Anisotropic exceptional points of arbitrary order

TL;DR

This paper discovers and characterizes anisotropic exceptional points of arbitrary order in non-Hermitian systems, revealing universal critical exponents, geometric structures, and crossover behaviors.

Contribution

It introduces the concept of anisotropic EPs of arbitrary order, analyzes their universal critical exponents, and explores their geometric configurations and transitions.

Findings

Eigenvalues and eigenvectors show distinct behaviors near anisotropic EPs.

Critical exponents of phase rigidity are universal and depend on the order of the EP.

EPs form ellipses and rings in parameter space, indicating complex topological structures.

Abstract

A pair of anisotropic exceptional points (EPs) of arbitrary order are found in a class of non-Hermitian random systems with asymmetric hoppings. Both eigenvalues and eigenvectors exhibit distinct behaviors when these anisotropic EPs are approached from two orthogonal directions in the parameter space. For an order-$N$ anisotropic EP, the critical exponents $\nu$ of phase rigidity are $(N-1)/2$ and $N-1$, respectively. These exponents are universal within the class. The order-$N$ anisotropic EPs split and trace out multiple ellipses of EPs of order $2$ in the parameter space. For some particular configurations, all the EP ellipses coalesce and form a ring of EPs of order $N$. Crossover to the conventional order-$N$ EPs with $\nu=(N-1)/N$ is discussed.