Topological Transition of Dirac Points in a Microwave Experiment

By means of a microwave tight-binding analogue experiment of a graphenelike lattice, we observe a topological transition between a phase with a pointlike band gap characteristic of massless Dirac fermions and a gapped phase. By applying a controlled anisotropy on the structure, we investigate the transition directly via density of states measurements. The wave function associated with each eigenvalue is mapped and reveals new states at the Dirac point, localized on the armchair edges. We find that with increasing anisotropy, these new states are more and more localized at the edges.

Figure 1

Left panels: Schematics of the hexagonal lattices. ${\mathbf{a}}_{\mathbf{1}}$ and ${\mathbf{a}}_{\mathbf{2}}$ define the primitive cell of the Bravais lattice. $t$ (blue intersite links) and $t^{\prime }$ (red ones, along $y$ axis) correspond to the nearest-neighbor couplings. $d$ and $d^{\prime }$ are the corresponding distances. An anisotropy is introduced along the $y$ axis: $t^{\prime }$ increases when $d^{\prime }$ decreases. Right panels: Calculated density of states (DOS) for an infinite system with first, second, and third nearest-neighbor couplings given in the text. The red dashed line is the Dirac frequency ${\nu }_D={\nu }_0+3t_2$. The insets show the corresponding dispersion relation. (a)–(b) Regular honeycomb lattice, $\beta =1$. The two Dirac points are distinguishable. (c)–(d) Compressed lattice at the topological transition, $\beta =1.8$. The Dirac points merge. (e)–(f) $\beta =3.5$, a band gap opens (gray zone).

Figure 2

Experimental DOS for (a) the regular graphene ($\beta =1$), (b) and (c) anisotropic structures with $\beta =1.8$ and $\beta =3.5$, respectively. The dashed red lines indicate the Dirac frequency ${\nu }_D$. The gray zone shows the band gap with edge states at ${\nu }_D$.

Figure 3

(a) Experimental DOS for $\beta$ varying from 1 to 3.5. To remove edges states and highlight the band gap opening, the DOS have been averaged over bulk sites only. The arrows at $\beta =1$, 1.8, and 3.5 indicate the DOS plotted, respectively, in Figs. 2a, 2b, 2c, where the average was performed over all sites. (b) Calculated DOS for $\beta$ varying from 1 to 3.5 for an infinite system. The arrows at $\beta =1$, 1.8, and 3.5 indicate the DOS plotted in Figs. 1b, 1d, 1f, respectively. In (a) and (b), the gray zone shows the band gap opening, the dotted lines denote the linear dependence of the band edges, and the red dashed line indicates ${\nu }_D$.

Figure 4

Experimental wave function distributions at the Dirac frequency ${\nu }_D$ (red dashed line in Fig. 3) for $\beta$ ranging from 1.3 to 3.5.