Dynamic 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 Ξₙ(t)=4〈S₀^ξ(t)Sₙ^ξ〉, ξ=x,y, at zero temperature of the one-dimensional S=1/2 isotropic XY model (h=γ=0) and of the transverse Ising (TI) model at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=-JΣl^[(1+γ)SₗˣSₗ₊₁ˣ+(1-γ)SₗʸSₗ₊₁ʸ+hS₁ᶻ]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions Ξ₀(0) and on the singularities of their frequency-dependent Fourier transforms Φ₀^ξξ(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φ₀^ξξ(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 the TI case, the leading singularities in φ₀ˣˣ(ω) are alternately one-sided and two-sided power-law singularities, the first two of which (at ω=0,2J) are divergent. The dominant singularities in the XY case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at ω=0,J) are divergent. The singularities at higher frequencies in both models are finite and 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 (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the XY-like substances Cs₂CoCl₄ and PrCl₃ and the S=1/2 Ising-like substance CsCoCl₃{{·{}}2H}₂$O.