Eigenvalue topology of non-Hermitian band structures in two and three dimensions

In the band theory for non-Hermitian systems, the energy eigenvalues, which are complex, can exhibit nontrivial topology which is not present in Hermitian systems. In one dimension, it was recently noted theoretically and demonstrated experimentally that the eigenvalue topology is classified by the braid group. The classification of eigenvalue topology in higher dimensions, however, remained an open question. Here, we give a complete description of eigenvalue topology in two and three dimensional systems, including the gapped and gapless cases. We reduce the topological classification problem to a purely computational problem in algebraic topology. In two dimensions, the Brillouin zone torus is punctured by exceptional points, and each nontrivial loop in the punctured torus acquires a braid group invariant. These braids satisfy the constraint that the composite of the braids around the exceptional points is equal to the commutator of the braids on the fundamental cycles of the torus. In three dimensions, there are exceptional knots and links, and the classification depends on how they are embedded in the Brillouin zone three-torus. When the exceptional link is contained in a contractible ball, the classification can be expressed in terms of the knot group of the link. Our results provide a comprehensive understanding of non-Hermitian eigenvalue topology in higher dimensional systems, and should be important for the further explorations of topologically robust open quantum and classical systems.

Figure 1

Eigenvalue topology in two dimensions. (a) The Brillouin zone torus is shown as its fundamental polygon. In this example, there are two exceptional points. The nontrivial loops in the twice-punctured torus are the fundamental cycles $a$ and $b$, as well as the loops $e_i$ around the exceptional points. These loops satisfy the relation $ab{a}^{-1}{b}^{-1}=e_1{e}_2$. (b) The eigenvalues of the Hamiltonian define an element of the braid group for each nontrivial loop in the twice-punctured torus. The number of bands is the number of strands in the braid, which lives in a 3D space where two dimensions are the real and imaginary parts of the energy and the third dimension is the coordinate along a path in the Brillouin zone. In this three-band example, $a={\sigma }_1$ and $b={\sigma }_2$ are positive half twists; on the other hand, $e_1={\sigma }_2^{-1}$ and $e_2={\sigma }_1$. We can verify that ${\sigma }_1{\sigma }_2{\sigma }_1^{-1}{\sigma }_2^{-1}={\sigma }_2^{-1}{\sigma }_1$ using the braid relation ${\sigma }_2{\sigma }_1{\sigma }_2={\sigma }_1{\sigma }_2{\sigma }_1$: ${\sigma }_1{\sigma }_2{\sigma }_1^{-1}{\sigma }_2^{-1}={\sigma }_2^{-1}\left({\sigma }_2{\sigma }_1{\sigma }_2\right){\sigma }_1^{-1}{\sigma }_2^{-1}={\sigma }_2^{-1}\left({\sigma }_1{\sigma }_2{\sigma }_1\right){\sigma }_1^{-1}{\sigma }_2^{-1}={\sigma }_2^{-1}{\sigma }_1$. This can also be seen intuitively by “pulling” the ends of the two composite braids.

Figure 2

Eigenvalue topology in three dimensions. Four different exceptional links are shown in panels (a)-(d): the unlink, the Hopf link, the trefoil knot, and a pair of nontrivial cycles. In the first three cases (a)-(c), the exceptional link $\mathrm{\Delta }$ is contained in a small ball; the nontrivial loops are the fundamental cycles $a,b,$ and $c$, as well as the generators $e_i$ of the knot group ${\pi }_1({\mathbb{R}}^3-\mathrm{\Delta })$. The relations are $[a,b]=[b,c]=[c,a]=1$, together with any relations coming from the knot group. In the final case (d), the exceptional link is contained in a solid torus rather than a ball, leading to the relations $[a,b]=e_1{e}_2, [b,c]=[c,a]=1$.