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Journal of the Physical Society of Japan

Longitudinal Spin Correlation in Spin-1/2 DimerizedXYChain. II. Finite Temperature

Kiyomi Okamoto

Published Jun 15, 1989• 3 citations
Publisher

Abstract

We exactly calculate the long-distance asymptotic behavior of the longitudinal spin correlation \(R{\equiv}\langle S_{\imath}^{z}S_{\jmath}^{z}\rangle\) 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.

DOI
10.1143/jpsj.58.2004

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We exactly calculate the long-distance asymptotic behavior of the longitudinal spin correlation (R{\equiv}\langle S_{\imath}^{z}S_{\jmath}^{z}\rangle) 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.
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.
The long-time and long-distance asymptotic behavior of the xxx 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.
The long-time and long-distance asymptotic behavior of the xxx 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 investigate the long-distance asymptotic behavior of the dimer correlations in the spin-1/2 alternating XYXYXYchain 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.
We review some known properties of the "emptiness formation probability" correlation which we have calculated numerically for spin-1/2 XX chains with constant (homogeneous) or alternating (dimerized) nearestneighbor coupling and an external field (in z direction) for arbitrary temperature.The long-distance asymptotic behavior of this correlation is known to be Gaussian at zero temperature and exponential at finite temperature for the homogeneous chain.By simple analytical arguments the exponential behavior at finite temperature extends to the dimerized system.Numerical results for the dimerized chain confirm the exponential decay at finite temperature and show Gaussian decay at zero temperature.
The decay of (disorder-averaged) static spin correlation functions at T=0 for the one-dimensional spin-1/2 XXZ antiferromagnet with uniform longitudinal coupling JΔ and random transverse coupling Jλi is investigated by numerical calculations for ensembles of finite chains. At Δ=0 (XX model) the calculation is based on the Jordan-Wigner mapping to free lattice fermions for chains with up to N=100 sites. At Δ≠0 Lanczos diagonalizations are carried out for chains with up to N=22 sites. The longitudinal correlation function 〈Sz0Szr〉 is found to exhibit a power-law decay with an exponent that varies with Δ and, for nonzero Δ, also with the width of the λi-distribution. The results for the transverse correlation function 〈Sx0Sxr〉 show a crossover from power-law decay to exponential decay as the exchange disorder is turned on.