Non-Hermitian higher-order Dirac semimetals

In this Letter, we study three-dimensional non-Hermitian higher-order Dirac semimetals (NHHODSMs). Our focus is on ${C}_{4}$-symmetric non-Hermitian systems where we investigate inversion ($\mathcal{I}$) or time-reversal ($\mathcal{T}$) symmetric models of NHHODSMs having real bulk spectra. We show that they exhibit the striking property that the bulk and surfaces are anti-$\mathcal{P}\mathcal{T}$ and $\mathcal{P}\mathcal{T}$ symmetric, respectively, and so belong to two different topological classes realizing a non-Hermitian topological phase which we call a hybrid-$\mathcal{P}\mathcal{T}$ topological phase. Interestingly, while the bulk spectrum is still fully real, we find that exceptional Fermi rings (EFRs) appear connecting the two Dirac nodes on the surface. This provides a route to probe and utilize both the bulk Dirac physics and exceptional rings/points on equal footing. Moreover, particularly for $\mathcal{T}$-NHHODSMs, we also find real hinge arcs connecting the surface EFRs. We show that this higher-order topology can be characterized using a biorthogonal real-space formula of the quadrupole moment. Furthermore, by applying Hermitian ${C}_{4}$-symmetric perturbations, we discover various phases, particularly (i) an intrinsic $\mathcal{I}$-NHHODSM having hinge arcs and surface exceptional Fermi arcs, and (ii) a $\mathcal{T}$-symmetric skin-topological HODSM which possesses both topological and skin hinge modes. The interplay between non-Hermition and higher-order topology in this Letter paves the way toward uncovering rich phenomena and hybrid functionality that can be readily realized in experiment.