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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.25603 |
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| _version_ | 1866914425386041344 |
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| author | Mandal, Ipsita |
| author_facet | Mandal, Ipsita |
| contents | We develop a systematic framework for determining the nature of exceptional points of $n^{\rm th}$ order (EP$_n$s) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations in the Jordan-normal basis of the defective matrix at an EP$_n$, we show that the upper-$k$ Hessenberg structure of the perturbation directly dictates the leading-order eigenvalue- and eigenvector-splitting to be $\propto ε^{1/k}$, when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), we find that EP$_3$s in P- and C-symmetric systems are restricted to at most $\sim ε^{1/2}$ branch points, while PT-symmetric systems generically support EP$_3$s with the strongest possible singularities (viz. $\sim ε^{1/3}$). We illustrate these results with concrete three-dimensional models in which exceptional curves and surfaces emerge. We further show that fine-tuned perturbations can suppress the leading-order branch point to a less-singular splitting, which have implications for designing direction-dependent EP-based sensors. The appendix extends the analysis to four-band C- and P-symmetric models, establishing the existence of EP$_4$s with $\sim ε^{1/4}$ singularities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_25603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Puiseux series about exceptional singularities dictated by symmetry-allowed Hessenberg forms of perturbation matrices Mandal, Ipsita Mesoscale and Nanoscale Physics Optics Quantum Physics We develop a systematic framework for determining the nature of exceptional points of $n^{\rm th}$ order (EP$_n$s) in non-Hermitian (NH) systems, represented by complex square matrices. By expressing symmetry-preserving perturbations in the Jordan-normal basis of the defective matrix at an EP$_n$, we show that the upper-$k$ Hessenberg structure of the perturbation directly dictates the leading-order eigenvalue- and eigenvector-splitting to be $\propto ε^{1/k}$, when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), we find that EP$_3$s in P- and C-symmetric systems are restricted to at most $\sim ε^{1/2}$ branch points, while PT-symmetric systems generically support EP$_3$s with the strongest possible singularities (viz. $\sim ε^{1/3}$). We illustrate these results with concrete three-dimensional models in which exceptional curves and surfaces emerge. We further show that fine-tuned perturbations can suppress the leading-order branch point to a less-singular splitting, which have implications for designing direction-dependent EP-based sensors. The appendix extends the analysis to four-band C- and P-symmetric models, establishing the existence of EP$_4$s with $\sim ε^{1/4}$ singularities. |
| title | Puiseux series about exceptional singularities dictated by symmetry-allowed Hessenberg forms of perturbation matrices |
| topic | Mesoscale and Nanoscale Physics Optics Quantum Physics |
| url | https://arxiv.org/abs/2603.25603 |