Higher-order exceptional points unveiled by nilpotence and mathematical induction

Non-Hermitian systems can have peculiar degeneracies of eigenstates called exceptional points (EPs). An EP of n degenerate states is said to have order n, and higher-order EPs (HEPs) with n ≥ 3 exhibit intrinsic order-scaling responses potential for applications. However, traditional eigenvalue-based searches for HEPs are facing fundamental limitations in terms of complexity and implementation. Here, we propose a design for HEPs based on a simple property for matrices termed nilpotence and concise inductive procedure. The nilpotence guarantees a HEP with specified order and helps divide the problem. Our inductive routine can repeatedly double EP order starting from known designs, such as a well-known 2 × 2 parity-time-symmetric Hamiltonian. By applying our framework, we systematically design reciprocal photonic cavity systems operating at HEPs with up to n = 14 and find their unconventionally chiral, transparent, and enhanced responses. Our work pushes the investigation and application of HEPs to previously unexplored regimes in various physical systems. Higher-order exceptional points (HEPs) of non-Hermitian systems give order-scaling responses that promise applications, but traditional methods to find them face fundamental challenges. Here, the authors propose an algebraic framework that simplifies the search for HEPs, enables repeated doubling of their order, and is used to design photonic systems with HEPs of various order showing peculiar responses.