Exact Correlations in Topological Quantum Chains
Although free-fermion systems are considered exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or entanglement measures. We derive closed expressions for such quantities for a dense subclass of certain classes of topolo...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212033 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Exact Correlations in Topological Quantum Chains. Nick G. Jones and Ruben Verresen. SIGMA 19 (2023), 098, 54 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862704438732390400 |
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| author | Jones, Nick G. Verresen, Ruben |
| author_facet | Jones, Nick G. Verresen, Ruben |
| citation_txt | Exact Correlations in Topological Quantum Chains. Nick G. Jones and Ruben Verresen. SIGMA 19 (2023), 098, 54 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Although free-fermion systems are considered exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or entanglement measures. We derive closed expressions for such quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains called generalised cluster models. While there is a bijection between general models in these classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results: (1) we derive closed expressions for the string correlation functions—the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into ground state entanglement; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit—an independent and explicit construction for the BDI class is given in a concurrent work [Phys. Rev. Res. 3 (2021), 033265, 26 pages, arXiv:2105.12143]; (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. General models in these classes can be obtained by taking limits of the models we analyse, giving a further application of our results. To the best of our knowledge, these results constitute the first application of Day's formula and Gorodetsky's formula for Toeplitz determinants to many-body quantum physics.
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| first_indexed | 2026-03-18T19:38:26Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212033 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T19:38:26Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
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| spelling | Jones, Nick G. Verresen, Ruben 2026-01-23T10:09:10Z 2023 Exact Correlations in Topological Quantum Chains. Nick G. Jones and Ruben Verresen. SIGMA 19 (2023), 098, 54 pages 1815-0659 2020 Mathematics Subject Classification: 82B10; 81V74 arXiv:2105.13359 https://nasplib.isofts.kiev.ua/handle/123456789/212033 https://doi.org/10.3842/SIGMA.2023.098 Although free-fermion systems are considered exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or entanglement measures. We derive closed expressions for such quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains called generalised cluster models. While there is a bijection between general models in these classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results: (1) we derive closed expressions for the string correlation functions—the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into ground state entanglement; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit—an independent and explicit construction for the BDI class is given in a concurrent work [Phys. Rev. Res. 3 (2021), 033265, 26 pages, arXiv:2105.12143]; (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. General models in these classes can be obtained by taking limits of the models we analyse, giving a further application of our results. To the best of our knowledge, these results constitute the first application of Day's formula and Gorodetsky's formula for Toeplitz determinants to many-body quantum physics. We thank J. Bibo, B. Jobst, F. Pollmann, and A. Smith for many inspiring discussions on this subject and for collaboration on related work [43]. We are also grateful to A. B¨ottcher, A. Its, J. Keating, F. Mezzadri, N. Schuch, and S. Stevens for helpful correspondence and to A. Smith and the anonymous referees for valuable comments on the manuscript. This work was completed while N.G.J. held a Heilbronn Research Fellowship at the Mathematical Institute, University of Oxford, and the Heilbronn Institute for Mathematical Research, Bristol, UK. R.V. was supported by the Harvard Quantum Initiative Postdoctoral Fellowship in Science and Engineering and by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440, Ashvin Vishwanath). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Exact Correlations in Topological Quantum Chains Article published earlier |
| spellingShingle | Exact Correlations in Topological Quantum Chains Jones, Nick G. Verresen, Ruben |
| title | Exact Correlations in Topological Quantum Chains |
| title_full | Exact Correlations in Topological Quantum Chains |
| title_fullStr | Exact Correlations in Topological Quantum Chains |
| title_full_unstemmed | Exact Correlations in Topological Quantum Chains |
| title_short | Exact Correlations in Topological Quantum Chains |
| title_sort | exact correlations in topological quantum chains |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212033 |
| work_keys_str_mv | AT jonesnickg exactcorrelationsintopologicalquantumchains AT verresenruben exactcorrelationsintopologicalquantumchains |