Exceptional links and twisted Fermi ribbons in non-Hermitian systems

The generic nature of band touching points in three-dimensional band structures is at the heart of the rich phenomenology, topological stability, and novel Fermi arc surface states associated with Weyl semimetals. Here we report on the corresponding scenario emerging in systems effectively described by non-Hermitian Hamiltonians. Remarkably, three-dimensional non-Hermitian systems have generic band touching along one-dimensional closed contours, forming exceptional rings and links in reciprocal space. The associated Seifert surfaces support open “Fermi ribbons” where the real part of the energy gap vanishes, providing a novel class of higher-dimensional bulk generalizations of Fermi arcs which are characterized by an integer twist number. These results have possible applications to a plethora of physical settings, ranging from mechanical systems and optical metamaterials with loss and gain to heavy fermion materials with finite-lifetime quasiparticles. In particular, photonic crystals provide fertile ground for simulating the exuberant phenomenology of exceptional links and their concomitant Fermi ribbons.

Figure 1

Exceptional line topology in non-Hermitian systems. The top row (a)–(d) shows the surfaces $\nu =0$ in green and $\eta =0$ in blue; see Eq. (3) for a definition. Their intersections give the exceptional lines where the bands meet and the theory becomes defective, here displayed in red. When particle-hole symmetry is present, these correspond to $E=0$, i.e., zero energy and infinite lifetime. The bottom row (e)–(h) shows the corresponding Fermi surface topology. In (a) a non-Hermitian term is added to a single Weyl node so that it splits into an exceptional ring that is accompanied by a disk-shaped Fermi surface (e). In (b) $\eta ,\nu$ are tori with a relative translation, resulting in two exceptional rings and a Fermi surface in the form of a punctured disk. If both tilting and translation is applied to one of the tori, then the exceptional rings form a Hopf link (c). Finally, elliptic deformation—stretching in one direction and contracting in the other—in combination with bending gives rise to a twice-braided link (d). The Fermi surfaces corresponding to intersecting tori are Seifert surfaces [45] with integer twists (g)–(h). Thus, the Fermi surface connects the two exceptional lines, or equivalently, the exceptional lines are terminations points of the now-open Fermi surfaces, which we dub “Fermi ribbons.” The exact form of the models (b)–(d) is outlined in Eqs. (11, 12, 13, 14).

Figure 2

The three half-twist Möbius Fermi surface which terminates on a three-foil knot (a) is not an allowed nodal structure because it is not orientable. Still, these considerations do not rule out knots constructed from pairs of exceptional lines (b). In this case, it is a ribbon-shaped orientable Fermi surface, connecting the line pair which forms a three-foil knot. This dispersion is described by Eqs. (17) and (18).