Experimental demonstration of splitting rules for exceptional points and their characterization

In non-Hermitian systems, the eigenvalues near exceptional points of order N (EPNs) usually exhibit E A1/N dispersion/splitting under a small perturbation A. Such high sensitivity to perturbation makes them ideal candidates for sensors. However, E - A1/m dispersions with m = 1, 2, ... , N - 1 are also possible for EPNs. Using a transposed Jordan block (TJB) matrix H0 as a representative Hamiltonian, we present general rules that provide a unified understanding of the splitting behaviors of EPNs in diverse systems. Specifically, when the k-diagonal entries (H0)i,i+k of H0 are perturbed by A, we observe a dispersion of the form E - A1/m, where m = k + 1. The phase rigidity's exponent x and discriminant number y together can serve as topological invariants to completely characterize an EPN with E - A1/m splitting, and they are proportional, i.e., x = v/N = (N - 1)/m. The results are demonstrated experimentally using electrical circuits. Next, we demonstrate the applicability of the splitting rules to general non-TJB Hamiltonians that exhibit an EPN through an example. Moreover, we observe that the splitting of EPNs can lead to the emergence of lower-order EP structures, such as EP2 ellipses in the parameter space. We find that the EP2 ellipses exhibit two types of coalescence: with and without an increase in the EP order.