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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mi>n</a:mi> </a:math> -fold EPs ( <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mrow> <b:mi>EP</b:mi> <b:mi>n</b:mi> <b:mi mathvariant="normal">s</b:mi> </b:mrow> </b:math> ) generically emerge as special points on manifolds of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mi>EP</d:mi> <d:mi>m</d:mi> <d:mi mathvariant="normal">s</d:mi> </d:mrow> </d:math> , where <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:mrow> <f:mi>m</f:mi> <f:mo><</f:mo> <f:mi>n</f:mi> </f:mrow> </f:math> , <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>EP</g:mi> <g:mi>n</g:mi> <g:mi mathvariant="normal">s</g:mi> </g:mrow> </g:math> 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 <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mi>EP</i:mi> <i:mi>m</i:mi> <i:mi mathvariant="normal">s</i:mi> </i:mrow> </i:math> in direct connection to <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:mrow> <k:mi>EP</k:mi> <k:mi>n</k:mi> <k:mi mathvariant="normal">s</k:mi> </k:mrow> </k:math> . We find that simply counting the number of constraints defining the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>EP</m:mi> <m:mi>n</m:mi> <m:mi mathvariant="normal">s</m:mi> </m:mrow> </m:math> is not sufficient in the presence of similarities; the constraints can also be satisfied by the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:mrow> <o:mi>EP</o:mi> <o:mi>m</o:mi> </o:mrow> </o:math> 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 <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"> <p:mrow> <p:mi>EP</p:mi> <p:mi>m</p:mi> </p:mrow> </p:math> manifold to emerge in <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"> <q:mi>n</q:mi> </q:math> -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.