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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
Article

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 \(\alpha ^n\), \(|\alpha |<1\), 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.

Keywords

Toeplitz matrices Szegö–Widom theorem Casimir forces Topological superconductors 

Notes

Acknowledgements

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.

References

  1. 1.
    Böttcher, A., Silbermann, B.: Analysis of Toeplitz Operators, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  2. 2.
    Widom, H.: Asymptotic behavior of block Toeplitz matrices and determinants II. Adv. Math. 21(1), 1 (1976)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dubail, J., Santachiara, R., Emig, T.: Critical Casimir force between inhomogeneous boundaries. Eur. Phys. Lett. 112, 66004 (2015)ADSCrossRefGoogle Scholar
  4. 4.
    Dubail, J., Santachiara, R., Emig, T.: Conformal field theory of critical Casimir forces between surfaces with alternating boundary conditions in two dimensions. J. Stat. Mech. 2017, 033201 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    McCoy, B., Wu, T.T.: Theory of Toeplitz determinants and the spin correlations of the two-dimensional ising model. Phys. Rev. 155, 438 (1967)ADSCrossRefGoogle Scholar
  6. 6.
    Szegö, G.: On Certain Hermitian Forms Associated with the Fourier Series of a Positive Function, pp. 223–237. Communications du Séminaire Mathématique de l’Université de Lund, Tome (1952)zbMATHGoogle Scholar
  7. 7.
    Böttcher, A.: One more proof of the Borodin-Okounkov formula for Toeplitz determinants. Integr. Equ. Oper. Theory 41(1), 123 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Basor, E.L., Widom, H.: On a Toeplitz determinant identity of Borodin and Okounkov. Integr. Equ. Oper. Theory 37(4), 397 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kitaev, A.: Unpaired Majorana fermions in quantum wires. Phys. Usp. 44, 131 (2001)ADSCrossRefGoogle Scholar
  10. 10.
    Kitaev, A.: Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22 (2009)ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Ryu, S., Schnyder, A.P., Furusaki, A., Ludwig, A.W.W.: Topological insulators and superconductors: tenfold way and dimensional hierarchy. New J. Phys. 12, 065010 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Kennedy, R., Zirnbauer, M.: Bott periodicity for \(\mathbb{Z}_2\) symmetric ground states of gapped free-fermion systems. Commun. Math. Phys. 342, 909 (2016)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Basor, E., Bleher, P.: Exact solution of the classical dimer model on a triangular lattice: monomer-monomer correlations. Commun. Math. Phys. 356(2), 397–425 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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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