$PT$-invariant Weyl semimetals in gauge-symmetric systems

Weyl semimetals typically appear in systems in which either time-reversal ($T$) or inversion ($P$) symmetry is broken. Here we show that in the presence of gauge potentials these topological states of matter can also arise in fermionic lattices preserving both $T$ and $P$. We analyze in detail the case of a cubic lattice model with $\pi$ fluxes, discussing the role of gauge symmetries in the formation of Weyl points and the difference between the physical and the canonical $T$ and $P$ symmetries. We examine the robustness of this $PT$-invariant Weyl semimetal phase against perturbations that remove the chiral sublattice symmetries, and we discuss further generalizations. Finally, motivated by advances in ultracold-atom experiments and by the possibility of using synthetic magnetic fields, we study the effect of random perturbations of the magnetic fluxes, which can be compared to a local disorder in realistic scenarios.

Figure 1

Left: Effect of time-reversal and inversion symmetries on the Weyl cones of the Hamiltonian (6). A Weyl cone is mapped onto itself when applying the canonical $\mathcal{PT}$ symmetry. However, due to the gauge transformation ${\mathcal{U}}_y$, the physical $PT$ symmetry relates Weyl cones at different momenta. Right: Band structure in a diagonal slab geometry, with hard-wall boundaries in the $x-y$ direction and a thickness of 80 sites. The color scale indicates the amplitude of the eigenstates in the first and last eight sites from the boundaries. Weyl cones of opposite chirality are connected by zero-energy Fermi arcs (in red).

Figure 2

Spectrum for $\vec{B}=(3/4,1,1)\pi$ with $k_z=0$, displaying Weyl cones at zero energy. The inset shows the detail of a Weyl cone: the topological semimetal physics emerges clearly below an energy scale ${\epsilon }_W$, whereas above it the system has a metallic behavior.

Figure 3

Double-logarithmic plot of the density of states as a function of energy for different values of $U$. The DOS is computed for a cube with a linear size of 160 sites and is averaged over $\sim 150–200$ random flux realizations. For clarity, the green and blue curves are shifted vertically by multiplying with a constant factor (shown on the plot). The DOS close to the Fermi level is constant, indicating a diffusive-metal behavior. In the intermediate energy range and for small values of $U$ we observe an energy scaling characteristic of Weyl semimetals (dashed lines), $\rho \left(E\right)-\rho \left(0\right)={a\left|E\right|}^b$, with $b\approx 2$. Larger values of $U$ lead to an increase in the energy range associated with a diffusive metal and a reduction of the region associated with a topological semimetal.

Figure 4

States with energy $\left|E\right|<0.02\omega$ in the presence of a harmonic trap in the surface BZ. The color depicts their localization close to the hard-wall boundary. Here $L=50, \mu =\omega /2$, and $\mathrm{\Omega }$ is chosen to have the potential energy ranging from $-\omega /2$ to $+\omega /2$ from the center towards the surface. The Weyl points become extended regions of zero-energy bulk states, connected by curved Fermi arcs.