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

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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.

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Correspondence to T. Emig.

Appendix: More Examples of Block Toeplitz Matrices with \(E(\phi ) = 0\): Other Symmetry Classes of Topological Superconductors and Insulators

Appendix: More Examples of Block Toeplitz Matrices with \(E(\phi ) = 0\): Other Symmetry Classes of Topological Superconductors and Insulators

At this point, we would like to come back to Example 1. We note that it does not belong to symmetry class D because it has a symbol that is block anti-diagonal. In fact, in physics, there is another symmetry class for symbols like that in Example 1, called class BDI [10]. This class is the one of real anti-symmetric matrices A for which there exists a unitary matrix M that squares to the identity, \(M^2 = I_{Nn}\), such that

$$\begin{aligned} M^\dagger A M \, = \, - A , \end{aligned}$$

a property that encodes time-reversal symmetry. Such a property implies that, up to a change of basis, the symbol corresponding to A can be transformed in block anti-diagonal form

$$\begin{aligned} \phi (\theta ) \, = \, \left( \begin{array}{c|c} 0 &{} B(e^{i \theta }) \\ \hline - B^\dagger (e^{i \theta }) &{} 0 \end{array} \right) \, . \end{aligned}$$
(11)

with \(B^* (e^{-i \theta }) = B(e^{i \theta })\). In class BDI, it is not Kitaev’s \(\mathbb {Z}_2\) invariant that decides whether or not there are pairs of zero modes. Instead, the number of such pairs is counted by the (absolute value of the) winding of \(\det B(e^{i \theta })\), which we denote as \(\mathcal {I}_\mathrm{BDI}\). The winding \(\mathcal {I}_\mathrm{BDI}\) is an arbitrary integer number, and so the number of pairs of zero modes can be larger than 1. For instance, consider the

Example 1b

Let u be a real number, and

$$\begin{aligned} \phi (\theta ) \, = \, \left( \begin{array}{cc} 0 &{} 1-u e^{-i 2 \theta } \\ -1+u e^{i 2\theta } &{} 0 \end{array} \right) . \end{aligned}$$
(12)

The corresponding Toeplitz matrix

$$\begin{aligned} T_n(\phi ) \, = \, \left( \begin{array}{c|c|c|c|c|c} \begin{array}{cc} 0 &{} 1 \\ -1 &{} 0 \end{array} &{} 0 &{} \begin{array}{cc} 0 &{} 0 \\ u &{} 0 \end{array} &{} 0 &{} \dots &{} 0 \\ \hline 0 &{} \begin{array}{cc} 0 &{} 1 \\ -1 &{} 0 \end{array} &{} 0&{} \begin{array}{cc} 0 &{} 0 \\ u &{} 0 \end{array} &{} \dots &{} \\ \hline \begin{array}{cc} 0 &{} -u \\ 0 &{} 0 \end{array} &{} 0 &{} \begin{array}{cc} 0 &{} 1 \\ -1 &{} 0 \end{array} &{}0&{} \ddots &{} \vdots \\ \hline 0 &{} \begin{array}{cc} 0 &{} -u \\ 0 &{} 0 \end{array} &{} 0 &{} \ddots &{} \ddots &{} \begin{array}{cc} 0 &{} 0 \\ u &{} 0 \end{array} \\ \hline \vdots &{} \vdots &{} \ddots &{} \ddots &{} \ddots &{} 0 \\ \hline 0 &{} &{} \dots &{} \begin{array}{cc} 0 &{} -u \\ 0 &{} 0 \end{array} &{} 0 &{} \begin{array}{cc} 0 &{} 1 \\ -1 &{} 0 \end{array} \end{array} \right) \end{aligned}$$

has determinant 1, for all u. For \(u \notin \{ 1,-1 \}\), one has \(\det \phi (\theta ) \, = \, 1 + u^2 -2 u \cos 2\theta > 0\). Kitaev’s \(\mathbb {Z}_2\) index is equal to \(\mathcal {I}_\mathrm{D} \, =\, +1\) but is does not provide sufficient information on the existence of zero modes. Instead, the proper index to use is \( \mathcal {I}_\mathrm{BDI}\), the winding of \(\det B (e^{i \theta }) = 1 - u e^{-i 2 \theta }\), which equals \(- 2\) if \(|u| > 1\). Therefore, there are two pairs of zero modes in this example, when \(|u|>1\).

The factorization corresponding to Example 2 in Sect. 1 also illustrates why there is difficulty with the original Szegö theorem. However here the special structure does allow one to find the formulas. Here is the factorization:

$$\begin{aligned} \phi = P_{1} P_{2} P_{3} P_{4} \end{aligned}$$

where

$$\begin{aligned} P_{1} = \left( \begin{array}{c@{\quad }c}1&{}0\\ {} &{}\\ 0&{}1- e^{-i 2\theta }/u\end{array}\right) , \quad P_{2} = \left( \begin{array}{c@{\quad }c}0&{}u\\ {} &{}\\ u&{}0\end{array}\right) , \quad P_{3} = \left( \begin{array}{c@{\quad }c}e^{i2\theta }&{}0\\ {} &{}\\ 0&{}e^{-i2\theta }\end{array}\right) \end{aligned}$$

and

$$\begin{aligned} P_{4} = \left( \begin{array}{c@{\quad }c}1&{}0\\ {} &{}\\ 0&{}-1 + e^{i2\theta }/u\end{array}\right) . \end{aligned}$$

The above examples illustrate that, in order to determine whether a block-Toeplitz matrix has zero modes, one needs to carefully identify the symmetries of the matrix. Different symmetries will lead to different topological invariants that count the zero modes, and to different asymptotics of block-Toeplitz determinants. In the three example considered so far, we dealt with real antisymmetric matrices, related to 1d superconductors. Looking at the topological classification of topological insulators and superconductors in Refs. [10, 11], it is clear that there are other symmetry classes that host non-trivial topological phases of 1d insulators or superconductors (in 1d, those classes are: BDI, D, DIII, CII, AIII), and therefore are associated to block Toeplitz matrices with a particular structure that will correspond to cases where \(E(\phi ) = 0\) in the Szegö–Widom theorem.

We give one last example, related to a 1d insulator in symmetry class AIII. This is the class of hermitian matrices H that, up to a change of basis, are of block anti-diagonal form

$$\begin{aligned} \left( \begin{array}{c|c} 0 &{} B(e^{i\theta }) \\ \hline B^\dagger (e^{i \theta }) &{} 0 \end{array} \right) \, . \end{aligned}$$

Similarly to the class BDI, this may be viewed as a consequence of the existence of a unitary matrix M that squares to the identity, \(M^2 = I_{Nn}\), such that

$$\begin{aligned} M^\dagger H M \, = \, - H , \end{aligned}$$
(13)

a property often dubbed “chiral symmetry” or “sub-lattice symmetry” in physics. In class AIII, the number of pairs of zero modes is counted by the winding of \(\det B(e^{i \theta })\), which defines an index \(\mathcal {I}_{\mathrm{AIII}}\) that takes integer values. An example belonging to class AIII is the following.

Example 3

Let \(\zeta \in \mathbb {C}\), and

$$\begin{aligned} \phi (\theta ) \, = \, \left( \begin{array}{cc} 0 &{} 1+ \zeta ^* e^{-i \theta } \\ 1+ \zeta e^{i \theta } &{} 0 \end{array} \right) , \end{aligned}$$

associated to the Toeplitz matrix

$$\begin{aligned} T_n(\phi ) = \left( \begin{array}{c|c|c|c|c|c} \begin{array}{cc} 0 &{} 1 \\ 1 &{} 0 \end{array} &{} \begin{array}{cc} 0 &{} 0 \\ \zeta &{} 0 \end{array} &{} 0 &{} &{} \dots &{} 0 \\ \hline \begin{array}{cc} 0 &{} \zeta ^* \\ 0 &{} 0 \end{array} &{} \begin{array}{cc} 0 &{} 1 \\ 1 &{} 0 \end{array} &{} \begin{array}{cc} 0 &{} 0 \\ \zeta &{} 0 \end{array} &{} 0 &{} \dots &{} \\ \hline 0 &{} \begin{array}{cc} 0 &{} \zeta ^* \\ 0 &{} 0 \end{array} &{} \begin{array}{cc} 0 &{} 1 \\ 1 &{} 0 \end{array} &{} \begin{array}{cc} 0 &{} 0 \\ \zeta &{} 0 \end{array} &{} 0 &{} \vdots \\ \hline &{} 0 &{} \begin{array}{cc} 0 &{} \zeta ^* \\ 0 &{} 0 \end{array} &{} \ddots &{} \ddots &{} \vdots \\ \hline \vdots &{} \vdots &{} 0 &{} \ddots &{} \ddots &{} \begin{array}{cc} 0 &{} 0 \\ \zeta &{} 0 \end{array} \\ \hline 0 &{} &{} &{} \dots &{} \begin{array}{cc} 0 &{} \zeta ^* \\ 0 &{} 0 \end{array} &{} \begin{array}{cc} 0 &{} 1 \\ 1 &{} 0 \end{array} \end{array} \right) \, . \end{aligned}$$

that has determinant 1. One has \(\det \phi ( \theta ) > 0\) for \(|\zeta | \ne 1\). It is easy to compute \(G(\phi )\) of the Szegö–Widom theorem; one finds \(G(\phi ) = \mathrm{max}[1, |\zeta |^2]\). Thus \(E(\phi )\) is in fact zero if \(|\zeta | > 1\). This is consistent with the observation that the index \(\mathcal {I}_\mathrm{AIII}\) (the winding of \(\det B(e^{i \theta })\)) equals \(-1\) for \(|\zeta |>1\).

The factorization for this example is almost identical to Example 1b except that \(2 \theta \) is replaced by \(\theta \). And as a final remark, the determinants of both of these examples follow from the remarks on page 5. Because of the specific structure of the matrices, both reduce to finding the asymptotics of two scalar functions with non-zero winding.

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Basor, E., Dubail, J., Emig, T. et al. Modified Szegö–Widom Asymptotics for Block Toeplitz Matrices with Zero Modes. J Stat Phys 174, 28–39 (2019). https://doi.org/10.1007/s10955-018-2177-8

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Keywords

  • Toeplitz matrices
  • Szegö–Widom theorem
  • Casimir forces
  • Topological superconductors