Collective edge modes of a quantum Hall ferromagnet in graphene

Physical Review B

Bilayer graphene (BLG) offers a rich platform for broken symmetry states stabilized by interactions. In this work we study the phase diagram of BLG in the quantum Hall regime at filling factor ν = 0 within the Hartree-Fock approximation. In the simplest non-interacting situation this system has eight (nearly) degenerate Landau levels near the Fermi energy, characterized by spin, valley, and orbital quantum numbers. We incorporate in our study two effects not previously considered: (i) the nonperturbative effect of trigonal warping in the single-particle Hamiltonian, and (ii) short-range SU(4) symmetry-breaking interactions that distinguish the energetics of the orbitals. We find within this model a rich set of phases, including ferromagnetic, layer-polarized, canted antiferromagnetic, Kekulé, a "spin-valley entangled" state, and a "broken U(1) × U(1)" phase. This last state involves independent spontaneous symmetry breaking in the layer and valley degrees of freedom, and has not been previously identified. We present phase diagrams as a function of interlayer bias D and perpendicular magnetic field B ⊥ for various interaction and Zeeman couplings, and discuss which are likely to be relevant to BLG in recent measurements. Experimental properties of the various phases and transitions among them are also discussed.

Physical Review B

Monolayer graphene at charge neutrality in a quantizing magnetic field is a quantum Hall ferromagnet. Due to the spin and valley (near) degeneracies, there is a plethora of possible ground states. Previous theoretical work, based on a stringent ultra short-range assumption on the symmetryallowed interactions, predicts a phase diagram with distinct regions of spin-polarized, canted antiferromagnetic, inter-valley coherent, and charge density wave order. While early experiments suggested that the system was in the canted antiferromagnetic phase at a perpendicular field, recent scanning tunneling studies universally find Kekulé bond order, and sometimes also charge density wave order. Recently, it was found that if one relaxes the stringent assumption mentioned above, a phase with coexisting canted antiferromagnetic and Kekulé order exists in the region of the phase diagram believed to correspond to real samples. In this work, starting from the continuum limit appropriate for experiments, we present the complete phase diagram of ν = 0 graphene in the Hartree-Fock approximation, using generic symmetry-allowed interactions, assuming translation invariant ground states up to an intervalley coherence. Allowing for a sublattice potential (valley Zeeman coupling), we find numerous phases with different types of coexisting order. We conclude with a discussion of the physical signatures of the various states.

Low Temperature Physics, 2008

We analyze a gap equation for the propagator of Dirac quasiparticles and conclude that in graphene in a magnetic field, the order parameters connected with the quantum Hall ferromagnetism dynamics and those connected with the magnetic catalysis dynamics necessarily coexist (the latter have the form of Dirac masses and correspond to excitonic condensates). This feature of graphene could lead to important consequences, in particular, for the existence of gapless edge states. Solutions of the gap equation corresponding to recently experimentally discovered novel plateaus in graphene in strong magnetic fields are described.

The total energy of interacting Dirac electrons in the lowest Landau level is found, within the same many-body theoretical framework, to exhibit jointly strong cusps at angular momenta corresponding to quantum Hall fractions and an insulating energy gap (that increases with the magnetic field) at the Dirac point. The model employs single-particle basis functions obeying the infinite-mass boundary condition at the sample edge, and initial volume-type states transform into edge-type states with increasing field. A single-ring rotating-Wigner-molecule correlated many-body state forms as a result of the interelectron repulsion. These results are contrasted with those corresponding to a boundless graphene sheet and a two-dimensional electron gas in semiconductor heterostructures.

Modern Physics Letters B, 2012

We investigate the quantum Hall effect in graphene. We argue that in graphene in presence of an external magnetic field there is dynamical generation of mass by a rearrangement of the Dirac sea. We show that the mechanism breaks the lattice valley degeneracy only for the n = 0 Landau levels and leads to the new observed ν = ±1 quantum Hall plateaus. We suggest that our result can be tested by means of numerical simulations of planar Quantum Electro Dynamics with dynamical fermions in an external magnetic fields on the lattice.

Nature Communications, 2017

Quantum Hall effect provides a simple way to study the competition between single particle physics and electronic interaction. However, electronic interaction becomes important only in very clean graphene samples and so far the trilayer graphene experiments are understood within non-interacting electron picture. Here, we report evidence of strong electronic interactions and quantum Hall ferromagnetism seen in Bernal-stacked trilayer graphene. Due to high mobility ∼500,000 cm2V−1s−1 in our device compared to previous studies, we find all symmetry broken states and that Landau-level gaps are enhanced by interactions; an aspect explained by our self-consistent Hartree–Fock calculations. Moreover, we observe hysteresis as a function of filling factor and spikes in the longitudinal resistance which, together, signal the formation of quantum Hall ferromagnetic states at low magnetic field.

Physical Review Letters, 2014

We report on transport measurements of dual-gated, single-layer graphene devices in the quantum Hall regime, allowing for independent control of the filling factors in adjoining regions. Progress in device quality allows us to study scattering between edge states when the four-fold degeneracy of the Landau level is lifted by electron correlations, causing edge states to be spin and/or valley polarized. In this new regime, we observe a dramatic departure from the equilibration seen in more disordered devices: edge states with opposite spins propagate without mixing. As a result, the degree of equilibration inferred from transport can reveal the spin polarization of the ground state at each filling factor. In particular, the first Landau level is shown to be spin-polarized at half-filling, providing an independent confirmation of a conclusion of Ref. . The conductance in the bipolar regime is strongly suppressed, indicating that co-propagating edge states, even with the same spin, do not equilibrate along PN interfaces. We attribute this behavior to the formation of an insulating ν = 0 stripe at the PN interface.

Physical Review B, 2013

Motivated by the recent experiments indicating a spin-unpolarized ν = 0 quantum Hall state in graphene, we theoretically investigate the ground state based on the many-body problem projected onto the n = 0 Landau level. For an effective model with the on-site Coulomb repulsion and antiferromagnetic exchange couplings, we show that the ground state is a doubly degenerate spin-resolved chiral condensate in which all the zero-energy states with up spin are condensed into one chirality, while those with down spin to the other. This can be exactly shown for an Ising-type exchange interaction. The charge gap due to the on-site repulsion in the ground state is shown to grow linearly with the magnetic field, in qualitative agreement with the experiments.

We consider a straight one-dimensional potential step created across a graphene flake. Charge and spin transport through such a potential step are studied in the presence of both intrinsic and extrinsic (Rashba) spin-orbit coupling (SOC). At normal incidence electrons are completely reflected when the Rashba interaction (with strength $\lambda_R$) is dominant whereas they are perfectly transmitted if the two types of SOC are exactly balanced. At normal incidence, the transmission probability of the step is thus controlled continuously from 0 to 1 by tuning the ratio of the two types of SOC. Besides the transport of charge in the direction normal to the barrier, we show the existence of a spin transport along the barrier. The magnitude of the spin Hall current is determined by a subtle interplay between the height of the potential step and the position of Fermi energy. It is demonstrated that contributions from inter-band matrix elements and evanescent modes are dominant in spin tran...

Physical Review B, 2006

Graphene exhibits quantum Hall ferromagnetism in which an approximate SU (4) symmetry involving spin and valley degrees of freedom is spontaneously broken. We construct a set of integer and fractional quantum Hall states that break the SU (4) spin/valley symmetry, and study their neutral and charged excitations. Several properties of these ferromagnets can be evaluated analytically in the SU (4) symmetric limit, including the full collective mode spectrum at integer fillings. By constructing explicit wave functions we show that the lowest energy skyrmion states carry charge ±1 for any integer filling, and that skyrmions are the lowest energy charged excitations for graphene Landau level index |n| ≤ 3. We also show that the skyrmion lattice states which occur near integer filling factors support four gapless collective mode branches in the presence of full SU (4) symmetry. Comparisons are made with the more familiar SU (2) quantum Hall ferromagnets studied previously.