Dirac Semimetals in Two Dimensions

Graphene is famous for being a host of 2D Dirac fermions. However, spin-orbit coupling introduces a small gap, so that graphene is formally a quantum spin Hall insulator. Here we present symmetry-protected 2D Dirac semimetals, which feature Dirac cones at high-symmetry points that are not gapped by spin-orbit interactions and exhibit behavior distinct from both graphene and 3D Dirac semimetals. Using a two-site tight-binding model, we construct representatives of three possible distinct Dirac semimetal phases and show that single symmetry-protected Dirac points are impossible in two dimensions. An essential role is played by the presence of nonsymmorphic space group symmetries. We argue that these symmetries tune the system to the boundary between a 2D topological and trivial insulator. By breaking the symmetries we are able to access trivial and topological insulators as well as Weyl semimetal phases.

Figure 1

A nonsymmorphic symmetry $\{g|\mathbf{t}\}$ leads to band crossings on a $g$ invariant line or plane in momentum space. (a) Without other symmetries, pairs of bands intersect an odd number of times as they cross the BZ. (b) With time-reversal symmetry, with ${\mathrm{\Theta }}^2=+1$, the crossing occurs at the zone boundary $\mathbf{G}/2$, where $e^{i\mathbf{G}·\mathbf{t}}=-1$. For ${\mathrm{\Theta }}^2=-1$, Kramers pairs at $\mathbf{k}=0$ and $\mathbf{G}/2$ connect as in (c), leading to a line node in an invariant plane or a Weyl node on an invariant line. The labels indicate the eigenvalues $\pm \lambda e^{i\mathbf{k}·\mathbf{t}}$ of $\{g|\mathbf{t}\}$. (d) With inversion and time-reversal symmetry, all states are degenerate (they are offset for clarity), and the crossing occurs at $\mathbf{G}/2$.

Figure 2

Energy bands for structures with Dirac points protected by inversion symmetry, with lattice structure and BZ shown above. Dirac points and nodal lines in the BZ are marked in cyan. (a) The $\sqrt{2}\times \sqrt{2}$ supercell of a square lattice with nodal lines along the BZ edge protected by $\{E|\mathbf{t}\}$. (b) Crinkling the lattice breaks $\{E|\mathbf{t}\}$, leaving nonsymmorphic symmetries and Dirac points at $X_1$, $X_2$, and $M$. (c) Distorting in the $⟨11⟩$ directions eliminates screw axes $\{C_{2x}|\frac{1}{2}0\}$ and $\{C_{2y}|0\frac{1}{2}\}$ (as well as $C_4$), gapping the corner Dirac point. (d) Alternatively, displacing one of the sites along $⟨10⟩$ breaks $\{M_{\widehat{z}}|\frac{1}{2}\frac{1}{2}\}$, leaving Dirac points at the corner and one edge.

Figure 3

Without inversion, nonsymmorphic symmetries protect nodal lines or Weyl points. This may be achieved by altering each site to be bipartite and asymmetric (e.g., heterodimers). (a) Breaking inversion in case I, while preserving $\{M_{\widehat{z}}|\frac{1}{2}\frac{1}{2}\}$, leads to nodal lines that circle the former Dirac points at $X_{1,2}$. (b) Breaking inversion in case II, preserving $\{C_{2\widehat{x}}|\frac{1}{2}0\}$, results in Weyl points on the $C_{2\widehat{x}}$ invariant lines.

Figure 4

By further lowering the symmetry, the Dirac semimetal in Fig. 2 can be driven into either trivial (I) or topological (TI) insulator phases. Lattice displacements in (a) give rise to the phase diagram in (b).