Theory for the spectral splitting exponent of exceptional points

TL;DR

The paper addresses how the spectral splitting at an $N$-th order exceptional point scales with a perturbation without solving the full spectrum. It develops a general framework based on the Jordan block form, showing that the leading exponent $\alpha$ in $\Delta E \sim \epsilon^{\alpha}$ is determined by the perturbation’s matrix-position pattern, with $\alpha=1/k$ for integers $k$ and $1\le k\le N$. Through analytical treatment of four solvable cases and numerical validation in a fourth-order SUSY EP model, the authors derive a direct perturbation-structure design principle: the smallest index sum $j=p+q$ among nonzero perturbation elements fixes the splitting exponent as $\alpha=1/(N-j)$, enabling targeted EP responses. This framework provides a practical route to engineer EP-based sensors with tailored sensitivity, and suggests broad applicability to diverse non-Hermitian systems.

Abstract

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 $N$th-order EP scales with perturbation strength $ε$ over a wide range, from $ε$ to $ε^{1/N}$. 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.

Figures (2)

• Figure 1: Absolute eigenenergy splitting $|\Delta E|$ for the 4-site non-Hermitian SUSY array under five different perturbations $H_{p1}$ to $H_{p5}$. The blue, magenta, green, orange and red dots represent numerical results for $H_{p1}$, $H_{p2}$, $H_{p3}$, $H_{p4}$ and $H_{p5}$, respectively. Solid curves crossing the dots correspond to the analytical predictions $|\Delta E| = \epsilon^{1/4}$, $\epsilon^{1/3}$, $\epsilon^{1/2}$, $\epsilon$, and $0$, respectively. Parameters are $\omega_0 = 0$, $\gamma=1$, and $J=1$.
• Figure 2: Spectral splitting induced by (a) $\hat{H}_{p6}$ and (b) $\hat{H}_{p7}$. The orange dots represent numerical results. The three lines in (a) correspond to $\ln|\Delta E| = \frac{1}{4} \ln \epsilon + C$ with $C = 0.07$, $0.008$, and $-0.09$, respectively. The two lines in (b) correspond to $\ln|\Delta E| = \frac{1}{2} \ln \epsilon + \frac{1}{2} \ln \frac{3\pm \sqrt{5}}{2}$. Parameters are $\omega_0 = 0$, $\gamma=1$, and $J=1$.