Zero modes, energy gap, and edge states of anisotropic honeycomb lattice in a magnetic field

Abstract

We present systematic study of zero modes and gaps by introducing effects of anisotropy of hopping integrals for a tight-binding model on the honeycomb lattice in a magnetic field. The condition for the existence of zero modes is analytically derived. From the condition, it is found that a tiny anisotropy for graphene is sufficient to open a gap around zero energy in a magnetic field. This gap behaves as a non-perturbative and exponential form as a function of the magnetic field. The non-analytic behavior with respect to the magnetic field can be understood as tunneling effects between energy levels around two Dirac zero modes appearing in the honeycomb lattice, and an explicit form of the gap around zero energy is obtained by the WKB method near the merging point of these Dirac zero modes. Effects of the anisotropy for the honeycomb lattices with boundaries are also studied. The condition for the existence of zero energy edge states in a magnetic field is analytically derived. On the basis of the condition, it is recognized that anisotropy of the hopping integrals induces abrupt changes of the number of zero energy edge states, which depend on the shapes of the edges sensitively.

36 pages, 20 figures; added discussion on experiments in Sec.VI, cited Refs.[35]-[40], and reworded Sec.IV D

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Cited by in corpus (8)

• Merging of Dirac points in a two-dimensional crystal
• Su-Schrieffer-Heeger chain with one pair of PT-symmetric defects
• A semi-Dirac point in the Hofstadter spectrum
• Self-similar occurrence of massless Dirac particles in graphene under magnetic field
• Unusual electronic properties of clean and disordered zigzag graphene nanoribbons
• Quantum Hall effective action for anisotropic Dirac semi-metal
• Topological Mott transition in a two band model of spinless fermions with on-site Coulomb repulsion
• Dirac fermion spectrum of the fractional quantum Hall states