Topological qubits in graphenelike systems

Physica Scripta, 2012

We review the classification of all the 36 possible gap-opening instabilities in graphene, i.e., the 36 relativistic masses of the two-dimensional Dirac Hamiltonian when the spin, valley, and superconducting channels are included. We then show that in graphene it is possible to realize an odd number of Majorana fermions attached to vortices in superconducting order parameters if a proper hierarchy of mass scales is in place.

2020

Two-dimensional (2D) materials, composed of single atomic layers, have attracted vast research interest since the breakthrough discovery of graphene. One major benefit of such systems is the simple ability to tune the chemical potential by back-gating, in-principle enabling to vary the Fermi level through the charge neutrality point, thus tuning between electron and hole doping. For 2D Superconductors, this means that one may potentially achieve the strongly-coupled superconducting regime described by Bose Einstein Condensation physics of small bosonic tightly bound electron pairs. Furthermore, it should be possible to access both electron and hole based superconductivity in a single system. However, in most 2D materials, an insulating gap opens up around the charge neutrality point, thus preventing approach to this regime. Graphene is unique in this sense since it is a true semi-metal in which the un-gapped Dirac point is protected by the symmetries. In this work we show that singl...

Physical Review B, 2013

We show that carbon nanotubes (CNT) are good candidates for realizing one-dimensional topological superconductivity with Majorana fermions localized near the end points. The physics behind topological superconductivity in CNT is novel and is mediated by a recently reported curvatureinduced spin-orbit coupling which itself has a topological origin. In addition to the spin-orbit coupling, an important new requirement for a robust topological state is broken chirality symmetry about the nanotube axis. We use topological arguments to show that, for recently realized strengths of spin-orbit coupling and broken chirality symmetry, a robust topological gap ∼ 500 mK is achievable in carbon nanotubes.

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

Physical Review B, 2010

We study the superconducting instabilities of a single species of two-dimensional Rashba-Dirac fermions, as it pertains to the surface of a three-dimensional time-reversal symmetric topological band insulator. We also discuss the similarities as well as the differences between this problem and that of superconductivity in two-dimensional time-reversal symmetric noncentrosymmetric materials with spin-orbit interactions. The superconducting order parameter has both s-wave and p-wave components, even when the superconducting pair potential only transfers either pure singlet or pure triplet pairs of electrons in and out of the condensate, a corollary to the nonconservation of spin due to the spin-orbit coupling. We identify one single superconducting regime in the case of superconductivity in the topological surface states (Rashba-Dirac limit), irrespective of the relative strength between singlet and triplet pair potentials. In contrast, in the Fermi limit relevant to the noncentrosymmetric materials we find two regimes depending on the value of the chemical potential and the relative strength between singlet and triplet potentials. We construct explicitly the Majorana bound states in these regimes. In the single regime for the case of the Rashba-Dirac limit, there exists one and only one Majorana fermion bound to the core of an isolated vortex. In the Fermi limit, there are always an even number (0 or 2 depending on the regime) of Majorana fermions bound to the core of an isolated vortex. In all cases, the vorticity required to bind Majorana fermions is quantized in units of the flux quantum, in contrast to the half flux in the case of two-dimensional p x ± ip y superconductors that break time-reversal symmetry.

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.

The European Physical Journal Special Topics, 2007

We discuss topological aspects of electronic properties of graphene, including edge effects, with the tight-binding model on a honeycomb lattice and its extensions to show the following: (i) Appearance of the pair of massless Dirac dispersions, which is the origin of anomalous properties including a peculiar quantum Hall effect (QHE), is not accidental to honeycomb, but is rather generic for a class of two-dimensional lattices that interpolate between square and π-flux lattices. Persistence of the peculiar QHE is interpreted as a topological stability. (ii) While we have the massless Dirac dispersion only around E = 0, the anomalous QHE associated with the Dirac cone unexpectedly persists for a wide range of the chemical potential. The range is bounded by van Hove singularities, at which we predict a transition to the ordinary fermion behavior accompanied by huge jumps in the QHE with a sign change. (iii) For edges we establish a coincidence between the quantum Hall effect in the bulk and the quantum Hall effect for the edge states, which is a manifestation of the topological bulk-edge correspondence. We have also explicitly shown that the E = 0 edge states in honeycomb in zero magnetic field persist in magnetic field.

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.

Physical Review B, 2011

The Bi2Se3 class of topological insulators has recently been shown to undergo a superconducting transition upon hole or electron doping (Cux-Bi2Se3 with TC =3.8 o K and Pdx-Bi2Te3 with TC =5 o K), raising the possibilities that these are the first known "topological superconductors" or realizes a superconducting state that can be potentially used as Majorana platforms (L. A. Wray et.al., Nature Phys. 6, 855-859 (2010)). We use angle resolved photoemission spectroscopy to examine the full details of the spin-orbital groundstates of these materials including Bi2Te3, observing that the spin-momentum locked topological surface states remain well defined and non-degenerate with respect to bulk electronic states at the Fermi level in the optimally doped superconductor and obtaining their experimental Fermi energies. The implications of this unconventional surface (that undergoes superconducting at lower temperatures) topology are discussed, and we also explore the possibility of realizing the same topology in superconducting variants of Bi2Te3 (with TC ∼ 5 o K). Characteristics of the experimentally measured three dimensional bulk states are examined in detail for these materials with respect to the superconducting state and topological properties, showing that a single Majorana fermion zero mode is expected to be bound at each superconducting vortex on the surface. Systematic measurements also reveal intriguing renormalization and charge correlation instabilities of the surface-localized electronic modes.

Physical Review B, 2020

Starting from the strong-coupling limit of an extended Hubbard model, we develop a spin-fermion theory to study the insulating phase and pairing symmetry of the superconducting phase in twisted bilayer graphene. Assuming that the insulating phase is an anti-ferromagnetic insulator, we show that fluctuations of the anti-ferromagnetic order in the conducting phase can mediate superconducting pairing. Using a self-consistent mean-field analysis, we find that the pairing wave function has a chiral d-wave symmetry. Consistent with this observation, we show explicitly the existence of chiral Majorana edge modes by diagonalizing our proposed Hamiltonian on a finite-sized system. These results establish twisted bilayer graphene as a promising platform to realize topological superconductivity. 1(b). This loss of monolayer valley quantum number explains why the experimentally observed Landau level degeneracy close to half filling is half that at charge neutrality. This breaking of valley symmetry (as opposed to spin symmetry) is supported by Hartree-Fock calculations [10], and because experiments suggests spin-singlet