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

Abstract

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.