# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1903.01737