The long-time and long-distance asymptotic behavior of the
x spin correlations at finite temperature in an anisotropic spin-1/2 XY chain is determined numerically. The decay of the correlations is exponential in both space and time. Similar exponential decay of correlations was already found earlier in the special case of the isotropic model, where analytical expressions for the decay rates could be derived via a mapping to a different model. While no such mapping is known for the anisotropic model, the asymptotic correlations can be very well approximated by a natural generalization of the known analytic results for the isotropic case.
We exactly calculate the long-distance asymptotic behavior of the longitudinal spin correlation
R≡⟨SzSz⟩ of the spin-1/2 dimerized XY chain at finite temperatures. The spin correlation R decays in the exponential law with the correlation length ξ( T ), which decreases with the increase of T . The present results are compared with those obtained by the use of the phase Hamiltonian method.
We investigate the long-distance asymptotic behavior of the dimer correlations in the spin-1/2 alternating
XYchain both at T=0 and at sufficiently low-temperatures. The correlations consist of the dimer long-range order part and the exponentially decaying one. Although the dimer long-range order takes the different values depending on the choice of the spin pairs, the behavior in the decaying term is same irrespective of the choice of spin pairs.
The long-distance asymptotic behavior of the longitudinal spin correlation < S i z S j z > in the ground-state of the spin-1/2 dimerized XY chain is exactly calculated. The spin correlation < S i z S j z > decays in the exponential law with the correlation length ζ=1/ln [(1+δ)/(1-δ)]. Here δ (0≦δ≦1) represents the degree of alternation: i.e., the stronger and the weaker spin couplings are expressed as J (1+δ) and J (1-δ), respectively. The correlation length ξ smoothly changes from ∞ (power decay) to 0 as we move away from δ=0 (no alternation) to δ=1 (independent spin pairs). The nearest neighbor spin correlation is also investigated. The present results are compared with those obtained by the use of the phase Hamiltonian method.
We consider the one-dimensional $X\ensuremath{-}Y$ model of Lieb, Schulz, and Mattis and study the asymptotic behavior of each of the three correlation functions $〈{{\ensuremath{\sigma}}_{0}}^{i}{{\ensuremath{\sigma}}_{N}}^{i}〉={{\ensuremath{\rho}}_{N}}^{i}$, where
i=x,
y, or
z. We study in detail the influence of $X\ensuremath{-}Y$ anisotropy by separately studying the correlation functions in both the isotropic and anisotropic cases at both nonzero and zero temperatures. For nonzero temperature we derive both low- and high-temperature expansions for all three correlations and show that these correlations go to zero exponentially as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The behavior near
T=0 is studied in the isotropic case by considering the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit with
TN fixed, while in the anisotropic case we must hold
T2N fixed as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. In this manner we obtain the
T=0 result that if the interaction is stronger in the
x direction, then ${{\ensuremath{\rho}}_{N}}^{x}$ approaches a constant exponentially while ${{\ensuremath{\rho}}_{N}}^{y}$ approaches zero exponentially as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We finally show that in the isotropic case at
T=0 that ${{\ensuremath{\rho}}_{N}}^{x}={{\ensuremath{\rho}}_{N}}^{y}\ensuremath{\rightarrow}{N}^{\frac{\ensuremath{-}1}{2}}$. In all cases, at least the first two terms of the asymptotic series are explicitly given.