Anisotropic exceptional points of arbitrary order and exceptional ellipses

Two anisotropic exceptional points (EPs) of arbitrarily high order are found in a class of random non-Hermitian systems, where the non-Hermiticity emerges from non-reciprocal hoppings. Both eigenvalues and phase rigidity show different asymptotic forms near the anisotropic EP in two orthogonal directions in the parameter space, making them anisotropic EPs. The critical exponents of phase rigidity follow universal rules near an anisotropic EP, and the exponents depend on the dimension of the Hamiltonian as well as the approaching direction, but are independent of the random configurations. We found multiple ellipses formed by EPs of order two converge to the two high-order EPs in the parameter space. A ring of high-order EPs is formed when all ellipses coalesce for some particular configurations.