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

We compute exactly the transverse time-dependent spin-spin correlation functions 〈Sx1(0)SxR+1(t)〉 and 〈Sy1(0)SyR+1(t)〉 at zero temperature for the one-dimensional XY model that is defined by the hamiltonian HN = - ΣNi=1 [(1 + γ)SxiSxi+1 + (1 − γ)SyiSyi+1 + hSzi]. We then analyze these correlation functions in two scaling limits: (a) γfixed, h → 1, R → ∞, t → ∞ such that ‖(h − 1)/γ‖[R2 − γ2t2]12 is fixed, and (b) h fixed less than one, γ → 0+, R → ∞, t → ∞ such that γ[R2 − (1 − h2)t2]12 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.