Figure 1

Graphene honeycomb lattice.Reuse & Permissions

Figure 2

Energy spectra when $t^{\prime }=t$ (top) and $t^{\prime }=2t$ (bottom). The latter corresponds to the merging of Dirac points into a single point $M$. The green dashed line depicts the Brillouin zone.Reuse & Permissions

Figure 3

The zero-energy wave function components for $t^{\prime }=\alpha t$ with $\alpha ⩾2$. The wave function is zero for all sites for which no value is specified. The black square denotes the vacancy. The direction of anisotropy is the $y$ direction. The two dashed lines are the boundaries of the upper and lower half planes, respectively.Reuse & Permissions

Figure 4

Snapshot of the LDOS obtained using exact diagonalization. We plot the zero-energy impurity state slightly above the merging point ($\alpha =2.1$) when a small gap opens in the spectrum. The highest-intensity site (in blue) corresponds to the $B$ site right below the impurity.Reuse & Permissions

Figure 5

Diagrammatic perturbative expansion of the generalized Green's function to all orders in the impurity potential.Reuse & Permissions

Figure 6

Fourier transform of the LDOS correction $\Delta \rho$ when $t^{\prime }=t$ (first column), $t^{\prime }=1.5t$ (second column), and $t^{\prime }=2t$ (third column). The energy is $\omega =0.15t$ for the first row and $\omega =0.80t$ for the second row.Reuse & Permissions

Figure 7

The LDOS correction as a function of position in the vicinity of the impurity for a critical value of anisotropy ($t^{\prime }=2t$). The second line presents a series of zoom ins of the plots outlined on the first line. In the first column, we compare $\Delta \rho$ obtained using the full $T$-matrix approximation to the one obtained by the tight-binding method for an energy $\omega =-0.20t$. Note that, consistent with the low-energy expansion given by Eq. (25), the FOs are dephased by $\pi$ between the two sublattices. The second column presents a comparison between the correction to the LDOS $\Delta \rho$ along the $x=0$ direction obtained by the full $T$-matrix approximation (full lines) and by the low-energy expansion (dotted lines) for $\omega =-0.20t$. In blue, we plot the LDOS on the $A$ sublattice comprising the impurity ($y=0$), whereas, in green, we plot the LDOS on the $B$ sublattice. In the third column, we plot the LDOS along the $y=0$ direction obtained by the full $T$-matrix approximation for $\omega =-0.20t$. The blue curve is obtained using the full $T$-matrix approximation, whereas, the black one is obtained in the continuum approximation.Reuse & Permissions

Figure 8

Correction of the LDOS $\Delta \rho$ (in arbitrary units) for $t^{\prime }=t$ (first column) and $t^{\prime }=2t$ (second and third columns), obtained using the full $T$-matrix approximation (first and second columns) and the low-energy expansion (third column). The energy is $\omega =0.15t$.Reuse & Permissions

Figure 9

Contour integral.Reuse & Permissions