Topological aspects of graphene

arXiv (Cornell University), 2006

Inspired by a recent discovery of a peculiar integer quantum Hall effect (QHE) in graphene, we study QHE on a honeycomb lattice in terms of the topological quantum number, with twofold interests: First, how the zero-mass Dirac QHE around the center of the tight-binding band crosses over to the ordinary finite-mass fermion QHE around the band edges. Second, how the bulk QHE is related with the edge QHE for the entire spectrum including Dirac and ordinary behaviors. We find the following: (i) The zero-mass Dirac QHE (with σxy = ∓(2N + 1)e 2 /h, N : integer) persists, surprisingly, up to the van Hove singularities, at which the ordinary fermion behavior abruptly takes over. Here a technique developed in the lattice gauge theory enabled us to calculate the behavior of the topological number over the entire spectrum. This result indicates a robustness of the topological quantum number, and should be observable if the chemical potential can be varied over a wide range in graphene. (ii) To see if the honeycomb lattice is singular in producing the anomalous QHE, we have systematically surveyed over square↔honeycomb ↔π-flux lattices, which is scanned by introducing a diagonal transfer t ′. We find that the massless Dirac QHE [propto(2N + 1)] forms a critical line, that is, the presence of Dirac cones in the Brillouin zone is preserved by the inclusion of t ′ and the Dirac region sits side by side with ordinary one persists all through the transformation. (iii) We have compared the bulk QHE number obtained by an adiabatic continuity of the Chern number across the square↔honeycomb↔ π-flux transformation and numerically obtained edge QHE number calculated from the whole energy spectra for sample with edges, which shows that the bulk QHE number coincides, as in ordinary lattices, with the edge QHE number throughout the lattice transformation.

Physical Review B, 2006

Inspired by a recent discovery of a peculiar integer quantum Hall effect (QHE) in graphene, we study QHE on a honeycomb lattice in terms of the topological quantum number, with twofold interests: First, how the zero-mass Dirac QHE around the center of the tight-binding band crosses over to the ordinary finite-mass fermion QHE around the band edges. Second, how the bulk QHE is related with the edge QHE for the entire spectrum including Dirac and ordinary behaviors. We find the following: (i) The zero-mass Dirac QHE (with σxy = ∓(2N + 1)e 2 /h, N : integer) persists, surprisingly, up to the van Hove singularities, at which the ordinary fermion behavior abruptly takes over. Here a technique developed in the lattice gauge theory enabled us to calculate the behavior of the topological number over the entire spectrum. This result indicates a robustness of the topological quantum number, and should be observable if the chemical potential can be varied over a wide range in graphene. (ii) To see if the honeycomb lattice is singular in producing the anomalous QHE, we have systematically surveyed over square↔honeycomb ↔π-flux lattices, which is scanned by introducing a diagonal transfer t ′. We find that the massless Dirac QHE [propto(2N + 1)] forms a critical line, that is, the presence of Dirac cones in the Brillouin zone is preserved by the inclusion of t ′ and the Dirac region sits side by side with ordinary one persists all through the transformation. (iii) We have compared the bulk QHE number obtained by an adiabatic continuity of the Chern number across the square↔honeycomb↔ π-flux transformation and numerically obtained edge QHE number calculated from the whole energy spectra for sample with edges, which shows that the bulk QHE number coincides, as in ordinary lattices, with the edge QHE number throughout the lattice transformation.

Journal of Physics: Conference Series, 2011

Topological aspects of graphene are reviewed focusing on the massless Dirac fermions with/without magnetic field. Doubled Dirac cones of graphene are topologically protected by the chiral symmetry. The quantum Hall effect of the graphene is described by the Berry connection of a manybody state by the filled Landau levels which naturally possesses non-Abelian gauge structures. A generic principle of the topologically non trivial states as the bulk-edge correspondence is applied for graphene with/without magnetic field and explain some of the characteristic boundary phenomena of graphene. 1

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.

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.

Physical Review B, 2008

Motivated by recent experiments and a theoretical analysis of the gap equation for the propagator of Dirac quasiparticles, we assume that the physics underlying the recently observed removal of sublattice and spin degeneracies in graphene in a strong magnetic field is connected with the generation of both Dirac masses and spin gaps. The consequences of such a scenario for the existence of the gapless edge states with zigzag and armchair boundary conditions are discussed. In the case of graphene on a half-plane with a zigzag edge, there are gapless edge states in the spectrum only when the spin gap dominates over the mass gap. In the case of an armchair edge, however, the existence of the gapless edge states depends on the specific type of mass gaps.

Low Temperature Physics, 2008

We review recent results concerning the spectrum of edge states in the quantum Hall effect in graphene. In particular, a special attention is payed to the derivation of the conditions under which gapless edge states exist in the spectrum of graphene with zigzag and armchair edges. We find that in the case of a half-plane or a ribbon with a zigzag edges, there are gapless edge states only when a spin gap dominates over a Dirac mass gap. In the case of a half-plane with an armchair edge, the existence of the gapless edge states depends on the specific type of Dirac mass gaps. The implications of these results for the dynamics in the quantum Hall effect in graphene are discussed.

Solid State Communications, 2009

There are two types of edge states in graphene with/without magnetic field. One is a quantum Hall edge state, which is topologically protected against small perturbation. The other is a chiral zero mode that is localized near the boundary with/without magnetic field. The latter is also topological but is guaranteed to be zero energy by the chiral symmetry, which is also responsible for massless Dirac like dispersion. Conceptual roles of the edge states are stressed and reviewed from a view point of the bulk-edge correspondence and the topological order.

Physical Review B, 2006

We study the integer and fractional quantum Hall effect on a honeycomb lattice at half-filling (graphene) in the presence of disorder and electron-electron interactions. We show that the interactions between the delocalized chiral edge states (generated by the magnetic field) and Andersonlocalized surface states (created by the presence of zig-zag edges) lead to edge reconstruction. As a consequence, the point contact tunneling on a graphene edge has a non-universal tunneling exponent, and the Hall conductivity is not perfectly quantized in units of e 2 /h. We argue that the magnetotransport properties of graphene depend strongly on the strength of electron-electron interactions, the amount of disorder, and the details of the edges.

PRB 69, 075104 (04); V.P. Gusynin, S.G. Sharapov, PRB 71, 125124 (05), PRL 95, 146801 (05); cond-mat/0512157; V.P. Gusynin, S.G. Sharapov, J.P. Carbotte, cond-mat/0603267.