Realizing exceptional points of any order in the presence of symmetry

Exceptional points (EPs) appear as degeneracies in the spectrum of non-Hermitian matrices at which the eigenvectors coalesce. In general, an EP of order n may find room to emerge if 2( n − 1) real constraints are imposed. Our results show that these constraints can be expressed in terms of the determinant and traces of the non-Hermitian matrix. Our findings further reveal that the total number of constraints may reduce in the presence of unitary and antiunitary symmetries. Additionally, we draw generic conclusions for the low-energy dispersion of the EPs. Based on our calculations, we show that in odd dimensions the presence of sublattice or pseudo-chiral symmetry enforces n th order EPs to disperse with the ( n − 1)th root. For two-, three- and four-band systems, we explicitly present the constraints needed for the occurrence of EPs in terms of system parameters and classify EPs based on their low-energy dispersion relations. we present a case study in the following. We start with a traceless non-Hermitian four-band

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