Flat band in disorder-driven non-Hermitian Weyl semimetals

We study the interplay of disorder and band-structure topology in a Weyl semimetal with a tilted conical spectrum around the Weyl points. The spectrum of particles is given by the eigenvalues of a non-Hermitian matrix, which contains contributions from a Weyl Hamiltonian and complex self-energy due to electron elastic scattering on disorder. We find that the tilt-induced matrix structure of the self-energy gives rise to either a flat band or a nodal line segment at the interface of the electron and hole pockets in the bulk band structure of type-II Weyl semimetals depending on the Weyl cone inclination. For the tilt in a single direction in momentum space, each Weyl point expands into a flat band lying on the plane, which is transverse to the direction of the tilt. The spectrum of the flat band is fully imaginary and is separated from the in-plane dispersive part of the spectrum by the “exceptional nodal ring” where the matrix of the Green's function in momentum-frequency space is defective. The tilt in two directions might shrink a flat band into a nodal line segment with “exceptional edge points.” We discuss the connection to the non-Hermitian topological theory.

Figure 1

The real part of the spectrum $E_s(k_x,0,k_z)$ of a disordered type-II Weyl semimetal. The tilting is along ${\mathit{e}}_z$ in momentum space and $s=-1$, see Eq. (8) in the text for details. The electron and hole pockets touch along the line segment at which the spectrum is purely imaginary and single valued at the edge points. The line segment is a diameter of the circle flat band, which lies on the plane $k_z=0$ as shown in the right panel of Fig. 2.

Figure 2

The real part of the spectrum $E_s(k_x,k_y,0)$ of a Weyl semimetal in which the tilting of the Weyl cone is along ${\mathit{e}}_z$ in momentum space. (Left) The band-touching Weyl point in the absence of disorder. (Right) In the disordered type-II Weyl semimetal the Weyl point expands into a circle flat band, which is defined by $k_x^2+k_y^2\le 1/4{\tau }^2{C}^2$, where $\tau$ is the mean free time and $C$ is the tilt parameter of the Weyl cone.

Figure 3

The real and imaginary parts of the spectrum $E_s\left(\mathbf{k}\right)$ at $k_z=0$ as a function of momentum $k_{\perp }=\sqrt{k_x^2+k_y^2}$ of a disordered type-II Weyl semimetal. The spectrum is purely imaginary within the region of momenta $0\le k_{\perp }\le 1/2\left|C\right|\tau$, where $\tau$ is the mean free time and $C$ is the tilt parameter. At $k_{\perp }=1/2\left|C\right|\tau$ the spectrum has a single imaginary value of $-i/2\tau$.