Unconventional topological insulators from extended topological band degeneracies

A general and beautiful picture for the realization of topological insulators is that the mass term of the Dirac model has a nodal surface wrapping one Dirac point. We show that this geometric picture based on Dirac points can be generalized to extended band degeneracies with nontrivial topological charges. As the nontrivial topological charges force the extended band degeneracies to be created or annihilated in pairs, when the nodal surface of mass wraps one such extended band degeneracy, the resulting gapped phase must be topologically nontrivial since it cannot adiabatically be deformed into a topologically trivial atomic insulator without closing the energy gap. We use nodal lines which carry a nontrivial $Z_2$ monopole charge in three dimensions to illustrate the physics. Notably, because the wrapping surfaces for an extended band degeneracy are diverse, we find that this generalization unearths topological insulators with unconventional patterns of boundary states.

Figure 1

(a)–(e) The evolution of $Z_2$-monopole-charge nodal lines in the $k_z=0$ plane. (a) No nodal lines exist, (b) two concentric nodal lines emerge, (c) one of the nodal lines shrinks to a point, (d) the point re-expands to a line, (e) the size of nodal lines increases further. (f) Energy spectra for a geometry with open boundary condition in the $z$ direction and periodic boundary condition in both $x$ and $y$ directions. $k_y=0, t_0=2.5$, and the system size in the $z$ direction is $L_z=100$. Characteristic drumhead surface states appear within the projections of the nodal lines. Common parameters: ${\lambda }_{x,y,z}=t=1$.

Figure 2

(a) A sphere-form MNS (yellow) wraps the nodal line (red) with a smaller size. (b) Between the two dashed purple lines, the in-gap dispersions are doubly degenerate, corresponding to the existence of a single Dirac cone on each $z$-normal surface. Outside, the blue and red lines are without degeneracy and represent boundary states on the lower and upper surface, respectively. (c) Each $x$-normal or $y$-normal surface also harbors a single Dirac cone, here we have only presented the results for $x$-normal surfaces because the system has rotational symmetry in the $xy$ plane. (d) When the MNS wraps both nodal lines (the inset), the configuration realizes a trivial insulator but with surface Dirac cones coexisting with bulk states on the $z$-normal surfaces. The parameters are ${\lambda }_{x,y,z}=t=1$ and $t_0=m_0=2.5$ in (a)–(c); ${\lambda }_z=t=1, {\lambda }_x={\lambda }_y=0.5, t_0=2.5$, and $m_0=1.5$ in (d); $L_z=100$ in (b) and (d), and $L_x=100$ in (c).

Figure 3

(a) A torus-form MNS (yellow) wraps the nodal line (red) with a smaller size. (b) Between the two dashed purple lines, the dispersions in red and blue are doubly degenerate, which correspond to trivial boundary states on the upper and lower $z$-normal surfaces, respectively. Outside, the in-gap dispersions are without degeneracy and correspond to topological boundary states. (c) Each $x$-normal surface (also for $y$-normal surfaces) harbors a single Dirac cone. (d) Boundary states on the $z$-normal surfaces for the configuration that the torus-form MNS encloses the nodal line with a larger size (the inset). The parameters are ${\lambda }_{x,y,z}=t=1, t_0=2.5$, and $m_0=1.5$ in (a)–(c); ${\lambda }_z=t=1, {\lambda }_x={\lambda }_y=0.5, t_0=2.5, B_0=2.0$ and $\delta =0.2$ in (d); $L_z=100$ in (b) and (d), and $L_x=100$ in (c).

Figure 4

(a) Nodal lines from $H_{\mathrm{NL}}$ (red) and $H_{\mathrm{M}}$ (yellow) form link structure. (b) The in-gap dispersions in red (up-moving states) and blue (down-moving states) are of fourfold degeneracy, corresponding to four pairs of gapless helical states localized at the four hinges. The left inset is a schematic diagram for the distribution of the gapless helical states, and the right inset shows the probability density profiles of the four branches of up-moving states at $k_z=0.1$. The parameters are ${\lambda }_z=t=1, {\lambda }_x={\lambda }_y=0.5, t_0=2.5, B_0=2.0, \delta =0.3$, and $L_x=L_y=20$.