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 (EPns) generically emerge as special points on manifolds of EPms, where m<n, EPns 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 EPms in direct connection to EPns. We find that simply counting the number of constraints defining the EPns is not sufficient in the presence of similarities; the constraints can also be satisfied by the EPm 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 in- duced spectral symmetries not always allow for any EPm 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.