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The electronic properties of a tight-binding model which possesses two types of hopping matrix element (or on-site energy) arranged in a Fibonacci sequence are studied. The wave functions are either self-similar (fractal) or chaotic and show "critical" (or "exotic") behavior. Scaling analysis for the self-similar wave functions at the center of the band and also at the edge of the band is performed. The energy spectrum is a Cantor set with zero Lebesque measure. The density of states is singularly concentrated with an index ae which takes a value in the range [cte'", az'"]. The fractal dimensions f(ae) of these singularities in the Cantor set are calculated. This function f(ae) represents the global scaling properties of the Cantor-set spectrum.

Topological stability of the edge states is investigated for non-Hermitian systems. We examine two classes of non-Hermitian Hamiltonians supporting real bulk eigenenergies in weak non-Hermiticity: SU(1,1) and SO(3,2) Hamiltonians. As an SU(1,1) Hamiltonian, the tight-binding model on the honeycomb lattice with imaginary on-site potentials is examined. Edge states with ReE=0 and their topological stability are discussed by the winding number and the index theorem, based on the pseudo-anti-Hermiticity of the system. As a higher symmetric generalization of SU(1,1) Hamiltonians, we also consider SO(3,2) models. We investigate non-Hermitian generalization of the Luttinger Hamiltonian on the square lattice, and that of the Kane-Mele model on the honeycomb lattice, respectively. Using the generalized Kramers theorem for the time-reversal operator Theta with Theta^2=+1 [M. Sato et al., arXiv:1106.1806], we introduce a time-reversal invariant Chern number from which topological stability of gapless edge modes is argued.Comment: 29 pages, 19 figures, typos fixe

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Tight binding electrons on the honeycomb lattice are studied where nearest neighbor hoppings in the three directions are ta, t b and tc, respectively. For the isotropic case, namely for ta = t b = tc, two zero modes exist where the energy dispersions at the vanishing points are linear in momentum k. Positions of zero modes move in the momentum space as ta, t b and tc are varied. It is shown that zero modes exist ifThe density of states near a zero mode is proportional to |E| but it is propotional to |E| at the boundary of this condition PACS numbers: 81.05. Uw, 73.43.Cd The integer quantum Hall effect has been observed in graphene 1,2 when the carriers are changed by the gate voltage. The quantization of the Hall effect is observed as σ xy = 2n e 2 h with n = ±1, ±3 · · · , where the factor 2 comes from the spin degrees of freedom. These quantum numbers are unusual, since in a usual case n = 0, ±1, ±2, · · · . This unusual quantum Hall effect was discussed in terms of relativistic Dirac theory 3 . However it is more natural to be explained by the realization of the quantum Hall effect in periodic systems 4 in the presence of zero modes 5,6 . We will call zero modes instead of massless Dirac excitations in this paper because we do not consider relativistic particles. The energy spectrum and the density of states of the honeycomb lattice near half filling and in zero or small magnetic field are similar to these in the square lattice near half filling in a very strong magnetic field about half flux quantum per each unit cell 5 .At zero carrier concentration (i.e. half-filled electrons), the resistivity ρ xx is close to the quantum value h/(4e 2 ) = 6.45kΩ independent of temperature 1 , which has been also attributed to the zero modes 1,2,7 .The existence of zero modes has also been proposed for the quasi-two-dimensional organic conductor α-(BEDT-TTF) 2 I 3 . The conductivity under pressure is almost constant in a wide range of temperature 8 . Pertinent numerical computations performed by Kobayashi et al.9 found that, for certain range of parameters, the Fermi surfaces become points and the density of states is proportional to energy at 3/4 filling of electrons. The existence of zero modes was also confirmed by the band structure calculation 10,11 . The unit cell for the model of α-(BEDT-TTF) 2 I 3 has four non-equivalent sites. Katayama et al. 12 studied simpler model with two sites in the unit cell and they obtained a condition for zero modes.In this Letter we study a tight binding model on the honeycomb lattice and obtain the condition of t a , t b and t c for the existence of zero modes.Unit cell of the honeycomb lattice contains two sublattices as shown in Fig. 1a. The Bravais lattice is a ) and v2 = (). Three nearest neighbor hoppings are ta, t b and tc b. The red hexagon is a Brillouin zone for the honeycomb lattice. The reciprocal lattice vectors are G1 = () and G2 = ( ). White circles are Γ points. Brillouin zone can also be taken by the green diamond.triangular lattice withwhere a is a distance bet...