Topological obstructed atomic limit insulators by annihilating Dirac fermions

We show that annihilating a pair of Dirac fermions implies a topological transition from the critical semimetallic phase to an obstructed atomic limit insulator phase instead of a trivial insulator. This is shown to happen because of branch cuts in the phase of the wave functions, leading to nontrivial Zak phase along certain directions. To this end, we study their ${\mathbb{Z}}_2$ invariant and also study the phase transition using entanglement entropy. We use low-energy Hamiltonians and numerical result from model systems to show this effect. These transitions are observed in realistic materials, including strained graphene and buckled honeycomb group V (Sb/As).

Figure 1

Phase diagram of Eq. (1) and the corresponding lower-dimensional eigenstates where the origin is at the center. Green and red dots in II and IV indicate the winding number of $\pm 1$. Red band denotes the surface state

Figure 2

[(a1) and (a2)] The phases ${\theta }_{\pm }$ of the eigenstates indicated on top for the semimetallic phases II and IV; [(b1) and (b2)] phases for the eigenstates for the insulating phases I and III. The central inset shows the eigenvalues in each limit.

Figure 3

(a) Possible combinations of parity eigenvalues for different phases. [(b) and (c)] Distinguishing phase II and IV based on position of Dirac cone and corresponding edge states. [(b1) and (c1)] shows the possible positions of Dirac crossing for the same parity arrangement. [(b2) and (c2)] show their respective 1-dimensional band structure when the system is cut along the ${\mathbf{b}}_2$ direction while (b3) and (c3) shows the same edge spectrum when cut along the ${\mathbf{b}}_1$ direction.

Figure 4

(a) Entanglement entropy $S$ (rescaled from $[0,log(2\left)\right]\to [0,1]$) through the path shown in the inset for strained graphene (b) $\partial S$ (black) and ${\partial }^{2}S$ (red) for the corresponding path in (a) ($S$ is shown in the background as a dashed line for clarity). (c) $S$ and (d) $\partial S$ throughout the parameter space for strained graphene, black and white lines in left and right panel show the transition points from semimetal to insulator in bulk.

Figure 5

$S$ (green) and ${\partial }_{\delta }S$ (red) for ${\mathcal{H}}_{\mathrm{SSH}}$.

Figure 6

Left to right: Lattice model, $log(E_1-E_2)$ in $\mathit{k}$, red regions indicate Dirac cone locations, and bulk band structure.

Figure 7

Left: Bulk band structure, edge spectrum, $\log (E_2-E_1)$, and phase of ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ for phases $\mathrm{I}, \mathrm{II}, \mathrm{III}$, and $\mathrm{IV}$ given in text with parameters shown in Table 1. Right: Phase $\mathrm{IV}\to \mathrm{IVa}$ and $\mathrm{III}\to \mathrm{IIIa}$ on breaking chiral symmetry but preserving inversion.

Figure 8

Transition to OAL by annihilating Dirac points; $+$ and $-$ represent the parity eigenvalues. Top: Trivial phase (I) where the system annihilates the points at $X_2$. Bottom: Nontrivial phase (I) where the system annihilates the points at $\mathrm{\Gamma }$.

Figure 9

Top left: Anisotropic graphene lattice. Top right: $\log (E-1-E_2)$ for the isotropic case $t_1=t_2=t_3=1$. Middle/bottom: Edge spectrum at $t$ values given in Table 2.