Asymptotic behavior of Toeplitz determinants with a delta function singularity. (17th December 2020)

• Record Type: Journal Article
• Authors: Marić, Vanja; Franchini, Fabio
• Is Part Of: Journal of physics. Volume 54:Number 2(2021)
• Publication Date: 2020-12-17
• Subjects: Wiener–Hopf method -- spin-correlation functions in excited states -- frustrated quantum XY chain -- asymptotic behavior of Toeplitz determinants -- delta function symbols of Toeplitz determinants -- perturbation of a Toeplitz matrix -- correction to Szego theorem; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/abcd55
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We find the asymptotic behaviors of Toeplitz determinants with symbols which are a sum of two contributions: one analytical and non-zero function in an annulus around the unit circle, and the other proportional to a Dirac delta function. The formulas are found by using the Wiener–Hopf procedure. The determinants of this type are found in computing the spin-correlation functions in low-lying excited states of some integrable models, where the delta function represents a peak at the momentum of the excitation. As a concrete example of applications of our results, using the derived asymptotic formulas we compute the spin-correlation functions in the lowest energy band of the frustrated quantum XY chain in zero field, and the ground state magnetization.