The electron states of gapped pseudospin-1 fermions of the α − T3 lattice in the Coulomb field of a charged impurity are studied. The free α − T3 model has three dispersive bands with two energy gaps between them depending on the parameter Θ which controls the coupling of atoms of honeycomb lattice with atoms in the center of each hexagon, thus, interpolating between graphene Θ = 0 and the dice model Θ = π/4. The middle band becomes flat one with zero energy in the dice model. The bound electron states are found in the two cases: the centrally symmetric potential well and a regularized Coulomb potential of the charged impurity. As the charge of impurity increases, bound state energy levels descend from the upper and central continua and dive at certain critical charges into the central and lower continuum, respectively. In the dice model, it is found that the flat band survives in the presence of a potential well, however, is absent in the case of the Coulomb potential. The analytical results are presented for the energy levels near continuum boundaries in the potential well. For the genuine Coulomb potential, we present the recursion relations that determine the coefficients of the series expansion of wave functions of bound states. It is shown that the condition for the termination of the series expansion gives two equations relating energy and charge values. Hence, analytical solutions can exist for a countably infinite set of values of impurity charge at fixed Θ.

Boundary conditions for the two-dimensional fermions in ribbons of the hexagonal lattice are studied in the dice model whose energy spectrum in infinite system consists of three bands with one completely flat band of zero energy. Like in graphene the regular lattice terminations are of the armchair and zigzag types. However, there are four possible zigzag edge terminations in contrast to graphene where only one type of zigzag termination is possible. Determining the boundary conditions for these lattice terminations, the energy spectra of pseudospin-1 fermions in dice model ribbons with zigzag and armchair boundary conditions are found. It is shown that the energy levels for armchair ribbons display the same features as in graphene except the zero energy flat band inherent to the dice model. In addition, unlike graphene, there are no propagating edge states localized at zigzag boundary and there are specific zigzag terminations which give rise to bulk modes of a metallic type in dice model ribbons. We find that the existence of the flat zero-energy band in the dice model is very robust and is not affected by the zigzag and armchair boundaries. PACS numbers: 81.05.ue, 73.22.Pr I. INTRODUCTIONAfter the experimental discovery of graphene [1] there was an explosion of activity in the study of materials with relativistic like spectrum of quasiparticles whose dynamics is governed by the Dirac or Weyl equation. In addition to graphene, they are topological insulators [2,3] and 3D Dirac and Weyl semimetals [4][5][6]. However, the properties and energy dispersion of the electron states in condensed matter systems are constrained by the crystal space group rather than the Poincare group. This gives rises to the possibility of fermionic excitations with no analogues in high-energy physics. Indeed, it was proposed [7] that the three non-symmorphic space groups host fermionic excitations with three-fold degeneracies. The corresponding touchings of three bands are topologically non-trivial and either carry a Chern number ±2 or sit at the critical point separating the two Chern insulators.The triply degenerate fermions with nodal points located closely to the Fermi surface were predicted in the RERh 6 Ge 4 (RE = (Y, La, Lu)) [8] and NaCu 3 Te 2 [9] compounds. They are expected to occur at a high symmetry point in the Brillouin zone and are protected by nonsymmorphic symmetry [7]. Latter they were suggested also to exist at a symmetric axis [10]- [15]. Experimentally, the three-component fermions were observed in MoP and WC [16,17]. The triply degenerate topological semimetals provide an interesting platform for studying exotic physical properties such as the Fermi arcs, transport anomalies, and topological Lifshitz transitions. The pairing problem in materials with three bands crossing was studied in Ref. [18]. A pressure induced superconductivity was reported in MoP [19].Certain lattice systems possess strictly flat bands [20] (for a recent review of artificial flat band systems, see Ref. [21]). The dice model provides the his...

The electronic states on a finite width α−T3 ribbon in a magnetic field are studied in the framework of low-energy effective theory. Both zigzag and armchair types of boundary conditions are analyzed. The analytical solutions are compared with the results of numerical tight-binding calculations. It is found that the flat band of zero energy survives for all types of boundary conditions. The analytical estimates for the spectral gap in a weak magnetic field are discussed. For zigzag type boundary conditions the approximate expressions for the edge and bulk electron states in the strong magnetic field are found. I. INTRODUCTIONAfter the experimental discovery of graphene [1] the systems with relativisticlike quasiparticle spectrum attracted a great interest. In addition, it was shown [2] that in crystals with special space groups one can obtain fermionic excitations with no analogues in high-energy physics. One of the remarkable features of such quasiparticles is a possibility to possess strictly flat bands [3] (for a recent review of artificial flat band systems, see Ref.[4]). The dice model is probably historically the first example of such a system with a flat band which hosts pseudospin-1 fermions [5].Recently the α − T 3 model attracted a significant interest as an interpolation between graphene and dice model [6]. The α − T 3 model is a tight-binding model of two-dimensional fermions living on the T 3 (or dice) lattice where atoms are situated at both the vertices of a hexagonal lattice and the hexagons centers [5,7]. The parameter α describes the relative strength of the coupling between the honeycomb lattice sites and the central site. Since the α − T 3 model has three sites per unit cell, the electron states in this model are described by three-component pseudospin-1 fermions. It is natural then that the spectrum of the model is comprised of three bands. The two of them form a Dirac cone as in graphene, and the third band is completely flat and has zero energy [6]. All three bands meet at the K and K ′ points, which are situated at the corners of the Brillouin zone. The T 3 lattice has been experimentally realized in Josephson arrays [8] and its optical realization by laser beams was proposed in Ref. [9]. Recently, a 2D model for Hg 1−x Cd x Te at critical doping has been shown to map onto the α − T 3 model with an intermediate parameter α = 1/ √ 3 [10]. The presence of completely flat energy band results in surprisingly strong paramagnetic response in a magnetic field in the dice model (α = 1) [6]. The minimal conductivity and topological Berry winding were analyzed in threeband semi-metals in Ref. [11]. The dynamic polarizability of the dice model was calculated in the random phase approximation [12] and it was found that the plasmon branch due to strong screening in the flat band is pinched to the point ω = |k| = µ. In addition, the singular nature of the Lindhard function leads to a much faster decay of the Friedel oscillations. Recently several physical quantities have been studied in the α − T 3 latt...

In the low-energy two-band as well as four-band continuum models, we study the supercritical instability in gapped bilayer graphene in the field of a charged impurity. It is found that the screening effects are crucially important in bilayer graphene. If they are neglected, then the critical value for the impurity charge as the lowest-energy bound state dives into the lower continuum tends to zero as the gap $\Delta$ vanishes. If the screened Coulomb interaction is considered, then the critical charge tends to a finite value for $\Delta \to 0$. The different scalings of the kinetic energy of quasiparticles and the Coulomb interaction with respect to the distance to the charged impurity ensure that the wave function of the electron bound state does not shrink toward the impurity as its charge increases. This results in the absence of the fall-to-center phenomenon in bilayer graphene although the supercritical instability is realized.Comment: 15 pages, 6 figure

Square-octagon lattice underlies the description of a family of two-dimensional materials such as tetragraphene. In the present paper we show that the tight-binding model of square-octagon lattice contains both conventional and high-order van Hove points. In particular, the spectrum of the model contains flat lines along some directions composed of high-order saddle points. Their role is analyzed by calculating the orbital susceptibility of electrons. We find that the presence of van Hove singularities (VHS) of different kinds in the density of states leads to strong responses: paramagnetic for ordinary singularities and more complicated for high-order singularities. It is shown that at doping level of high-order VHS the orbital susceptibility as a function of hoppings ratio α reveals the dia-to paramagnetic phase transition at α ≈ 0.94. This is due to the competition of paramagnetic contribution of high-order VHS and diamagnetic contribution of Dirac cones. The results for the tight-binding model are compared with the low-energy effective pseudospin-1 model near the three band touching point.