Designing Dirac points in two-dimensional lattices

We present a framework to elucidate the existence of accidental contacts of energy bands, particularly those called Dirac points which are the point contacts with linear energy dispersions in their vicinity. A generalized von Neumann–Wigner theorem we propose here gives the number of constraints on the lattice necessary to have contacts without fine tuning of lattice parameters. By counting this number, one could search for the candidate of Dirac systems without solving the secular equation. The constraints can be provided by any kinds of symmetry present in the system. The theory also enables the analytical determination of a $\mathit{k}$-point having accidental contact by selectively picking up only the degenerate solution of the secular equation. By using these frameworks, we demonstrate that the Dirac points are feasible in various two-dimensional lattices, e.g., the anisotropic Kagomé lattice under inversion symmetry is found to have contacts over the whole lattice parameter space. Spin-dependent cases, such as the spin-density-wave state in LaOFeAs with reflection symmetry, are also dealt with in the present scheme.

Figure 1

(a) Schematic illustration of the merging of two Dirac points along the direction parallel and perpendicular to $\delta \mathit{q}$ (direction along which the Dirac points merge). (b) Merging of four Dirac points consisting of upper and lower pairs along the direction parallel to $\delta \mathit{q}$’s of each pair. (c) Sketches of energy bands when the band overlap is present (left panel) and absent (right panel).

Figure 2

Schematic illustration of (a) bond-centered and (b) site-centered inversions. Cross symbols indicate the inversion centers, circles and squares denote the lattice sites, and the shaded regions are the unit cells. Pairs of inequivalent sites which exchange by inversion are connected by broken lines. Pairs of inequivalent bonds connecting the inequivalent sites which do not exchange by inversion are shown as solid lines. Panel (c) is the systems with asymmetric bonds whose Hamiltonian is invariant under the space-time inversion which transforms the phase of site 1 at the inversion center. Panel (d) is the mixture of bond-centered and site-centered sites.

Figure 3

Representative lattice structure with bond-centered inversion of $n_{\mathrm{s}}=3,4$. Filled and open circles and squares represent the lattice sites, $\mu =1,2,3,4$, where 1 and 2 belong to subspace $S_{\mathrm{A}}$ and others to $S_{\mathrm{B}}$. Indices of bonds ($t_i$) represented by solid and broken lines are given in the left panel. Panel (d) is the model lattice of $\alpha$-ET${}_2$I${}_3$ which is simplified to panel (c) by removing bonds with small transfer integrals. The lattice structure of panel (c) is reduced to that of panel (b) when site 4 is removed. If we take the vertical bonds in (f) as uniform, as $t_3=t_3^{\prime }=t_4=t_4^{\prime }$, we find the lattice in (g) which has glide reflection symmetry ($y_o/2$-translation in the vertical direction and reflection with respect to $y$ axis) and affords essential line degeneracy.

Figure 4

Examination of the point contacts of the lattice structure in Fig. 3b. (a) $t_2/t_1$-$t_3/t_1$ diagram; the Dirac points are stable in the shaded region at $W_{\mu }=0,(\mu =1-3)$. These stable regions shift to those indicated by hatches when $W_3/t_1=1$−1, respectively. (b) Examples of band structures with Dirac points in the shaded region in (a), which are viewed from two different directions, $(\theta ,\varphi )=({90}^{°},{180}^{°})$ and $({90}^{°},{90}^{°})$. The polar angles, $\theta$ and $\varphi$, are defined in ($k_x$,$k_y$,$\epsilon$) space in the panel, where arrows indicate the direction of a view.

Figure 5

Examination of the point contacts of the simplified $\alpha$-type lattice structure in Fig. 3d. $|t_3/t_1|$-$|t_4/t_1|$ diagram for (a) $t_1=-t_2$ and (b) $t_2=t_1$. Dirac points are stable in the shaded regions: (a) $\sqrt{(t_4/t_1){}^2-(t_3/t_1){}^2}⩽2$ and (b) $(t_3/t_1){}^2+(t_4/t_1){}^2⩽4$. Panels in (c) show four different examples of the band structures with Dirac points.

Figure 6

Representative lattice structure with site-centered inversion of $n_{\mathrm{s}}=3,4$, (a) Kagomé, (b) Lieb, and (c) anisotropic square lattices. The notation of lattice sites and the transfer integrals follow those given in Fig. 3.

Figure 7

Dirac points of the Kagomé lattice, which merge at $\Gamma$ and M points, but do not open a gap throughout the whole $(t_1,t_2,t_3)$-space, and under the on-site potentials. (a) Trajectories of point contacts between upper two bands in the anisotropic Kagomé lattice in Fig. 6a under the variation of $t_3/t_1$ for several choices of $0⩽t_2/t_1⩽2$. (b) Variation of $t_3/t_1$ along the M-$\Gamma$-K-M points which are indicated in bold lines in panel (a). (c) Variation of band structures along the $\Gamma$-M line with $t_2/t_1=1$. The last panel is the energy bands of the Lieb lattice. (d) Examples of band structure with and without on-site potential. (e) Merging of four Dirac points in the vicinity of M point under $t_2/t_1=1$, $t_3/t_1\to 0$. Their cross sections along the different merging directions of upper and lower pairs correspond to those in Fig. 1b.

Figure 8

(a) Decorated honeycomb lattice structure of $n_{\mathrm{s}}=3$ without the inversion symmetry but with reflection symmetry. (b) Upper shaded ($W_3/t⩾(t^{\prime }/t){}^2/3-3$) and lower hatched ($W_3/t⩽-(t^{\prime }/t){}^2+1$) regions afford Dirac points between the upper two and lower two bands, respectively, and the region (shaded $+$ hatched) at the center have the two sets of Dirac point. (c) Demonstration of the existence of Dirac points under the reflection symmetry, which emerge along the K-K’-$\Gamma$ lines in $\mathit{k}$-space. The contacts on K- and K’-points at $t^{\prime }=0$ isotropic honeycomb lattice) and those at $t^{\prime }/t\ne 0$ are essential and accidental ones, respectively. The upper and lower pairs merge at $\Gamma$- and $M$-points, respectively, at $t^{\prime }/t=\pm 3$ and $\pm 1$.

Figure 9

(a) Square lattice with original unit cell and magnetic unit cell of SDW state. Arrows represent spins which have $(\pi ,0)$ periodicity. Magnetic unit cell with (b) $n_{\mathrm{s}}$ $=$ 2 and (c) $n_{\mathrm{s}}=4$. For the case in (c), the Hamiltonian is invariant under the reflection against either the $\mathit{xz}$ or $\mathit{yz}$ plane, both of which exchange the two orbitals, $d_{\mathit{XZ}}$ and $d_{\mathit{YZ}}$. (d) Energy band of the self-consistent SDW solution for parameters, $t_{\nu \mu }=-0.2$ (nearest neighbor sites), $(t_{13}^{\prime },t_{24}^{\prime })=(0.3,0.15)$ (next-nearest neighbor sites in the (1,1)-direction), $U=1.2$, $U^{\prime }=0.9$, and $J=0.15$. The left panel is the cross-section at $k_x=0$ as a function of $k_y$, along which the Dirac point emerges (symmetry axis).