Antiferromagnetic Dirac semimetals in two dimensions

The search for symmetry-protected two-dimensional (2D) Dirac semimetals analogous to graphene is important both for fundamental and practical interest. The 2D Dirac cones are protected by crystalline symmetries and magnetic ordering may destroy their robustness. Here we propose a general framework to classify stable 2D Dirac semimetals in spin-orbit coupled systems having the combined time-reversal and inversion symmetries, and show the existence of the stable Dirac points in 2D antiferromagnetic semimetals. Compared to 3D Dirac semimetals which fall into two distinct classes, Dirac semimetals in 2D with combined time-reversal and inversion symmetries belong to a single class which is closely related to the nonsymmorphic space-group symmetries. We further provide a concrete model in antiferromagnetic semimetals which supports symmetry-protected 2D Dirac points. The symmetry breaking in such systems leads to 2D chiral topological states such as quantum anomalous Hall insulator and chiral topological superconductor phases.

Figure 1

A nonsymmorphic symmetry $\left\{g\right|\mathbf{t}\}$ ensures band crossing on a $g$ invariant line at the BZ boundary. The doubly degenerate bands are artificially split for clarity. (a) With $\mathcal{P}$ and $\mathcal{T}$, and $\mathcal{P}={\tau }_1$, Bloch states $\left(\right|u_{\mathbf{k}}^{+}\rangle ,\mathcal{T}|u_{\mathbf{k}}^{-}\rangle )$ and $\left(\right|u_{\mathbf{k}}^{-}\rangle ,\mathcal{T}|u_{\mathbf{k}}^{+}\rangle )$ carry different representations of $\left\{g\right|\mathbf{t}\}$ ($C_{2\widehat{x}}$, for example) and cross at TRIM. (b) Without $\mathcal{P}$ and $\mathcal{T}$, but with combined $\mathcal{PT}, \left(\right|u_{\mathbf{k}}^{+}\rangle ,\mathcal{PT}|u_{\mathbf{k}}^{+}\rangle )$ and $\left(\right|u_{\mathbf{k}}^{-}\rangle ,\mathcal{PT}|u_{\mathbf{k}}^{-}\rangle )$ have different eigenvalues of $\left\{C_{2\widehat{x}}\right|\frac{1}{2}\frac{1}{2}\}$ and cross avoidably.

Figure 2

Energy band with DPs protected by nonsymmorphic symmetries in an AFM lattice respecting $\mathcal{PT}$. The DPs are marked as green dots in the BZ. (a) DPs are gapped and cannot be protected by $\left\{C_{2\widehat{x}}\right|\frac{1}{2}0\}$ and $\left\{C_{2\widehat{y}}\right|0\frac{1}{2}\}$ when $\widehat{\mathbf{n}}=\widehat{z}$. (b) Two DPs located along the $X_1-M$ line are protected by $\left\{M_{\widehat{x}}\right|\frac{1}{2}0\}$ solely when $\widehat{\mathbf{n}}=\widehat{x}$. (c) Distortion in the $\langle 11\rangle$ direction eliminates $\left\{M_{\widehat{x}}\right|\frac{1}{2}0\}$ but keeps $\left\{M_{\widehat{z}}\right|\frac{1}{2}\frac{1}{2}\}$, gapping the DPs. (d) Alternatively, $B$ sites shifted in the $y$ direction breaks $\left\{M_{\widehat{z}}\right|\frac{1}{2}\frac{1}{2}\}$, but DPs still remain protected.

Figure 3

Symmetry-breaking phases. (a) Breaking $\mathcal{PT}$ by SIA, while preserving $\left\{M_{\widehat{x}}\right|\frac{1}{2}0\}$, leads to Weyl points (marked as yellow and green dots) on the $k_x=\pi$ line. (b) Breaking both $\mathcal{PT}$ and $\left\{M_{\widehat{x}}\right|\frac{1}{2}0\}$ results in the QAH insulator. In the edge spectrum, the green and red lines are edge states located at upper and lower edges, respectively.

Figure 4

(a) Nonsymmorphic symmetry-protected 2D DSMs with DPs located at the BZ boundary. (b) 2D DPs with accidental degeneracy appear at the phase transition between QSH and NI controlled by parameter $m$.

Figure 5

The electrically tunable QAH state in both $\mathcal{PT}$ and $\left\{M_{\widehat{x}}\right|\frac{1}{2}0\}$ broken AFM DSMs. The parameters $V=0.2$ and others are the same as in Fig. 3, which leads to a NI phase.