disorder and orbital-magnetic-field effects.

On-Chip Time-Domain Terahertz Spectroscopy of Superconducting Films below the Diffraction Limit.

Free-space time domain THz spectroscopy accesses electrodynamic responses in a frequency regime ideally matched to interacting condensed matter systems. However, THz spectroscopy is challenging when samples are physically smaller than the diffraction limit of ∼0.5 mm, as is typical, for example, in van der Waals materials and heterostructures. Here, we present an on-chip, time-domain THz spectrometer based on semiconducting photoconductive switches with a bandwidth of 200 to 750 GHz. We measure the optical conductivity of a 7.5-µm wide NbN film across the superconducting transition, demonstrating spectroscopic signatures of the superconducting gap in a sample smaller than 2% of the Rayleigh diffraction limit. Our spectrometer features an interchangeable sample architecture, making it ideal for probing superconductivity, magnetism, and charge order in strongly correlated van der Waals materials.

Imaging the Breakdown of Ohmic Transport in Graphene.

Ohm's law describes the proportionality of the current density and electric field. In solid-state conductors, Ohm's law emerges due to electron scattering processes that relax the electrical current. Here, we use nitrogen-vacancy center magnetometry to directly image the local breakdown of Ohm's law in a narrow constriction fabricated in a high mobility graphene monolayer. Ohmic flow is visible at room temperature as current concentration on the constriction edges, with flow profiles entirely determined by sample geometry. However, as the temperature is lowered below 200 K, the current concentrates near the constriction center. The change in the flow pattern is consistent with a crossover from diffusive to viscous electron transport dominated by electron-electron scattering processes that do not relax current.

Isospin magnetism and spin-polarized superconductivity in Bernal bilayer graphene.

In conventional superconductors, Cooper pairing occurs between electrons of opposite spin. We observe spin-polarized superconductivity in Bernal bilayer graphene when doped to a saddle-point van Hove singularity generated by large applied perpendicular electric field. We observe a cascade of electrostatic gate-tuned transitions between electronic phases distinguished by their polarization within the isospin space defined by the combination of the spin and momentum-space valley degrees of freedom. Although all of these phases are metallic at zero magnetic field, we observe a transition to a superconducting state at finite B‖ ≈ 150mT applied parallel to the two-dimensional sheet. Superconductivity occurs near a symmetry breaking transition, and exists exclusively above the B‖-limit expected of a paramagnetic superconductor with the observed transition temperature TC ≈ 30mK, consistent with a spin-triplet order parameter.

Quantum Oscillations in Two-Dimensional Insulators Induced by Graphite Gates.

We demonstrate a mechanism for magnetoresistance oscillations in insulating states of two-dimensional (2D) materials arising from the interaction of the 2D layer and proximal graphite gates. We study a series of devices based on different 2D systems, including mono- and bilayer T_{d}-WTe_{2}, MoTe_{2}/WSe_{2} moiré heterobilayers, and Bernal-stacked bilayer graphene, which all share a similar graphite-gated geometry. We find that the 2D systems, when tuned near an insulating state, generically exhibit magnetoresistance oscillations corresponding to a high-density Fermi surface, in contravention of naïve band theory. Simultaneous measurement of the resistivity of the graphite gates shows that the oscillations of the sample layer are precisely correlated with those of the graphite gates. Further supporting this connection, the oscillations are quenched when the graphite gate is replaced by a low-mobility metal, TaSe_{2}. The observed phenomenon arises from the oscillatory behavior of graphite density of states, which modulates the device capacitance and, as a consequence, the carrier density in the sample layer even when a constant electrochemical potential is maintained between the sample and the gate electrode. Oscillations are most pronounced near insulating states where the resistivity is strongly density dependent. Our study suggests a unified mechanism for quantum oscillations in graphite-gated 2D insulators based on electrostatic sample-gate coupling.

Superconductivity in rhombohedral trilayer graphene.

To access superconductivity via the electric field effect in a clean, two-dimensional device is a central goal of nanoelectronics. Recently, superconductivity has been realized in graphene moiré heterostructures1-4; however, many of these structures are not mechanically stable, and experiments show signatures of strong disorder. Here we report the observation of superconductivity-manifesting as low or vanishing resistivity at sub-kelvin temperatures-in crystalline rhombohedral trilayer graphene5,6, a structurally metastable carbon allotrope. Superconductivity occurs in two distinct gate-tuned regions (SC1 and SC2), and is deep in the clean limit defined by the ratio of mean free path and superconducting coherence length. Mapping of the normal state Fermi surfaces by quantum oscillations reveals that both superconductors emerge from an annular Fermi sea, and are proximal to an isospin-symmetry-breaking transition where the Fermi surface degeneracy changes7. SC1 emerges from a paramagnetic normal state, whereas SC2 emerges from a spin-polarized, valley-unpolarized half-metal17 and violates the Pauli limit for in-plane magnetic fields by at least one order of magnitude8,9. We discuss our results in view of several mechanisms, including conventional phonon-mediated pairing10,11, pairing due to fluctuations of the proximal isospin order12, and intrinsic instabilities of the annular Fermi liquid13,14. Our observation of superconductivity in a clean and structurally simple two-dimensional metal provides a model system to test competing theoretical models of superconductivity without the complication of modelling disorder, while enabling new classes of field-effect controlled electronic devices based on correlated electron phenomena and ballistic electron transport.

Half- and quarter-metals in rhombohedral trilayer graphene.

Ferromagnetism is most common in transition metal compounds where electrons occupy highly localized d orbitals. However, ferromagnetic order may also arise in low-density two-dimensional electron systems1-5. Here we show that gate-tuned van Hove singularities in rhombohedral trilayer graphene6 drive spontaneous ferromagnetic polarization of the electron system into one or more spin and valley flavours. Using capacitance and transport measurements, we observe a cascade of transitions tuned to the density and electronic displacement field between phases in which quantum oscillations have fourfold, twofold or onefold degeneracy, associated with a spin- and valley-degenerate normal metal, spin-polarized 'half-metal', and spin- and valley-polarized 'quarter-metal', respectively. For electron doping, the salient features of the data are well captured by a phenomenological Stoner model7 that includes valley-anisotropic interactions. For hole filling, we observe a richer phase diagram featuring a delicate interplay of broken symmetries and transitions in the Fermi surface topology. Finally, we introduce a moiré superlattice using a rotationally aligned hexagonal boron nitride substrate5,8. Remarkably, we find that the isospin order is only weakly perturbed, with the moiré potential catalysing the formation of topologically nontrivial gapped states whenever itinerant half- or quarter-metal states occur at half- or quarter-superlattice band filling. Our results show that rhombohedral graphene is an ideal platform for well-controlled tests of many-body theory, and reveal magnetism in moiré materials4,5,9,10 to be fundamentally itinerant in nature.

Isospin Pomeranchuk effect in twisted bilayer graphene.

In condensed-matter systems, higher temperatures typically disfavour ordered phases, leading to an upper critical temperature for magnetism, superconductivity and other phenomena. An exception is the Pomeranchuk effect in 3He, in which the liquid ground state freezes upon increasing the temperature1, owing to the large entropy of the paramagnetic solid phase. Here we show that a similar mechanism describes the finite-temperature dynamics of spin and valley isospins in magic-angle twisted bilayer graphene2. Notably, a resistivity peak appears at high temperatures near a superlattice filling factor of -1, despite no signs of a commensurate correlated phase appearing in the low-temperature limit. Tilted-field magnetotransport and thermodynamic measurements of the in-plane magnetic moment show that the resistivity peak is connected to a finite-field magnetic phase transition3 at which the system develops finite isospin polarization. These data are suggestive of a Pomeranchuk-type mechanism, in which the entropy of disordered isospin moments in the ferromagnetic phase stabilizes the phase relative to an isospin-unpolarized Fermi liquid phase at higher temperatures. We find the entropy, in units of Boltzmann's constant, to be of the order of unity per unit cell area, with a measurable fraction that is suppressed by an in-plane magnetic field consistent with a contribution from disordered spins. In contrast to 3He, however, no discontinuities are observed in the thermodynamic quantities across this transition. Our findings imply a small isospin stiffness4,5, with implications for the nature of finite-temperature electron transport6-8, as well as for the mechanisms underlying isospin ordering and superconductivity9,10 in twisted bilayer graphene and related systems.

Experimental Determination of the Energy per Particle in Partially Filled Landau Levels.

We describe an experimental technique to measure the chemical potential µ in atomically thin layered materials with high sensitivity and in the static limit. We apply the technique to a high quality graphene monolayer to map out the evolution of µ with carrier density throughout the N=0 and N=1 Landau levels at high magnetic field. By integrating µ over filling factor ν, we obtain the ground state energy per particle, which can be directly compared to numerical calculations. In the N=0 Landau level, our data show exceptional agreement with numerical calculations over the whole Landau level without adjustable parameters as long as the screening of the Coulomb interaction by the filled Landau levels is accounted for. In the N=1 Landau level, a comparison between experimental and numerical data suggests the importance of valley anisotropic interactions and reveals a possible presence of valley-textured electron solids near odd filling.

Linear Magnetoelectric Phase in Ultrathin MnPS_{3} Probed by Optical Second Harmonic Generation.

The transition metal thiophosphates MPS_{3} (M=Mn, Fe, Ni) are a class of van der Waals stacked insulating antiferromagnets that can be exfoliated down to the ultrathin limit. MnPS_{3} is particularly interesting because its Néel ordered state breaks both spatial-inversion and time-reversal symmetries, allowing for a linear magnetoelectric phase that is rare among van der Waals materials. However, it is unknown whether this unique magnetic structure of bulk MnPS_{3} remains stable in the ultrathin limit. Using optical second harmonic generation rotational anisotropy, we show that long-range linear magnetoelectric type Néel order in MnPS_{3} persists down to at least 5.3 nm thickness. However an unusual mirror symmetry breaking develops in ultrathin samples on SiO_{2} substrates that is absent in bulk materials, which is likely related to substrate induced strain.

Tuning superconductivity in twisted bilayer graphene.

Materials with flat electronic bands often exhibit exotic quantum phenomena owing to strong correlations. An isolated low-energy flat band can be induced in bilayer graphene by simply rotating the layers by 1.1°, resulting in the appearance of gate-tunable superconducting and correlated insulating phases. In this study, we demonstrate that in addition to the twist angle, the interlayer coupling can be varied to precisely tune these phases. We induce superconductivity at a twist angle larger than 1.1°-in which correlated phases are otherwise absent-by varying the interlayer spacing with hydrostatic pressure. Our low-disorder devices reveal details about the superconducting phase diagram and its relationship to the nearby insulator. Our results demonstrate twisted bilayer graphene to be a distinctively tunable platform for exploring correlated states.

Observation of fractional Chern insulators in a van der Waals heterostructure.

Topologically ordered phases are characterized by long-range quantum entanglement and fractional statistics rather than by symmetry breaking. First observed in a fractionally filled continuum Landau level, topological order has since been proposed to arise more generally at fractional fillings of topologically nontrivial Chern bands. Here we report the observation of gapped states at fractional fillings of Harper-Hofstadter bands arising from the interplay of a magnetic field and a superlattice potential in a bilayer graphene-hexagonal boron nitride heterostructure. We observed phases at fractional filling of bands with Chern indices [Formula: see text] Some of these phases, in [Formula: see text] and [Formula: see text] bands, are characterized by fractional Hall conductance-that is, they are known as fractional Chern insulators and constitute an example of topological order beyond Landau levels.

Helical edge states and fractional quantum Hall effect in a graphene electron-hole bilayer.

Helical 1D electronic systems are a promising route towards realizing circuits of topological quantum states that exhibit non-Abelian statistics. Here, we demonstrate a versatile platform to realize 1D systems made by combining quantum Hall (QH) edge states of opposite chiralities in a graphene electron-hole bilayer at moderate magnetic fields. Using this approach, we engineer helical 1D edge conductors where the counterpropagating modes are localized in separate electron and hole layers by a tunable electric field. These helical conductors exhibit strong non-local transport signals and suppressed backscattering due to the opposite spin polarizations of the counterpropagating modes. Unlike other approaches used for realizing helical states, the graphene electron-hole bilayer can be used to build new 1D systems incorporating fractional edge states. Indeed, we are able to tune the bilayer devices into a regime hosting fractional and integer edge states of opposite chiralities, paving the way towards 1D helical conductors with fractional quantum statistics.

Electrically Tunable Multiterminal SQUID-on-Tip.

We present a new nanoscale superconducting quantum interference device (SQUID) whose interference pattern can be shifted electrically in situ. The device consists of a nanoscale four-terminal-four-junction SQUID fabricated at the apex of a sharp pipet using a self-aligned three-step deposition of Pb. In contrast to conventional two-terminal-two-junction SQUIDs that display optimal sensitivity when flux biased to about a quarter of the flux quantum, the additional terminals and junctions allow optimal sensitivity at arbitrary applied flux, thus eliminating the magnetic field "blind spots". We demonstrate spin sensitivity of 5 to 8 µB/Hz1/2 over a continuous field range of 0 to 0.5 T with promising applications for nanoscale scanning magnetic imaging.

Visualization of superparamagnetic dynamics in magnetic topological insulators.

Quantized Hall conductance is a generic feature of two-dimensional electronic systems with broken time reversal symmetry. In the quantum anomalous Hall state recently discovered in magnetic topological insulators, time reversal symmetry is believed to be broken by long-range ferromagnetic order, with quantized resistance observed even at zero external magnetic field. We use scanning nanoSQUID (nano-superconducting quantum interference device) magnetic imaging to provide a direct visualization of the dynamics of the quantum phase transition between the two anomalous Hall plateaus in a Cr-doped (Bi,Sb)2Te3 thin film. Contrary to naive expectations based on macroscopic magnetometry, our measurements reveal a superparamagnetic state formed by weakly interacting magnetic domains with a characteristic size of a few tens of nanometers. The magnetic phase transition occurs through random reversals of these local moments, which drive the electronic Hall plateau transition. Surprisingly, we find that the electronic system can, in turn, drive the dynamics of the magnetic system, revealing a subtle interplay between the two coupled quantum phase transitions.