Braid-protected topological band structures with unpaired exceptional points

We demonstrate the existence of topologically stable unpaired exceptional points (EPs), and construct simple non-Hermitian (NH) tight-binding models exemplifying such remarkable nodal phases. While fermion doubling, i.e., the necessity of compensating the topological charge of a stable nodal point by an antidote, rules out a direct counterpart of our findings in the realm of Hermitian semimetals, here we derive how nonommuting braids of complex energy levels may stabilize unpaired EPs. Drawing on this insight, we reveal the occurrence of a single, unpaired EP, manifested as a non-Abelian monopole in the Brillouin zone of a minimal three-band model. This third-order degeneracy represents a sweet spot within a larger topological phase that cannot be fully gapped by any local perturbation. Instead, it may only split into simpler (second-order) degeneracies that can only gap out by pairwise annihilation after having moved around inequivalent large circles of the Brillouin zone. Our results imply the incompleteness of a topological classification based on winding numbers, due to non-Abelian representations of the braid group intertwining three or more complex energy levels, and provide insights into the topological robustness of non-Hermitian systems and their non-Abelian phase transitions.

Figure 1

Illustration of key results. (a) The braiding of complex energy levels as a function of lattice momentum is at the heart of our analysis. (b) In the two-dimensional Brillouin zone, NH phases are characterized by two braid elements $(a,b)$ along the noncontractible loops. For gapped systems, $a$ and $b$ commute (group commutator $ab{a}^{-1}{b}^{-1}=[a,b]=\mathbb{1}$). (c) Minimal model of a band structure with a nontrivial such commutator, hosting two unpaired EP2s. (d) While the unpaired EP2s may merge into a single EP3, the nontrivial braid topology implies that the system remains gapless. EPs are denoted by red dots and base points of loops by $*$ in all panels.

Figure 2

Composition rule for braids. (a) Schematic: Braids around degeneracies must add up to the boundary braid. (b) Boundary braid: Spectrum of Eq. (5) shown along the BZ boundary, counterclockwise from $(-\pi ,-\pi )$. Associating ${\sigma }_i$ to a counterclockwise exchange of eigenvalues $E_i,E_{i+1}$ in the complex plane allows us to read off the boundary braid as ${\sigma }_1^{\phantom{}}{\sigma }_2^{\phantom{}}{\sigma }_1^{-1}{\sigma }_2^{-1}$. (c) Spectrum of Eq. (5) shown on the entire BZ. The orange path corresponds to the path traversed in (a) and (b) that gives rise to the boundary braid. The two degeneracies marked in red carry charges ${\sigma }_2^{-1}$ and ${\sigma }_1$, which add up to the boundary charge [see Eq. (4)]. (d) Tuning this model to fuse the two EPs [see Eq. (6)] cannot annihilate them if the perturbation conserves the boundary charge, which is why we call these degeneracies unpaired. Instead, a single nontrivial EP3 remains as a non-Abelian monopole that compensates the entire braid charge of the boundary.

Figure 3

Non-Hermitian phase transition. (a) To gap out the system, any contractible loop (fuchsia) in the BZ must carry a trivial braid. The only way to change nontrivial braids (dashed) into trivial ones (dotted) continuously is to move an EP across the loop. (b) Schematic representation of the spectrum of Eq. (6) at $\delta =1$. The single nontrivial EP3 is marked in red, and the green lines illustrate the real-gap closings which form nontrivial loops around the BZ. (c) Schematic representation of the spectrum for the interpolation Eq. (7) between the unpaired model hosting unpaired EPs and a gapped phase. Initially the BZ boundaries (solid black lines) carry braid charges ${\sigma }_1$ (${\sigma }_2$) in the $k_x$ ($k_y$) direction. At $s=3/4$ the two EPs merge at the boundary, breaking the $k_x$ braid charge into a trivial braid; the corresponding boundary is then marked with dotted lines. For $s>3/4$, all braids along $k_y=\text{const}$ are trivial, while loops along $k_x=\text{const}$ correspond to braids of type ${\sigma }_2$. The boundary braid in that regime is then ${\sigma }_2{\sigma }_2^{-1}=\mathbb{1}$. (d) Same as (c) for interpolation Eq. (8) between nontrivial and trivial gapped phases for increasing interpolation parameter $t$. A single twofold degeneracy is created at $t=2/3$ and subsequently splits into two EP2s. These merge at the boundary for $t=6/7$. For $t>6/7$ the system is topologically trivial and gapped; all loops in this system correspond to the trivial braid. Note that the last panel in (c) and the first panel in (d) describe the same system.