Fig. 3
Homotopy between loops. Illustration of equivalence between homotopic loops in the parameter space of a generic Hamiltonian. a Loop a encloses two exceptional points (EPs) associated with matrices Mo and Mp. b Loop ⓑ encloses the same two EPs yet it cannot be deformed into loop ⓐ without crossing EP associated with Mr. Consequently, it has a different matrix product (assuming not accidental equivalence). On the other hand, loops ⓒ and ⓓ in c and d can be deformed into loop ⓐ without crossing any EP. As a result, they are equivalent (have the same matrix product) as shown in the text. e A peculiar case of free homotopy is presented. Loop ⓔ is homotopic with loop ⓐ for the starting point z but not for z′. As a result, the two loops are equivalent for the former point but not for the latter. The discussion here is very generic and can be extended easily to any other configuration of EPs and branch cuts (BCs). As a side note, we emphasize that the choice of the BCs is not unique. However, while different partitioning will lead to a new set of matrices, the final results and the topological relations between the loops are invariant. Black dots represent EPs, red lines are the BCs, and the blue loops are the encircling trajectories
