The analytically tractable zoo of similarity-induced exceptional structures

Exceptional points (EPs) are non-Hermitian spectral degeneracies marking a simultaneous coalescence of eigenvalues and eigenvectors. Despite the fact that multiband $n$-fold EPs (EP$n$s) generically emerge as special points on manifolds of EP$m$s, where $m<n$, EP$n$s as well as their topological properties have hitherto been studied as isolated objects. In this work we address this issue and carefully map out the emerging properties of multifold exceptional structures in three and four dimensions under the influence of one or multiple generalized similarities, revealing diverse combinations of EP$m$s in direct connection to EP$n$s. We find that simply counting the number of constraints defining the EP$n$s is not sufficient in the presence of similarities; the constraints can also be satisfied by the EP$m$-manifolds obeying certain spectral symmetries in the complex eigenvalue plane, reducing their dimension beyond what is expected from counting the number of constraints. Furthermore, the induced spectral symmetries not always allow for any EP$m$-manifold to emerge in $n$-band systems, making the plethora of exceptional structures deviate further from naive expectations. We illustrate our findings in simple periodic toy models. By relying on similarity relations instead of the less general symmetries, we simultaneously cover several physically relevant scenarios, ranging from optics and topolectrical circuits, to open quantum systems. This makes our predictions highly relevant and broadly applicable in modern research, as well as experimentally viable within various branches of physics.