$PT$-Symmetric Real Dirac Fermions and Semimetals

Recently, Weyl fermions have attracted increasing interest in condensed matter physics due to their rich phenomenology originated from their nontrivial monopole charges. Here, we present a theory of real Dirac points that can be understood as real monopoles in momentum space, serving as a real generalization of Weyl fermions with the reality being endowed by the $PT$ symmetry. The real counterparts of topological features of Weyl semimetals, such as Nielsen-Ninomiya no-go theorem, 2D subtopological insulators, and Fermi arcs, are studied in the $PT$ symmetric Dirac semimetals and the underlying reality-dependent topological structures are discussed. In particular, we construct a minimal model of the real Dirac semimetals based on recently proposed cold atom experiments and quantum materials about $PT$ symmetric Dirac nodal line semimetals.

Figure 1

Spectra of the real Dirac semimetal. (a) Bulk spectrum with $k_y=0$. (b) Spectrum of a slab with 100 unit cells along $\widehat{\mathbf{y}}$ direction. (c) The cross section of (b) at $k_x=0$. (d) The cross section of (b) at $k_z=0$. Parameters: ${\alpha }_{+}=0.6$, ${\alpha }_{-}=0.4$, $t_x=0.2$, $t_y=0.5$, and $m=0.1$.

Figure 2

The windings of Wilson loops around $k_x$ for 2D subsystems with given $k_z$. Each Wilson loop is computed along a large circle parametrized by $k_y$ for fixed $k_x$ and $k_z$. Panel (a) for the 2D subsystem with $k_z=0$ has unit winding number, but (b) for $k_z=\pi$ has zero winding number.

Figure 3

Spectra with $PT$ being violated. (a) and (b) are gapped after breaking the $PT$ symmetry by removing ($\uparrow A$) and ($\downarrow B$) on the top layer and ($\downarrow A$) and ($\uparrow B$) on the bottom layer, compared with (c) and (d) in Fig. 1 in the main text, respectively.