Fermion Doubling Theorems in Two-Dimensional Non-Hermitian Systems for Fermi Points and Exceptional Points

The fermion doubling theorem plays a pivotal role in Hermitian topological materials. It states, for example, that Weyl points must come in pairs in three-dimensional semimetals. Here, we present an extension of the doubling theorem to non-Hermitian lattice Hamiltonians. We focus on two-dimensional non-Hermitian systems without any symmetry constraints, which can host two different types of topological point nodes, namely, (i) Fermi points and (ii) exceptional points. We show that these two types of protected point nodes obey doubling theorems, which require that the point nodes come in pairs. To prove the doubling theorem for exceptional points, we introduce a generalized winding number invariant, which we call the “discriminant number.” Importantly, this invariant is applicable to any two-dimensional non-Hermitian Hamiltonian with exceptional points of arbitrary order and, moreover, can also be used to characterize nondefective degeneracy points. Furthermore, we show that a surface of a three-dimensional system can violate the non-Hermitian doubling theorems, which implies unusual bulk physics.

Figure 1

(a) In the 2D BZ, Fermi points (black points) are located at the crossings of the red line [$\mathrm{Re}{f}_{\mu }(\mathit{k})=0$] and the blue line [$\mathrm{Im}{f}_{\mu }(\mathit{k})=0$]. The gray arrows display the orientation of the vector field $[\mathrm{Re}{f}_{\mu }(\mathit{k}),\mathrm{Im}{f}_{\mu }(\mathit{k})]$. (a2) Shows that a small perturbation does not destroy the Fermi points. (b) The integration paths $\mathrm{\Gamma }({\mathit{k}}_F^j)$ (b1) of Eq. (3) [or Eq. (7)] can be continuously deformed to the boundary of the BZ (b3), without passing through any singular point (b2). The blue (red) dots represent Fermi points, or degeneracy points, with positive (negative) topological charge. (c) The absolute value of the complex energy bands $|E(\mathit{k})|$ (red and blue surfaces) and vector field of the characteristic polynomial $[\mathrm{Re}{f}_{\mu }(\mathit{k}),\mathrm{Im}{f}_{\mu }(\mathit{k})]$ (gray arrows) for the two-band model (5). The black and green points are Fermi points and exceptional points, respectively. The black arc represents the branch cut that connects the two exceptional points.

Figure 2

The complex energy spectra for the DP examples. (a),(b) Shows that the Hamiltonian (10) possesses two NDPs, which are the ends of the branch cut (solid lines). (c),(d) Shows that in the Hamiltonian (11) an EP and a NDP do not connect to any branch cut, although these points are located at the crossing of the energy bands (dashed lines).