Ab initio analysis of the topological phase diagram of the Haldane model

We present an ab initio analysis of a continuous Hamiltonian that maps into the celebrated Haldane model. The tunneling coefficients of the tight-binding model are computed by means of two independent methods—one based on the maximally localized Wannier functions, and the other through analytic expressions in terms of gauge-invariant properties of the spectrum—that provide a remarkable agreement and allow one to accurately reproduce the exact spectrum of the continuous Hamiltonian. By combining these results with the numerical calculation of the Chern number, we are able to draw the phase diagram in terms of the physical parameters of the microscopic model. Remarkably, we find that only a small fraction of the original phase diagram of the Haldane model can be accessed, and that the topological insulator phase is suppressed in the deep tight-binding regime.

Figure 1

Bravais lattice associated to the honeycomb potential in Eq. (2). Black and white circles refer to minima of type $A$ and $B$, respectively. The elementary cell is highlighted in gray. The various tunneling coefficients are indicated for the site of type $A$ in the central cell. The system is invariant under discrete translations generated by the Bravais vectors ${\mathit{a}}_{1/2}$ and under rotations of $\theta =2\pi /3$ radians around any vertex of the lattice. The rotational symmetry implies that next-to-nearest tunneling amplitudes $t_1$ along the same direction are conjugate pairs (solid and dashed lines in red); from the latter follows the equivalence of the hopping amplitudes separated by $2\pi /3$ radians. When sites $A$ and $B$ are degenerate, the system is also invariant under rotations by $\pi$ radians around the center of any elementary cell.

Figure 2

Density plot (in logarithmic scale) of the square of the real (a) and imaginary (b) parts of the MLWF for sublattice $A$, for $s=5,\alpha =0.1$, and ${\chi }_A=0$. The solid and dashed lines denote the unit cell and the honeycomb lattice of the scalar potential, respectively. In the latter, the corners of the hexagons mark the minima of the scalar lattice potential labeled either as $A$ or $B$.

Figure 3

Comparison of the four tight-binding coefficients as calculated from the MLWFs (solid red lines) and the analytical formulas of Eqs. (37, 38, 39, 40) (blue triangles) and Eqs. (41)–(44) (green circles). Results are shown as a function of $\alpha$, keeping fixed values $s=5$ and ${\chi }_A=0.001$. The gray area in the figures denotes the region where the system behaves as a topological insulator with $C\ne 0$ (see text and Sec. 4).

Figure 4

Relative deviations from the average values of (a) the phase, $1-{\phi }_{A,B}/\phi$, and (b) the magnitude of the next-to-nearest tunneling coefficient, $1-|t_{1A,B}|/|t_1|$, for ${\chi }_A=0.001,s=5$. Results calculated using the MLWFs.

Figure 5

Topological phase diagram of the Haldane model as a function of $\phi$ and $\epsilon /|t_1|$. The main figure is a zoom for $\phi \in [-0.45,0.45]$, while the inset illustrates the full nominal diagram, with $\phi \in [-\pi ,\pi ]$. Large (green), medium-size (red), and small (blue) dots correspond to nonzero Chern numbers calculated ab initio for $s=5,7$, and 9, respectively. The sign of the Chern number is equal to the sign of the phase. The solid (black) line denotes the analytical boundary $\epsilon /|t_1|=3\sqrt{3}sin\phi$. The vertical dashed lines delimit the physically accessible regions.

Figure 6

Topological phase diagram of the continuous Hamiltonian in Eq. (1), as a function of $\alpha$ and ${\chi }_A$, for three different values of the scalar potential amplitude $s$. The nontrivial topological state is indicated by big (green) dots for $s=5$, medium (red) dots for $s=7$, and small (blue) dots for $s=9$. The black dashed lines represent a guide to the eye for the phase boundaries for each value of $s$.

Figure 7

Behavior of the gaps ${\delta }_{+}$ (squares, solid line) and ${\delta }_{-}$ (triangles, dashed lines) as a function of $\alpha$, for $s=5$. The three panels correspond to ${\chi }_A=2\times {10}^{-4}$ (a), ${10}^{-3}$ (b), and $1.9\times {10}^{-3}$ (c). The latter corresponds to the maximal value of $|{\chi }_A|$ for which the system can be in the topological insulating phase. The gray shaded area corresponds to the region where the system is a topological insulator $\left(C=1\right)$, whereas the white background identifies a normal insulating state $\left(C=0\right)$. The inset in panel (b) shows the behavior of the gap ${\delta }_{-}$ at ${\mathit{k}}_D^{-}$ (dashed blue line) and at ${\overline{\mathit{k}}}^{-}\simeq {\mathit{k}}_D^{-}+1.68\times {10}^{-3}(1/2,\sqrt{3}/2)k_L$ (red continuous line), around the point $\alpha \approx 1.8$.

Figure 8

Gap created at the Dirac points by time-reversal symmetry breaking, for three different values of the scalar potential amplitude $s$, fixed value ${\chi }_A=2\times {10}^{-4}$. Solid (squares) and dashed lines (triangles) denote ${\delta }_{+}$ and ${\delta }_{-}$, respectively. The vertical dashed-dot-dot lines denote the gap closing points for the two lowest values of $s$.

Figure 9

Spread of the MLWFs as a function of $\alpha$ for fixed values $s=5,{\chi }_A=1\times {10}^{-3}$. The spread is decomposed into its gauge-invariant $\left({\mathrm{\Omega }}_I\right)$, band diagonal $\left({\mathrm{\Omega }}_D\right)$, and band off-diagonal $\left({\mathrm{\Omega }}_{OD}\right)$ terms. Note the ${10}^3$ factor in the case of ${\mathrm{\Omega }}_D$ and ${\mathrm{\Omega }}_{OD}$.