Exceptional point induced by hyperbolicity in an electrostatic spherical shell
Exceptional points (EPs) are non--Hermitian degeneracies at which both eigenvalues and eigenvectors coalesce, usually engineered through balanced gain and loss or non--reciprocity. In this work, we show that the radial electrostatic problem of an anisotropic core--shell nanoparticle realizes an EP without any of the alluded conditons. Written in the logarithmic radial coordinate, the problem is a Cauchy--Euler equation that maps onto a damped oscillator and, equivalently, onto a two-site Hatano--Nelson Hamiltonian, so that the non-Hermiticity is emulated in space rather than in time and the sole control parameter is the degree of dielectric anisotropy, $\varepsilon_t/\varepsilon_r$. The shell is passive, reciprocal, and lossless, yet it hosts an EP and a non--Hermitian phase transition governed by hyperbolicity alone. Beyond a hyperbolicity threshold the electric field inside the shell turns from monotonic decay into spatial oscillation, producing hotspots at intermediate radii which can be exploited to tailor fluorescence, strong coupling, sensing, or cloaking. More broadly, this establishes hyperbolic media as a platform to simulate non--Hermitian physics without gain, loss, or non--reciprocity.