We study the band structure of electrons hopping on a honeycomb lattice with 1/q (q integer) flux quanta through each elementary hexagon. In the nearest neighbor hopping model the two bands that eventually form the n = 0 Landau level have 2q zero energy Dirac touchings. In this work, we study the conditions needed for these Dirac points and their stability to various perturbations. We prove that these touchings and their locations are guaranteed by a combination of an anti-unitary particle-hole symmetry and the lattice symmetries of the honeycomb structure. We also study the stability of the Dirac touchings to one-body perturbations that explicitly lower the symmetry.
We consider Bloch electrons on the honeycomb lattice under a uniform magnetic field with $2 πp/q$ flux per cell. It is shown that the problem factorizes to two triangular lattices. Treating magnetic translations as Heisenberg-Weyl group and by the use of its irreducible representation on the space of theta functions, we find a nested set of Bethe equations, which determine the eigenstates and energy spectrum. The Bethe equations have simple form which allows to consider them further in the limit $p, q \to \infty$ by the technique of Thermodynamic Bethe Ansatz and analyze Hofstadter problem for…
We rely on a recent method for determining edge spectra and we use it to compute the Chern numbers for Hofstadter models on the honeycomb lattice having rational magnetic flux per unit cell. Based on the bulk-edge correspondence, the Chern number $σ_H$ is given as the winding number of an eigenvector of a $2 \times 2$ transfer matrix, as a function of the quasi-momentum $k \in (0,2 π)$. This method is computationally efficient (of order $O(n^4)$ in the resolution of the desired image). It also shows that for the honeycomb lattice the solution for $σ_H $ for flux $p/q$ in the $r$-th gap conform…
We perform an analytical study of the energy and entanglement spectrum of non-interacting fermionic bilayer honeycomb lattices in the presence of trigonal warping in the energy spectrum, on-site energy difference and uniform magnetic field. Employing single particle correlation functions, we present an explicit form for layer-layer entanglement Hamiltonian whose spectrum is entanglement spectrum. We demonstrate that in the absence of trigonal warping, at zero on-site energy difference exact correspondence is established between entanglement spectrum and energy spectrum of monolayer which means…
The $α$-$T_3$ model extrapolates between the pseudospin $S=1/2$ honeycomb lattice of graphene and the pseudospin $S=1$ dice lattice via parameter $α$. We present calculations of the magnetic properties of this hybrid pseudospin model, namely the absorptive magneto-optical conductivity and the Hofstadter butterfly spectra. In the magneto-optics curves, signatures of the hybrid system corollary a doublet structure present in the peaks, resulting from differing Landau level energies in the $K$ and $K^{\prime}$ valleys. In the Hofstadter spectra, we detail the evolution of the Hofstadter butterfly…
In this work, we report original properties inherent to independent particles subjected to a magnetic field by emphasizing the existence of regular structures in the energy spectrum's outline. We show that this fractal curve, the well-known Hofstadter butterfly's outline, is associated to a specific sequence of Chern numbers that correspond to the quantized transverse conductivity. Indeed the topological invariant that characterizes the fundamental energy band depicts successive stairways as the magnetic flux varies. Moreover each stairway is shown to be labeled by another Chern number which m…
The emergence of Hofstadter butterflies for bosons in synthetic-gauge-field antiferromagnetic patterns is theoretically studied. We report on a specific tight-binding model of artificial antiferromagnetic structures incorporating both nearest and next-to-nearest neighbour tunnelings and allowing for the formation of the fractal spectra even with the vanishing gauge field flux through the lattice. The model is applied to square and honeycomb lattices. Possible experimental realization is suggested for the lattices of microring resonators connected by waveguides. Finally, the structure of the bu…
We address the energy spectrum of honeycomb lattice with various defects or impurities under a perpendicular magnetic field. We use a tight-binding Hamiltonian including interactions with the nearest neighbors and investigate its energy structure for two different choices of point defects or impurities. In the first case, we fix a unit cell consisting of 8 lattice points and survey the energy eigenvalues in the presence of up to 2 point defects. Then it turns out that the existence of the fractal energy structure, called Hofstadter's butterfly, depends on the choice of defect pairs. In the sec…
We consider a tight-binding model on the honeycomb lattice in a magnetic field. For special values of the hopping integrals, the dispersion relation is linear in one direction and quadratic in the other. We find that, in this case, the energy of the Landau levels varies with the field B as E_n(B) ~ [(n+γ)B]^{2/3}. This result is obtained from the low-field study of the tight-binding spectrum on the honeycomb lattice in a magnetic field (Hofstadter spectrum) as well as from a calculation in the continuum approximation at low field. The latter links the new spectrum to the one of a modified quar…
As a topological insulator, the quantum Hall (QH) effect is indexed by the Chern and spin-Chern numbers $\mathcal{C}$ and $\mathcal{C}_{\text{spin}}$. We have only $\mathcal{C}_{\text{spin}}=0$ or $\pm \frac{1}{2}$ in conventional QH systems. We investigate QH effects in generic monolayer honeycomb systems. We search for spin-resolved characteristic patterns by exploring Hofstadter's butterfly diagrams in the lattice theory and fan diagrams in the low-energy Dirac theory. The Chern and spin-Chern numbers are calculated based on the bulk-edge correspondence in the lattice theory and on the Kubo…
We consider two-dimensional Fermi systems with quadratic band touching and $C_3$ symmetry, as realizable in Bernal-stacked honeycomb bilayers. Within a renormalization-group analysis, we demonstrate the existence of a quantum critical point at a finite value of the density-density interactions, separating a semimetallic disordered phase at weak coupling from a gapped ordered phase at strong coupling. The latter may be characterized by, for instance, antiferromagnetic, quantum anomalous Hall, or charge density wave order. In the semimetallic phase, each point of quadratic band touching splits i…
We show that the new quantum oscillations of the magnetization can occur when the Fermi surface consists of points (massless Dirac points) or even when the chemical potential is in a energy gap by studying the tight-binding electrons on a honeycomb lattice in a uniform magnetic field. The quantum oscillations of the magnetization as a function of the inverse magnetic field are known as the de Haas-van Alphen (dHvA) oscillations and the frequency is proportional to the area of the Fermi surface. The dominant period of the new oscillations corresponds to the area of the first Brillouin zone and …
We explore a new class of topologically stable zero energy modes which are protected by coexisting chiral and spatial symmetries. If a chiral symmetric Hamiltonian has an additional spatial symmetry such as reflection, inversion and rotation, the Hamiltonian can be separated into independent chiral-symmetric subsystems by the eigenvalue of the space symmetry operator. Each subsystem supports chiral zero energy modes when a topological index assigned to the block is nonzero. By applying the argument to Bloch electron systems, we detect band touching at symmetric points in the Brillouin zone. In…
In this work we use Floquet-Bloch theory to study the influence of circularly and linearly polarized light on two-dimensional band structures with semi-Dirac band touching points, taking the anisotropic nearest neighbor hopping model on the honeycomb lattice as an example. We find circularly polarized light opens a gap and induces a band inversion to create a finite Chern number in the two-band model. By contrast, linearly polarized light can either open up a gap (polarized in the quadratically dispersing direction) or split the semi-Dirac band touching point into two Dirac points (polarized i…
The notion of Dirac cones, wherein two or more bands become degenerate at a certain momentum, is the starting point for the study of topological phases. Dirac cones have been thoroughly explored in fermionic systems such as graphene, Weyl semimetals, etc. The underlying mathematical structure in these systems is a Clifford algebra -- a rule for identifying sets of matrices that span the Hamiltonian. This structure allows for the identification of suitable `mass' terms to open band gaps. In this article, we extend these ideas to bosonic systems. Due to the pseudo-orthogonal nature of eigenvecto…
Band structures of topological insulators are characterized by non-local topological invariants. Consequently, proposals for the experimental detection using local probes are rare. A recent paper [Slager et al., Phys. Rev. B 92, 085126 (2015)] has argued, based on theoretical results for a particular class of models, that insulators with topologically trivial and non-trivial band structures in two space dimensions display a qualitatively different response to point-like impurities. Here we present a comprehensive investigation of the impurity response of a large set of models of non-interactin…
We demonstrate the possibility of topological states for non-Dirac electrons. Specifically it is shown that, because of the $C_{\rm 3}$ crystal symmetry and time reversal symmetry, $p_x$ and $p_y$ orbits accommodated on triangular lattice exhibit a quadratic band touching at $Γ$ point at the Fermi level. When the atomic spin-orbit coupling (SOC) is taken into account, a gap is opened resulting in a quantum spin Hall effect state. As revealed explicitly by a $k\cdot p$ model, the topology is associated with a meron structure in the pseudo spin texture with vorticity two, a mechanism different f…
Dirac materials are of great interest as condensed matter realizations of the Dirac and Weyl equations. In particular, they serve as a starting point for the study of topological phases. This physics has been extensively studied in electronic systems such as graphene, Weyl- and Dirac semi-metals. In contrast, recent studies have highlighted several examples of Dirac-like cones in collective excitation spectra, viz. in phonon, magnon and triplon bands. These cannot be directly related to the Dirac or Weyl equations as they are bosonic in nature with pseudo-unitary band bases. In this article, w…
The extraordinary electronic properties of Dirac materials, the two-dimensional partners of Weyl semimetals, arise from the linear crossings in their band structure. When the dispersion around the Dirac points is tilted, the emergence of intricate transport phenomena has been predicted, such as modified Klein tunnelling, intrinsic anomalous Hall effects and ferrimagnetism. However, Dirac materials are rare, particularly with tilted Dirac cones. Recently, artificial materials whose building blocks present orbital degrees of freedom have appeared as promising candidates for the engineering of ex…
