Theory for the spectral splitting exponent of exceptional points

Exceptional points (EPs), singularities in non-Hermitian systems where eigenvalues and eigenstates coalesce, exhibit a dramatically enhanced response to perturbations compared to Hermitian degeneracies. This makes them exceptional candidates for sensing applications. The spectral splitting of an $N\mathrm{th}$-order EP scales with perturbation strength $\ensuremath{\epsilon}$ over a wide range, from $\ensuremath{\epsilon}$ to ${\ensuremath{\epsilon}}^{1/N}$. Although the exact scaling exponent can be determined, in principle, by solving the characteristic equation, this approach becomes analytically intractable for large $N$ and often fails to yield useful physical insight. In this work, we develop a theory to directly predict the scaling exponent from the matrix positions of the perturbation. By using the Jordan block structure of the unperturbed Hamiltonian, we show that the splitting exponent can be analytically determined when the matrix positions of the perturbation satisfy some specific conditions. Our analytical framework provides a useful design principle for engineering perturbations to achieve a desired spectral response, facilitating the development of EP-based sensors.