Dynamical correlation functions for one-dimensional quantum spin systems: New results based on a rigorous approach

We present new results on the time-dependent correlation functions Ξ_n(t) =4〈S^ξ₀(t)S^ξ _n〉, ξ=x,y at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising model (TI) at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=-J∑_l[(1+γ)S^x _lS^x_l+1 +(1-γ)S^y _lS^y_l+1 +hS^z_l]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions (ACF's) Ξ₀(t) and on the singularities of their frequency-dependent Fourier transforms φ^ξξ ₀(ω). We have also determined the latter functions by high-precision numerical calculations. The functions φ ^ξξ₀(ω), ξ=x,y have singularities at the infinite sequence of frequencies ω=mω₀, m=0, 1, 2, 3,. where ω₀=J for the XY model and ω₀=2J for the TI model. In both models the singularities in φ^xx₀ (ω) for m=0, 1 are divergent, whereas the nonanalyticities at higher frequencies become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis.