Non-Abelian nature of systems with multiple exceptional points

The defining characteristic of an exceptional point (EP) in the parameter space of a family of operators is that, upon encircling the EP, eigenstates are permuted. When one encircles multiple EPs, the question arises how to properly compose the effects of the individual EPs. This was thought to be ambiguous. We show that one can solve this problem by considering based loops and their deformations. The theory of fundamental groups allows us to generalize this technique to arbitrary degeneracy structures like exceptional lines in a three-dimensional parameter space. As permutations of three or more objects form a non-Abelian group, the next question that arises is whether one can experimentally demonstrate this noncommutative behavior. This requires at least two EPs of a family of operators that have at least three eigenstates. A concrete implementation in a recently proposed $\mathcal{PT}$ symmetric waveguide system is suggested as an example of how to experimentally check the composition law and show the non-Abelian nature of non-Hermitian systems with multiple EPs.

Figure 1

Two EPs ($\mathsf{x}$'s) encircled individually by loops ${\gamma }_1$ and ${\gamma }_2$, respectively, and a loop ${\gamma }_3$ which encircles both EPs.

Figure 2

Example of loops based at the central black dot that enclose two EPs, represented by $\mathsf{x}$'s. (a) Two paths around two EPs. (b) Deformed ${\gamma }_1{\gamma }_2$. (c) Deformed ${\gamma }_2{\gamma }_1$. The deformed ${\gamma }_1{\gamma }_2$ and ${\gamma }_2{\gamma }_1$ resemble each other but are not homotopic relative to the base point.

Figure 3

Aspects of a connecting path or bridge, both in parameter and in eigenvalue space. (a) Two loops with different base points. A bridge is needed to pull back information from one point to the other. This requires that the bridge be traversed twice and in opposite directions. (b) Example of eigenvalue paths induced from a bridge. Open circles represent eigenvalues at $x_1$ and have unprimed labels; filled circles, eigenvalues at $x_2$ and have primed labels. The bridge induces the relabeling $1\mapsto 2^{\prime }, 2\mapsto 1^{\prime }, 3\mapsto 3^{\prime }$.

Figure 4

Schematic of the experimental setup (see the text).

Figure 5

EP structure in the $\mathrm{Re}\left(z\right)=0$ plane.

Figure 6

EP and $\mathcal{PT}$ structure in parameter space. Solid lines show EPs; red squares and blue stripes represent broken and exact $\mathcal{PT}$ symmetry, respectively. The plot is symmetric in $\mathrm{Re}\left(z\right)\mapsto -\mathrm{Re}\left(z\right)$.

Figure 7

Paths near $(z,c)=(0,-1)$. The two blue lines are EP lines: the $z=0$ line consists of EP3s and the other line consists of EP2s. The black circle and dashed red figure eight are on the plane $c=-0.9$ and can be used to experimentally verify non-Abelian behavior (see the text).

Figure 8

Loops on plane $c=-0.9$ of the parameter space (left panels). All loops have the same base point, shown by the small black circle. The solid part of the loop is always traversed first, so before the dashed part of the loop. Orientations of the loops are indicated by arrows. EPs are represented by gray $\mathsf{x}$'s; the upper one is an EP2 and the lower one is an EP3. The complex energy planes (right panels) show the resulting paths of the three eigenvalues, each drawn with its own color and style. Labeling eigenvalues top to bottom, we can read off the induced permutation: (a) permutation (23); (b) permutation (132); (c) permutation $\left(13\right)=\left(132\right)\left(23\right)$; (d) Permutation $\left(12\right)=\left(23\right)\left(132\right)$; (e) permutation (13); (f) permutation (12). One observes that the $\mathrm{\Lambda }$ group of the system is isomorphic to $S_3$.

Figure 9

The tangent intersection as EPs of different orders: (a) EP2; (b) EP3.