Winding around non-Hermitian singularities

Non-Hermitian singularities are ubiquitous in non-conservative open systems. Owing to their peculiar topology, they can remotely induce observable effects when encircled by closed trajectories in the parameter space. To date, a general formalism for describing this process beyond simple cases is still lacking. Here we develop a general approach for treating this problem by utilizing the power of permutation operators and representation theory. This in turn allows us to reveal a surprising result that has so far escaped attention: loops that enclose the same singularities in the parameter space starting from the same point and traveling in the same direction, do not necessarily share the same end outcome. Interestingly, we find that this equivalence can be formally established only by invoking the topological notion of homotopy. Our findings are general with far reaching implications in various fields ranging from photonics and atomic physics to microwaves and acoustics.

2 Finally, loop 3 in Supplementary Fig. 1b was chosen to be a titled ellipse with the line connecting κ 0 and κ 0 as the major axis. This ellipse has semi-major axis a ≈ 0.3858, focal distance c = a − 0.002 and a rotating angle θ = arctan 1 4 . Therefore the parametric function of loop 3 is: Im[κ(τ )] = c y + a cos(ωτ ) sin θ + b sin(ωτ ) cos θ,

Supplementary Note 2. Varying gain/loss instead of coupling
To confirm that these control parameters (gain, loss, propagation constants and real coupling coefficients) provide enough degrees of freedom to observe the exotic effects discussed in the main text, we briefly investigate the encircling of EPs associated with the Hamiltonian and Im [γ]. Both loops encircle only EP 1 , yet they are topologically inequivalent since: Loop 2 cannot be deformed into loop 1 without crossing EP 2 ; and the permutation matrices M 1 and M 2 (which are, incidentally, similar to those defined in Eq. (5) associated with EP 1,2 in the main text) do not commute. Thus, indeed a system of four coupled waveguides with variable gain/loss and propagation constants can be used to study the topological equivalence between encircling loops in the parameter space.
Case study using complex propagation constants. In the example considered in the previous section, we have studied H as we vary the complex coupling coefficients.
While this is not impossible, it is rather difficult to achieve experimentally. An easier approach that lends itself to an easier experimental implementation is to change the complex propagation constant which corresponds to changing the real propagation constants and the gain/loss factors.
Here we confirm that the main features of this work can be still observed under these conditions.
a The exceptional points landscape of H in a two-dimensional parameter space spanned by Re [γ] and Im[γ]. One can identify two topologically inequivalent loops (blue lines) that encircle the exceptional point EP 1 . b and c show the eigenvalue exchange relations associated with these two loops, confirming their nonequivalence. Black dots represent exceptional points, red lines are the branch cuts. The colors along the eigenvalue trajectory indicate the branch at which the relevant eigenvalue is located.