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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.05073 |
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| _version_ | 1866914906886897664 |
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| author | Ammari, Habib Barandun, Silvio Liu, Ping Uhlmann, Alexander |
| author_facet | Ammari, Habib Barandun, Silvio Liu, Ping Uhlmann, Alexander |
| contents | Recently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non-reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of the eigenmodes and leads to correct spectral convergence properties. The results in this paper clarify and prove rigorously how the spectral properties of a finite structure are associated with those of the corresponding semi-infinitely or infinitely periodic lattices and give explicit characterisations of how to extend the Hermitian theory to non-reciprocal settings. Based on our theory, we characterise the generalised Brillouin zone for both open boundary conditions and periodic boundary conditions. Our results are consistent with the physical literature and give explicit generalisations to the $k$-Toeplitz matrix cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05073 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalised Brillouin Zone for Non-Reciprocal Systems Ammari, Habib Barandun, Silvio Liu, Ping Uhlmann, Alexander Mathematical Physics Materials Science Rings and Algebras Optics 35B34, 47B28, 35P25, 35C20, 81Q12, 15A18, 15B05 Recently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non-reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of the eigenmodes and leads to correct spectral convergence properties. The results in this paper clarify and prove rigorously how the spectral properties of a finite structure are associated with those of the corresponding semi-infinitely or infinitely periodic lattices and give explicit characterisations of how to extend the Hermitian theory to non-reciprocal settings. Based on our theory, we characterise the generalised Brillouin zone for both open boundary conditions and periodic boundary conditions. Our results are consistent with the physical literature and give explicit generalisations to the $k$-Toeplitz matrix cases. |
| title | Generalised Brillouin Zone for Non-Reciprocal Systems |
| topic | Mathematical Physics Materials Science Rings and Algebras Optics 35B34, 47B28, 35P25, 35C20, 81Q12, 15A18, 15B05 |
| url | https://arxiv.org/abs/2408.05073 |