Interference of Non-Hermiticity with Hermiticity at Exceptional Points
The recent growth in popularity of the non-Hermitian quantum Hamiltonians <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>λ</mi><mo>)</m...
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| Language: | English |
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MDPI AG
2022-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/20/3721 |
| _version_ | 1797471715813490688 |
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| author | Miloslav Znojil |
| author_facet | Miloslav Znojil |
| author_sort | Miloslav Znojil |
| collection | DOAJ |
| description | The recent growth in popularity of the non-Hermitian quantum Hamiltonians <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> with real spectra is strongly motivated by the phenomenologically innovative possibility of an access to the non-Hermitian degeneracies called exceptional points (EPs). What is actually presented in the present paper is a perturbation-theory-based demonstration of a fine-tuned nature of this access. This result is complemented by a toy-model-based analysis of the related details of quantum dynamics in the almost degenerate regime with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>≈</mo><msup><mi>λ</mi><mrow><mo>(</mo><mi>E</mi><mi>P</mi><mo>)</mo></mrow></msup></mrow></semantics></math></inline-formula>. In similar studies, naturally, one of the decisive obstacles is the highly nontrivial form of the underlying mathematics. Here, many of these obstacles are circumvented via several drastic simplifications of our toy models—i.a., our <i>N</i> by <i>N</i> matrices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>H</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are assumed real, tridiagonal and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula>-symmetric, and our <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is assumed to be split into its Hermitian and non-Hermitian components staying in interaction. This is shown to lead to several remarkable spectral features of the model. Up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>8</mn></mrow></semantics></math></inline-formula>, their description is even shown tractable non-numerically. In particular, it is shown that under generic perturbation, the “unfolding” removal of the spontaneous breakdown of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula>-symmetry proceeds via intervals of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> with complex energy spectra. |
| first_indexed | 2024-03-09T19:53:05Z |
| format | Article |
| id | doaj.art-6159e3d114ce4f4d8d6f94ac5364e8c1 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T19:53:05Z |
| publishDate | 2022-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-6159e3d114ce4f4d8d6f94ac5364e8c12023-11-24T01:06:12ZengMDPI AGMathematics2227-73902022-10-011020372110.3390/math10203721Interference of Non-Hermiticity with Hermiticity at Exceptional PointsMiloslav Znojil0Department of Physics, Faculty of Science, University of Hradec Králové, Rokitanského 62, 500 03 Hradec Králové, Czech RepublicThe recent growth in popularity of the non-Hermitian quantum Hamiltonians <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> with real spectra is strongly motivated by the phenomenologically innovative possibility of an access to the non-Hermitian degeneracies called exceptional points (EPs). What is actually presented in the present paper is a perturbation-theory-based demonstration of a fine-tuned nature of this access. This result is complemented by a toy-model-based analysis of the related details of quantum dynamics in the almost degenerate regime with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>≈</mo><msup><mi>λ</mi><mrow><mo>(</mo><mi>E</mi><mi>P</mi><mo>)</mo></mrow></msup></mrow></semantics></math></inline-formula>. In similar studies, naturally, one of the decisive obstacles is the highly nontrivial form of the underlying mathematics. Here, many of these obstacles are circumvented via several drastic simplifications of our toy models—i.a., our <i>N</i> by <i>N</i> matrices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>H</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are assumed real, tridiagonal and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula>-symmetric, and our <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is assumed to be split into its Hermitian and non-Hermitian components staying in interaction. This is shown to lead to several remarkable spectral features of the model. Up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>8</mn></mrow></semantics></math></inline-formula>, their description is even shown tractable non-numerically. In particular, it is shown that under generic perturbation, the “unfolding” removal of the spontaneous breakdown of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula>-symmetry proceeds via intervals of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> with complex energy spectra.https://www.mdpi.com/2227-7390/10/20/3721non-Hermitian quantum mechanics of the closed and open systemsnon-Hermitian and Hermitian components of the Hamiltoniancontrol of access to the exceptional point degeneracies and to the related quantum phase transitionsperturbation theory tractability of the matrix toy models exhibiting PT symmetry |
| spellingShingle | Miloslav Znojil Interference of Non-Hermiticity with Hermiticity at Exceptional Points Mathematics non-Hermitian quantum mechanics of the closed and open systems non-Hermitian and Hermitian components of the Hamiltonian control of access to the exceptional point degeneracies and to the related quantum phase transitions perturbation theory tractability of the matrix toy models exhibiting PT symmetry |
| title | Interference of Non-Hermiticity with Hermiticity at Exceptional Points |
| title_full | Interference of Non-Hermiticity with Hermiticity at Exceptional Points |
| title_fullStr | Interference of Non-Hermiticity with Hermiticity at Exceptional Points |
| title_full_unstemmed | Interference of Non-Hermiticity with Hermiticity at Exceptional Points |
| title_short | Interference of Non-Hermiticity with Hermiticity at Exceptional Points |
| title_sort | interference of non hermiticity with hermiticity at exceptional points |
| topic | non-Hermitian quantum mechanics of the closed and open systems non-Hermitian and Hermitian components of the Hamiltonian control of access to the exceptional point degeneracies and to the related quantum phase transitions perturbation theory tractability of the matrix toy models exhibiting PT symmetry |
| url | https://www.mdpi.com/2227-7390/10/20/3721 |
| work_keys_str_mv | AT miloslavznojil interferenceofnonhermiticitywithhermiticityatexceptionalpoints |