Transverse time-dependent spin correlation functions for the one-dimensional XY model at zero temperature

Abstract

We compute exactly the transverse time-dependent spin-spin correlation functions 〈 S x 1(0) S x R+1 ( t)〉 and 〈 S y 1(0) S y R+1 ( t)〉 at zero temperature for the one-dimensional XY model that is defined by the hamiltonian H N = - Σ N i=1 [(1 + γ)S x iS x i+1 + (1 − γ)S y iS y i+1 + hS z i] . We then analyze these correlation functions in two scaling limits: (a) γ fixed, h → 1, R → ∞, t → ∞ such that ‖(h − 1)/γ‖[R 2 − γ 2t 2] 1 2 is fixed, and (b) h fixed less than one, γ → 0 +, R → ∞, t → ∞ such that γ[R 2 − (1 − h 2)t 2] 1 2 is fixed. In these scaling regions we give both a perturbation expansion representation of the various scaling functions and we express these scaling functions in terms of a certain Painlevé transcendent of the third kind. From these representations we study both the small and large scaling variable limits in both the space-like and time-like regions.