Canted antiferromagnetic phase of the ν=0 quantum Hall state in bilayer graphene

Electron wave and quantum optics in graphene

In the last decade, graphene has become an exciting platform for electron optical experiments, in some aspects superior to conventional two-dimensional electron gases (2DEGs). A major advantage, besides the ultra-large mobilities, is the fine control over the electrostatics, which gives the possibility of realising gap-less and compact p-n interfaces with high precision. The latter host non-trivial states,e.g., snake states in moderate magnetic fields, and serve as building blocks of complex electron interferometers. Thanks to the Dirac spectrum and its non-trivial Berry phase, the internal (valley and sublattice) degrees of freedom, and the possibility to tailor the band structure using proximity effects, such interferometers open up a completely new playground based on novel device architectures. In this review, we introduce the theoretical background of graphene electron optics, fabrication methods used to realise electron-optical devices, and techniques for corresponding numerical simulations. Based on this, we give a comprehensive review of ballistic transport experiments and simple building blocks of electron optical devices both in single and bilayer graphene, highlighting the novel physics that is brought in compared to conventional 2DEGs. After describing the different magnetic field regimes in graphene p-n junctions and nanostructures, we conclude by discussing the state of the art in graphene-based Mach-Zender and Fabry-Perot interferometers.

Intra-Zero-Energy Landau Level Crossings in Bilayer Graphene at High Electric Fields

The highly tunable band structure of the zero-energy Landau level (zLL) of bilayer graphene makes it an ideal platform for engineering novel quantum states. However, the zero-energy Landau level at high electric fields has remained largely unexplored. Here we present magnetotransport measurements of bilayer graphene in high transverse electric fields. We observe previously undetected Landau level crossings at filling factors ν = -2, 1, and 3 at high electric fields. These crossings provide constraints for theoretical models of the zero-energy Landau level and show that the orbital, valley, and spin character of the quantum Hall states at high electric fields is very different from low electric fields. At high E, new transitions between states at ν = -2 with different orbital and spin polarization can be controlled by the gate bias, while the transitions between ν = 0 → 1 and ν = 2 → 3 show anomalous behavior.

Aharonov-Bohm Oscillations in Bilayer Graphene Quantum Hall Edge State Fabry-Pérot Interferometers

Bernal-stacked bilayer graphene exhibits a wealth of interaction-driven phenomena, including robust even-denominator fractional quantum Hall states. We construct Fabry-Pérot interferometers using a split-gate design and present measurements of the Aharonov-Bohm oscillations. The edge state velocity is found to be approximately 6 × 10⁴ m/s at filling factor ν = 2 and decreases with increasing filling factor. The dc bias and temperature dependence of the interference point to electron-electron interaction induced decoherence mechanisms. These results pave the way for the quest of fractional and non-Abelian braiding statistics in this promising device platform.

Interplay between topological valley and quantum Hall edge transport

An established way of realising topologically protected states in a two-dimensional electron gas is by applying a perpendicular magnetic field thus creating quantum Hall edge channels. In electrostatically gapped bilayer graphene intriguingly, even in the absence of a magnetic field, topologically protected electronic states can emerge at naturally occurring stacking domain walls. While individually both types of topologically protected states have been investigated, their intriguing interplay remains poorly understood. Here, we focus on the interplay between topological domain wall states and quantum Hall edge transport within the eight-fold degenerate zeroth Landau level of high-quality suspended bilayer graphene. We find that the two-terminal conductance remains approximately constant for low magnetic fields throughout the distinct quantum Hall states since the conduction channels are traded between domain wall and device edges. For high magnetic fields, however, we observe evidence of transport suppression at the domain wall, which can be attributed to the emergence of spectral minigaps. This indicates that stacking domain walls potentially do not correspond to a topological domain wall in the order parameter.

Gate-Tunable Magnetism and Giant Magnetoresistance in Suspended Rhombohedral-Stacked Few-Layer Graphene

Conventionally, magnetism arises from the strong exchange interaction among the magnetic moments of d- or f-shell electrons. It can also emerge in perfect lattices from nonmagnetic elements, such as that exemplified by the Stoner criterion. Here we report tunable magnetism in suspended rhombohedral-stacked few-layer graphene (r-FLG) devices with flat bands. At small doping levels (n ∼ 10¹¹ cm^-2), we observe prominent conductance hysteresis and giant magnetoconductance that exceeds 1000% as a function of magnetic fields. Both phenomena are tunable by density and temperature and disappear at n > 10¹² cm^-2 or T > 5 K. These results are confirmed by first-principles calculations, which indicate the formation of a half-metallic state in doped r-FLG, in which the magnetization is tunable by electric field. Our combined experimental and theoretical work demonstrate that magnetism and spin polarization, arising from the strong electronic interactions in flat bands, emerge in a system composed entirely of carbon atoms.

Coexistence of Canted Antiferromagnetism and Bond Order in ν=0 Graphene

Motivated by experimental studies of graphene in the quantum Hall regime, we revisit the phase diagram of a single sheet of graphene at charge neutrality. Because of spin and valley degeneracies, interactions play a crucial role in determining the nature of the ground state. We show that, generically within the Hartree-Fock approximation, in the regime of interest there is a region of coexistence between magnetic and bond orders in the phase diagram. We demonstrate this result both in continuum and lattice models, and argue that the coexistence phase naturally provides a possible explanation for unreconciled experimental observations on the quantum Hall effect in graphene.

Accurate Measurement of the Gap of Graphene/h-BN Moiré Superlattice through Photocurrent Spectroscopy

Monolayer graphene aligned with hexagonal boron nitride (h-BN) develops a gap at the charge neutrality point (CNP). This gap has previously been extensively studied by electrical transport through thermal activation measurements. Here, we report the determination of the gap size at the CNP of graphene/h-BN superlattice through photocurrent spectroscopy study. We demonstrate two distinct measurement approaches to extract the gap size. A maximum of ∼14  meV gap is observed for devices with a twist angle of less than 1°. This value is significantly smaller than that obtained from thermal activation measurements, yet larger than the theoretically predicted single-particle gap. Our results suggest that lattice relaxation and moderate electron-electron interaction effects may enhance the CNP gap in graphene/h-BN superlattice.

Scattering of Magnons at Graphene Quantum-Hall-Magnet Junctions

Motivated by recent nonlocal transport studies of quantum-Hall-magnet (QHM) states formed in monolayer graphene's N=0 Landau level, we study the scattering of QHM magnons by gate-controlled junctions between states with different integer filling factors ν. For the ν=1|-1|1 geometry we find that magnons are weakly scattered by electric potential variation in the junction region, and that the scattering is chiral when the junction lacks a mirror symmetry. For the ν=1|0|1 geometry, we find that kinematic constraints completely block magnon transmission if the incident angle exceeds a critical value. Our results explain the suppressed nonlocal-voltage signals observed in the ν=1|0|1 case. We use our theory to propose that valley waves generated at ν=-1|1 junctions and magnons can be used in combination to probe the spin or valley flavor structure of QHM states at integer and fractional filling factors.

Magnetic Exchange Field Modulation of Quantum Hall Ferromagnetism in 2D van der Waals CrCl3/Graphene Heterostructures

The efforts toward the experimental realization of spin-polarized transports in ideal materials or platforms, such as the magnetized graphene or various quantum Hall states, is a research hotspot in spintronics. Magnetic van der Waals materials open the door for exploring various physical phenomena, technologies, and integrating novel spintronic devices seamlessly within the 2D limit. Here, we demonstrate magnetic proximity effect in chromium trichloride (CrCl₃)/bilayer graphene (BLG) heterostructures by low-temperature transport measurements. An effective exchange field induced in BLG has been demonstrated by the Zeeman spin Hall effect via nonlocal measurements. Furthermore, the exchange field modulates the quantum Hall ground state of BLG and thus favors the formation of a canted antiferromagnetic (CAF) phase in an external perpendicular magnetic field (B_⊥). Asymmetric nonlocal magneto-transport behaviors are also observed at opposite B_⊥ directions, due to the asymmetric modulation on the exchange field by external B_⊥ directions. Our work suggests that the 2D magnetic van der Waals materials and graphene hybrid systems offer a unique platform for quantum Hall ferromagnetism physics.

Charge Neutral Current Generation in a Spontaneous Quantum Hall Antiferromagnet

The intrinsic Hall effect allows for the generation of a nondissipative charge neutral current, such as a pure spin current generated via the spin Hall effect. Breaking of the spatial inversion or time reversal symmetries, or the spin-orbit interaction is generally considered necessary for the generation of such a charge neutral current. Here, we challenge this general concept and present generation and detection of a charge neutral current in a centrosymmetric material with little spin-orbit interaction. We employ bilayer graphene, and find enhanced nonlocal transport in the quantum Hall antiferromagnetic state, where spontaneous symmetry breaking occurs due to the electronic correlation.

Widely Tunable Quantum Phase Transition from Moore-Read to Composite Fermi Liquid in Bilayer Graphene

We develop a proposal to realize a widely tunable and clean quantum phase transition in bilayer graphene between two paradigmatic fractionalized phases of matter: the Moore-Read fractional quantum Hall state and the composite Fermi liquid metal. This transition can be realized at total fillings ν=±3+1/2 and the critical point can be controllably accessed by tuning either the interlayer electric bias or the perpendicular magnetic field values over a wide range of parameters. We study the transition numerically within a model that contains all leading single particle corrections to the band structure of bilayer graphene and includes the fluctuations between the n=0 and n=1 cyclotron orbitals of its zeroth Landau level to delineate the most favorable region of parameters to experimentally access this unconventional critical point. We also find evidence for a new anisotropic gapless phase stabilized near the level crossing of n=0/1 orbits.

Metallic Phase and Temperature Dependence of the ν=0 Quantum Hall State in Bilayer Graphene

The ν=0 quantum Hall state of bilayer graphene is a fertile playground to realize many-body ground states with various broken symmetries. Here we report the experimental observations of a previously unreported metallic phase. The metallic phase resides in the phase space between the previously identified layer polarized state at large transverse electric field and the canted antiferromagnetic state at small transverse electric field. We also report temperature dependence studies of the quantum spin Hall state of ν=0. Complex nonmonotonic behavior reveals concomitant bulk and edge conductions and excitations. These results provide a timely experimental update to understand the rich physics of the ν=0 state.

Externally Controlled Magnetism and Band Gap in Twisted Bilayer Graphene

We theoretically study the effects of electron-electron interaction in twisted bilayer graphene in a transverse dc electric field. When the twist angle is not very small, the electronic spectrum of the bilayer consists of four Dirac cones inherited from each graphene layer. An applied bias voltage leads to the appearance of two holelike and two electronlike Fermi surface sheets with perfect nesting among electron and hole components. Such a band structure is unstable with respect to the exciton band-gap opening due to the screened Coulomb interaction. The exciton order parameter is accompanied by spin-density-wave order. The gap depends on the twist angle and can be varied by a bias voltage. This result correlates well with recent transport measurements [J.-B. Liu et al., Sci. Rep. 5, 15285 (2015)SRCEC32045-232210.1038/srep15285]. Our proposal allows the coexistence of (i) an externally controlled semiconducting gap and (ii) a nontrivial multicomponent magnetic order. This is interesting for both fundamental research and applications.

Effective Landau Level Diagram of Bilayer Graphene

The E=0 octet of bilayer graphene in the filling factor range of -4<ν<4 is a fertile playground for many-body phenomena, yet a Landau level diagram is missing due to strong interactions and competing quantum degrees of freedom. We combine measurements and modeling to construct an empirical and quantitative spectrum. The single-particlelike diagram incorporates interaction effects effectively and provides a unified framework to understand the occupation sequence, gap energies, and phase transitions observed in the octet. It serves as a new starting point for more sophisticated calculations and experiments.

Direct measurement of discrete valley and orbital quantum numbers in bilayer graphene

The high magnetic field electronic structure of bilayer graphene is enhanced by the spin, valley isospin, and an accidental orbital degeneracy, leading to a complex phase diagram of broken symmetry states. Here, we present a technique for measuring the layer-resolved charge density, from which we directly determine the valley and orbital polarization within the zero energy Landau level. Layer polarization evolves in discrete steps across 32 electric field-tuned phase transitions between states of different valley, spin, and orbital order, including previously unobserved orbitally polarized states stabilized by skew interlayer hopping. We fit our data to a model that captures both single-particle and interaction-induced anisotropies, providing a complete picture of this correlated electron system. The resulting roadmap to symmetry breaking paves the way for deterministic engineering of fractional quantum Hall states, while our layer-resolved technique is readily extendable to other two-dimensional materials where layer polarization maps to the valley or spin quantum numbers.

The phase diagram of bilayer graphene at high magnetic fields has been an outstanding question, with orders possibly between multiple internal quantum degrees of freedom. Here, Hunt et al. report the measurement of the valley and orbital order, allowing them to directly reconstruct the phase diagram.

Thermal Transport Signatures of Broken-Symmetry Phases in Graphene

In the half filled zero-energy Landau level of bilayer graphene, competing phases with spontaneously broken symmetries and an intriguing quantum critical behavior have been predicted. Here we investigate signatures of these broken-symmetry phases in thermal transport measurements. To this end, we calculate the spectrum of spin and valley waves in the ν=0 quantum Hall state of bilayer graphene. The presence of Goldstone modes enables heat transport even at low temperatures, which can serve as compelling evidence for spontaneous symmetry breaking. By varying external electric and magnetic fields, it is possible to determine the nature of the symmetry breaking. Temperature-dependent measurements may yield additional information about gapped modes.

Fingerprints of a Bosonic Symmetry-Protected Topological State in a Quantum Point Contact

In this work, we study the transport through a quantum point contact for bosonic helical liquid that exists at the edge of a bilayer graphene under a strong magnetic field. We identify "smoking gun" transport signatures to distinguish a bosonic symmetry-protected topological (BSPT) state from a fermionic two-channel quantum spin Hall (QSH) state in this system. In particular, a novel charge-insulator-spin-conductor phase is found for the BSPT state, while either the charge-insulator-spin-insulator or the charge-conductor-spin-conductor phase is expected for the two-channel QSH state. Consequently, a simple transport measurement will reveal the fingerprint of bosonic topological physics in bilayer graphene systems.

Bilayer Graphene as a Platform for Bosonic Symmetry-Protected Topological States

Bosonic symmetry protected topological (BSPT) states, the bosonic analogue of topological insulators, have attracted enormous theoretical interest in the last few years. Although BSPT states have been classified by various approaches, there is so far no successful experimental realization of any BSPT state in two or higher dimensions. In this paper, we propose that a two-dimensional BSPT state with U(1)×U(1) symmetry can be realized in bilayer graphene in a magnetic field. Here the two U(1) symmetries represent total spin S^{z} and total charge conservation, respectively. The Coulomb interaction plays a central role in this proposal-it gaps out all the fermions at the boundary, so that only bosonic charge and spin degrees of freedom are gapless and protected at the edge. Based on the above conclusion, we propose that the bulk quantum phase transition between the BSPT and trivial phase, which can be driven by applying both magnetic and electric fields, can become a "bosonic phase transition" with interactions. That is, only bosonic modes close their gap at the transition, which is fundamentally different from all the well-known topological insulator to trivial insulator transitions that occur for free fermion systems. We discuss various experimental consequences of this proposal.

Excitation gap of fractal quantum hall states in graphene

In the presence of a magnetic field and an external periodic potential the Landau level spectrum of a two-dimensional electron gas exhibits a fractal pattern in the energy spectrum which is described as the Hofstadter's butterfly. In this work, we develop a Hartree-Fock theory to deal with the electron-electron interaction in the Hofstadter's butterfly state in a finite-size graphene with periodic boundary conditions, where we include both spin and valley degrees of freedom. We then treat the butterfly state as an electron crystal so that we could obtain the order parameters of the crystal in the momentum space and also in an infinite sample. A phase transition between the liquid phase and the fractal crystal phase can be observed. The excitation gaps obtained in the infinite sample is comparable to those in the finite-size study, and agree with a recent experimental observation.

Multicomponent Quantum Hall Ferromagnetism and Landau Level Crossing in Rhombohedral Trilayer Graphene

Using transport measurements, we investigate multicomponent quantum Hall (QH) ferromagnetism in dual-gated rhombohedral trilayer graphene (r-TLG) in which the real spin, orbital pseudospin, and layer pseudospins of the lowest Landau level form spontaneous ordering. We observe intermediate QH plateaus, indicating a complete lifting of the degeneracy of the zeroth Landau level (LL) in the hole-doped regime. In charge neutral r-TLG, the orbital degeneracy is broken first, and the layer degeneracy is broken last and only in the presence of an interlayer potential U⊥. In the phase space of U⊥ and filling factor ν, we observe an intriguing "hexagon" pattern, which is accounted for by a model based on crossings between symmetry-broken LLs.

Insulating state in tetralayers reveals an even-odd interaction effect in multilayer graphene

Close to charge neutrality, the electronic properties of graphene and its multilayers are sensitive to electron-electron interactions. In bilayers, for instance, interactions are predicted to open a gap between valence and conduction bands, turning the system into an insulator. In mono and (Bernal-stacked) trilayers, which remain conducting at low temperature, interactions do not have equally drastic consequences. It is expected that interaction effects become weaker for thicker multilayers, whose behaviour should converge to that of graphite. Here we show that this expectation does not correspond to reality by revealing the occurrence of an insulating state close to charge neutrality in Bernal-stacked tetralayer graphene. The phenomenology-incompatible with the behaviour expected from the single-particle band structure-resembles that observed in bilayers, but the insulating state in tetralayers is visible at higher temperature. We explain our findings, and the systematic even-odd effect of interactions in Bernal-stacked layers of different thickness that emerges from experiments, in terms of a generalization of the interaction-driven, symmetry-broken states proposed for bilayers.

Ultracold Fermi gases with emergent SU(N) symmetry

We review recent experimental and theoretical progress on ultracold alkaline-earth Fermi gases with emergent SU(N) symmetry. Emphasis is placed on describing the ground-breaking experimental achievements of recent years. The latter include (1) the cooling to below quantum degeneracy of various isotopes of ytterbium and strontium, (2) the demonstration of optical Feshbach resonances and the optical Stern-Gerlach effect, (3) the realization of a Mott insulator of (173)Yb atoms, (4) the creation of various kinds of Fermi-Bose mixtures and (5) the observation of many-body physics in optical lattice clocks. On the theory side, we survey the zoo of phases that have been predicted for both gases in a trap and loaded into an optical lattice, focusing on two and three dimensional systems. We also discuss some of the challenges that lie ahead for the realization of such phases such as reaching the temperature scale required to observe magnetic and more exotic quantum orders. The challenge of dealing with collisional relaxation of excited electronic levels is also discussed.

Layer anti-ferromagnetism on bilayer honeycomb lattice

Bilayer honeycomb lattice, with inter-layer tunneling energy, has a parabolic dispersion relation, and the inter-layer hopping can cause the charge imbalance between two sublattices. Here, we investigate the metal-insulator and magnetic phase transitions on the strongly correlated bilayer honeycomb lattice by cellular dynamical mean-field theory combined with continuous time quantum Monte Carlo method. The procedures of magnetic spontaneous symmetry breaking on dimer and non-dimer sites are different, causing a novel phase transition between normal anti-ferromagnet and layer anti-ferromagnet. The whole phase diagrams about the magnetism, temperature, interaction and inter-layer hopping are obtained. Finally, we propose an experimental protocol to observe these phenomena in future optical lattice experiments.

Transport measurement of Landau level gaps in bilayer graphene with layer polarization control

Landau level (LL) gaps are important parameters for understanding electronic interactions and symmetry-broken processes in bilayer graphene (BLG). Here we present transport spectroscopy measurements of LL gaps in double-gated suspended BLG with high mobilities in the quantum Hall regime. By using bias as a spectroscopic tool, we measure the gap Δ for the quantum Hall (QH) state at filling factors ν = ±4 and -2. The single-particle Δ(ν=4) scales linearly with magnetic field B and is independent of the out-of-plane electric field E⊥. For the symmetry-broken ν = -2 state, the measured values of Δ(ν=-2) are ∼1.1 meV/T and 0.17 meV/T for singly gated geometry and dual-gated geometry at E⊥ = 0, respectively. The difference between the two values arises from the E⊥. dependence of Δ(ν=-2), suggesting that the ν = -2 state is layer polarized. Our studies provide the first measurements of the gaps of the broken symmetry QH states in BLG with well-controlled E⊥ and establish a robust method that can be implemented for studying similar states in other layered materials.

Observation of even denominator fractional quantum Hall effect in suspended bilayer graphene

We investigate low-temperature magneto-transport in recently developed, high-quality multiterminal suspended bilayer graphene devices, enabling the independent measurement of the longitudinal and transverse resistance. We observe clear signatures of the fractional quantum Hall effect with different states that are either fully developed, and exhibit a clear plateau in the transverse resistance with a concomitant dip in longitudinal resistance or incipient, and exhibit only a longitudinal resistance minimum. All observed states scale as a function of filling factor ν, as expected. An unprecedented even-denominator fractional state is observed at ν = -1/2 on the hole side, exhibiting a clear plateau in Rxy quantized at the expected value of 2h/e(2) with a precision of ∼0.5%. Many of our observations, together with a recent electronic compressibility measurement performed in graphene bilayers on hexagonal boron-nitride (hBN) substrates, are consistent with a recent theory that accounts for the effect of the degeneracy between the N = 0 and N = 1 Landau levels in the fractional quantum Hall effect and predicts the occurrence of a Moore-Read type ν = -1/2 state. Owing to the experimental flexibility of bilayer graphene, which has a gate-dependent band structure, can be easily accessed by scanning probes, and can be contacted with materials such as superconductors, our findings offer new possibilities to explore the microscopic nature of even-denominator fractional quantum Hall effect.

Evidence for a spontaneous gapped state in ultraclean bilayer graphene

At the charge neutrality point, bilayer graphene (BLG) is strongly susceptible to electronic interactions and is expected to undergo a phase transition to a state with spontaneously broken symmetries. By systematically investigating a large number of single-and double-gated BLG devices, we observe a bimodal distribution of minimum conductivities at the charge neutrality point. Although σ(min) is often approximately 2-3 e(2)/h (where e is the electron charge and h is Planck's constant), it is several orders of magnitude smaller in BLG devices that have both high mobility and low extrinsic doping. The insulating state in the latter samples appears below a transition temperature T(c) of approximately 5 K and has a T = 0 energy gap of approximately 3 meV. Transitions between these different states can be tuned by adjusting disorder or carrier density.

Transport spectroscopy of symmetry-broken insulating states in bilayer graphene

Bilayer graphene is an attractive platform for studying new two-dimensional electron physics, because its flat energy bands are sensitive to out-of-plane electric fields and these bands magnify electron-electron interaction effects. Theory predicts a variety of interesting broken symmetry states when the electron density is at the carrier neutrality point, and some of these states are characterized by spontaneous mass gaps, which lead to insulating behaviour. These proposed gaps are analogous to the masses generated by broken symmetries in particle physics, and they give rise to large Berry phase effects accompanied by spontaneous quantum Hall effects. Although recent experiments have provided evidence for strong electronic correlations near the charge neutrality point, the presence of gaps remains controversial. Here, we report transport measurements in ultraclean double-gated bilayer graphene and use source-drain bias as a spectroscopic tool to resolve a gap of ∼2 meV at the charge neutrality point. The gap can be closed by a perpendicular electric field of strength ∼15 mV nm(-1), but it increases monotonically with magnetic field, with an apparent particle-hole asymmetry above the gap. These data represent the first spectroscopic mapping of the ground states in bilayer graphene in the presence of both electric and magnetic fields.