When monolayer graphene is placed in a strong perpendicular magnetic field, Coulomb interactions among its Dirac-like charge carriers drive correlated phases: an insulating state at Landau-level filling factor nu=0 and a fractional quantum Hall effect (FQHE) at nu=1/3 and other fractions. The 2009 experiments that first observed the FQHE in suspended graphene left open the microscopic nature of these phases. This problem asks for the interaction-driven quantum Hall physics of monolayer graphene to be established microscopically: (1) which ground-state order produces the insulating nu=0 state and how its bulk and edge, electrical and thermal transport are reconciled with the candidate orders (quantum Hall ferromagnet, canted antiferromagnet, charge-density-wave/Kekule order, magnetic catalysis, pinned Wigner or skyrmion crystals); (2) direct quantitative determination of the quasiparticle excitation gap of the nu=1/3 (and neighboring) fractional liquids and its dependence on magnetic field, disorder, dielectric environment, and Landau-level mixing; (3) the extent to which higher Landau levels host correlated fractional states unique to relativistic carriers with approximate SU(4) spin-valley symmetry; and (4) the mechanism by which edge states become insulating in samples that otherwise show quantum Hall ferromagnetism. A 2024 thermal-transport experiment found vanishing bulk heat flow at nu=0 in monolayer graphene, contradicting the finite thermal conductance predicted for the expected ground states, so the central question of the nu=0 phase remains open.

Background

Graphene is a single atomic layer of carbon whose charge carriers behave, at low energies, as massless Dirac fermions: their energy is linear in momentum, E=ℏvF∣k∣E=\hbar v_F|k|, rather than quadratic. In a perpendicular magnetic field BB their spectrum forms highly degenerate Landau levels (discrete cyclotron orbits), of which the zeroth level, n=0n=0, is special: it is shared equally by electrons and holes and is fourfold degenerate in the two internal degrees of freedom of graphene — real spin and the valley index labelling the two inequivalent corners of the honeycomb-lattice Brillouin zone. The filling factor ν\nu counts how many of these internal flavors are filled. In ordinary two-dimensional semiconductor systems, strong field plus strong Coulomb interaction produces the fractional quantum Hall effect (FQHE): incompressible liquid ground states, most robustly at ν=1/3\nu=1/3, whose quasiparticle excitations carry fractional charge e/3e/3 and open an energy gap that appears experimentally as an activated (exponential-in-temperature) longitudinal resistance.

Because kinetic energy is quenched in Landau levels, the Coulomb interaction dominates, and in graphene the four spin-valley flavors generate an approximate SU(4) symmetry (the group of rotations acting simultaneously on spin and valley). Exchange can spontaneously break this symmetry, producing a quantum Hall ferromagnet: a state with a fixed spin-valley polarization whose low-energy excitations are topological spin textures (skyrmions) and which, in a partially filled Landau level, hosts spin-filtered counter-propagating edge channels. At charge neutrality, ν=0\nu=0, experiment instead shows an insulator whose resistance grows strongly with magnetic field. Competing theoretical explanations current at the time of the source paper were: (a) a quantum Hall ferromagnet in which two counter-propagating, oppositely spin-polarized edge states must be gapped by some mechanism that mixes spin or valley; (b) magnetic catalysis, in which interactions open a bulk Dirac-mass gap without edge states at all; and (c) pinned collective phases such as Wigner crystals (a crystalline arrangement of the electrons) or skyrmion crystals.

Subsequent theory sharpened the candidate list for ν=0\nu=0. Kharitonov (2012) derived a phase diagram for defect-free graphene in which four orders compete: a spin-polarized ferromagnet; a canted antiferromagnet, in which the spins of the two valleys order antiparallel and tilt away from the field direction; a charge-density wave, a valley polarization that also imbalances the two sublattices of the honeycomb lattice; and a Kekule distortion, a bond-order wave that doubles the unit cell and gaps the Dirac spectrum on the lattice scale. Landau-level mixing strongly renormalizes the valley anisotropy energies that select among these orders relative to the Zeeman energy. Thermal (heat) transport supplies a bulk-versus-edge discriminator that charge transport cannot: each gapless one-dimensional channel carries a universal quantum of thermal conductance, π2kB2T/3h\pi^2 k_B^2 T/3h per mode, at temperature TT, so heat flow measured while electrical conductance vanishes separates bulk from edge conduction. A 2024 thermal-transport measurement at ν=0\nu=0 in monolayer graphene on hexagonal boron nitride found the bulk heat flow to vanish, contradicting the finite thermal conductance predicted for the expected ground states (Delagrange et al., Nature Physics 20, 1927 (2024)).

In 2009 two groups observed the FQHE in graphene (Du et al.; Bolotin et al.), using suspended devices (fabricated so that the graphene sheet has no substrate beneath it, removing substrate-induced disorder and charge inhomogeneity) probed by two-terminal transport. Du et al. observed the FQHE at ν=1/3\nu=1/3 and higher fractions, with a nu=1/3 gap estimated only to order of magnitude from the temperature dependence of two-terminal resistance, and a field-induced insulating state at ν=0\nu=0 that competes with the fractional states and appears only in the highest-quality samples. The paper's discussion posed the unresolved questions that define this problem. A key complication for all of them is Landau-level mixing: the Coulomb scale e2/ϵℓe^2/\epsilon\ell (ℓ\ell the magnetic length, ϵ\epsilon the dielectric constant of the environment) is comparable to the Landau-level spacing ℏvF/ℓ\hbar v_F/\ell, so the projection onto a single Landau level that is exact in semiconductor systems is only approximate in graphene, with a dimensionless mixing strength κ=(e2/ϵvFℏ)\kappa=(e^2/\epsilon v_F\hbar) tunable mainly by the dielectric environment.

Problem Statement

Establish, microscopically and quantitatively, the interaction-driven quantum Hall phases of monolayer graphene and their transport, as follows. (1) Determine the microscopic origin of the insulating state at Landau-level filling factor ν=0\nu=0: which ground-state order (or, if the realized state is a mixture or a disorder-modified variant of these orders, the specific mixture and disorder class) is realized — a spin-polarized or canted quantum Hall ferromagnet with counter-propagating edge states gapped by spin/valley-mixing mechanisms, a magnetic-catalysis bulk mass gap without edge states, a charge-density-wave or Kekule order, or a pinned collective phase such as a Wigner or skyrmion crystal — as a function of Zeeman energy, Landau-level-mixing strength, dielectric environment, and disorder; and reconcile the bulk and edge, electrical and thermal transport signatures with the selected order, including the reported vanishing bulk heat flow. (2) Determine the quasiparticle excitation gap of the ν=1/3\nu=1/3 fractional quantum Hall liquid (and its neighbors) directly and quantitatively, and establish how it scales with magnetic field and with sample parameters (disorder, dielectric environment, Landau-level mixing), bringing measured activation gaps into quantitative agreement with microscopic theory. (3) Establish the extent to which higher Landau levels of graphene host new correlated fractional states, including states specific to relativistic carriers with approximate SU(4) spin-valley internal symmetry, and how sample disorder and edge structure stabilize or destroy these correlated phases. (4) Establish the mechanism by which edge states, when present, become insulating in samples that otherwise show signatures of quantum Hall ferromagnetism. These four components share one scientific objective: the microscopic identification of the correlated quantum Hall ground states of monolayer graphene and of their gap, order, and edge/bulk transport structure. A resolution requires components (1) and (2) jointly — the microscopic origin of the ν=0\nu=0 insulator and its reconciliation with transport, together with the quantitative theory-experiment agreement for the fractional excitation gaps — while components (3) and (4) are independently refutable sub-results; establishing any single component on its own does not complete the objective.

References

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