Asymptotic behavior of Toeplitz determinants with a delta function singularity

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.

References in corpus (7)

• Quantum transitions driven by one-bond defects in quantum Ising rings
• Quantum Ising chains with boundary fields
• Quantum Phase Transition induced by Topological Frustration
• The A-Cycle Problem for Transverse Ising Ring
• The Frustration of being Odd: How Boundary Conditions can destroy Local Order
• The Frustration of being Odd: Universal Area Law violation in local systems
• Rigorous proof for the non-local correlation functions in the antiferromagnetic seamed transverse Ising ring

Cited by in corpus (6)

• The quantum Ising chain for beginners
• Effects of defects in the XY chain with frustrated boundary conditions
• Resilience of the topological phases to frustration
• Topological Frustration can modify the nature of a Quantum Phase Transition
• Asymptotic behaviour of determinants through the expansion of the Moyal star product
• The fate of local order in topologically frustrated spin chains