Journal of Statistical Physics

, Volume 174, Issue 1, pp 28–39 | Cite as

Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes

  • E. Basor
  • J. Dubail
  • T. EmigEmail author
  • R. Santachiara


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 \(\alpha ^n\), \(|\alpha |<1\), when \(n\rightarrow \infty \), 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.


Toeplitz matrices Szegö–Widom theorem Casimir forces Topological superconductors 



We would like to thank E. Ardonne, K. Kozlowski, J.-M. Stéphan and D. Vodola for inspiring discussions. This work was partly supported by the A*MIDEX Project ANR-11-IDEX-0001-02 cofunded by the French program Investissements d’Avenir, managed by the French National Research Agency.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.American Institute of MathematicsSan JoseUSA
  2. 2.Laboratoire de Physique et Chimie Théoriques, CNRS UMR 7019Université de LorraineVandoeuvre-les-NancyFrance
  3. 3.Laboratoire de Physique Théorique et Modèles Statistiques, CNRS UMR 8626Université Paris-Saclay, Université Paris-SudOrsay CedexFrance
  4. 4.Joint MIT-CNRS Laboratory UMI 3466Massachusetts Institute of TechnologyCambridgeUSA

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