Merging of Dirac points in a two-dimensional crystal

We study under which general conditions a pair of Dirac points in the electronic spectrum of a two-dimensional crystal may merge into a single one. The merging signals a topological transition between a semimetallic phase and a band insulator. We derive a universal Hamiltonian that describes the physical properties of the transition, which is controlled by a single parameter, and analyze the Landau-level spectrum in its vicinity. This merging may be observed in the organic salt $\alpha -{\left(\text{BEDT-TTF}\right)}_2{\text{I}}_3$ or in an optical lattice of cold atoms simulating deformed graphene.

Figure 1

(Color online) Evolution of the spectrum when the quantity $\Delta$ is varied and changes in sign at the topological transition (arbitrary units). The low-energy spectrum stays linear in the $q_y$ direction.

Figure 2

(Color online) Potential profile $V\left(Y\right)={\left(\delta +Y^2\right)}^2-2Y$ and effective energy levels $E_n$ for (a) $\delta =-4$ and (b) $\delta =0$.

Figure 3

(Color online) Energy levels ${ϵ}_n\left(\delta \right)/{\left(m^{\ast }{c}^2{\omega }^2/2\right)}^{1/3}$ as a function of the dimensionless parameter $\delta \propto \Delta /B^{2/3}$. The dots on the $\delta =0$ axis indicate the semiclassical levels of the quartic Hamiltonian.[4]

Figure 4

(Color online) Top figures: (a) motion of the two Dirac points $\mathit{D}$ and $-\mathit{D}$ for the case $t_{00}=t_{01}=1$, $t_{10}=2+t_{11}$ under variation of $t_{11}$; (b) phase diagram. The $\Gamma$ and $X$ lines separate the semimetallic phase from the band insulator (gray). The Dirac points move from $X=\left(0,1\right)$ to the $\Gamma =\left(0,0\right)$ point when $t_{11}$ varies from 0 to $-2$. Bottom figures: (c) and (d) show the isoenergy curves, respectively, for $t_{11}=-1.5$ and $-2$.

Figure 5

Motion of the Dirac points for model (13), with $t_{00}=-1$, $t_{10}=-0.5$, $t_{01}=1.4$, while varying $-3<t_{11}<3$ (vertical axis).