Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes
Abstract The Szegö–Widom theorem provides an expression for the determinant of block Toeplitz matrices in the asymptotic limit of large matrix dimension n. We show that the presence of zero modes, i.e, eigenvalues that vanish as...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer US
2021
|
| Online Access: | https://hdl.handle.net/1721.1/131882 |
| _version_ | 1826212970126901248 |
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| author | Basor, E. Dubail, J. Emig, T. Santachiara, R. |
| author2 | MultiScale Materials Science for Energy and Environment, Joint MIT-CNRS Laboratory |
| author_facet | MultiScale Materials Science for Energy and Environment, Joint MIT-CNRS Laboratory Basor, E. Dubail, J. Emig, T. Santachiara, R. |
| author_sort | Basor, E. |
| collection | MIT |
| description | Abstract
The Szegö–Widom theorem provides an expression for the determinant of block Toeplitz matrices in the asymptotic limit of large matrix dimension n. We show that the presence of zero modes, i.e, eigenvalues that vanish as
$$\alpha ^n$$
α
n
,
$$|\alpha |<1$$
|
α
|
<
1
, when
$$n\rightarrow \infty $$
n
→
∞
, requires a modification of the Szegö–Widom theorem. A new asymptotic expression for the determinant of a certain class of block Toeplitz matrices with one pair of zero modes is derived. The result is inspired by one-dimensional topological superconductors, and the relation with the latter systems is discussed. |
| first_indexed | 2024-09-23T15:41:03Z |
| format | Article |
| id | mit-1721.1/131882 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:41:03Z |
| publishDate | 2021 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1318822023-02-17T21:11:20Z Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes Basor, E. Dubail, J. Emig, T. Santachiara, R. MultiScale Materials Science for Energy and Environment, Joint MIT-CNRS Laboratory Abstract The Szegö–Widom theorem provides an expression for the determinant of block Toeplitz matrices in the asymptotic limit of large matrix dimension n. We show that the presence of zero modes, i.e, eigenvalues that vanish as $$\alpha ^n$$ α n , $$|\alpha |<1$$ | α | < 1 , when $$n\rightarrow \infty $$ n → ∞ , requires a modification of the Szegö–Widom theorem. A new asymptotic expression for the determinant of a certain class of block Toeplitz matrices with one pair of zero modes is derived. The result is inspired by one-dimensional topological superconductors, and the relation with the latter systems is discussed. 2021-09-20T17:30:47Z 2021-09-20T17:30:47Z 2018-10-17 2020-09-24T21:38:37Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131882 en https://doi.org/10.1007/s10955-018-2177-8 Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Science+Business Media, LLC, part of Springer Nature application/pdf Springer US Springer US |
| spellingShingle | Basor, E. Dubail, J. Emig, T. Santachiara, R. Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes |
| title | Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes |
| title_full | Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes |
| title_fullStr | Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes |
| title_full_unstemmed | Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes |
| title_short | Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes |
| title_sort | modified szego widom asymptotics for block toeplitz matrices with zero modes |
| url | https://hdl.handle.net/1721.1/131882 |
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