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Journal of the Physical Society of Japan

Temperature Dependence of the Correlation Length of <SizSjz> in Spin-1/2 Alternating Heisenberg-XYAntiferromagnets in One Dimension

Kiyomi Okamoto

Published May 15, 1987• 6 citations
Publisher

Abstract

We conjecture the temperature dependence of the correlation length ξ of the spin-spin correlation function < S i z S j z > in spin-1/2 alternating Heisenberg- X Y antifer-romagnets. We use the phase representation and the self-consistent harmonic approx-imation to evaluate < S i z S j z >. The correlation length ξ as a furtction of the temperaturehas a peak at T ∼ ε g , where ε g is the energy gap between the ground state and the first excited state.

DOI
10.1143/jpsj.56.1627

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We conjecture the temperature dependence of the correlation length ξ of the spin-spin correlation function < S i z S j z > in spin-1/2 alternating Heisenberg- X Y antifer-romagnets. We use the phase representation and the self-consistent harmonic approx-imation to evaluate < S i z S j z >. The correlation length ξ as a furtction of the temperaturehas a peak at T ∼ ε g , where ε g is the energy gap between the ground state and the first excited state.
We calculate the correlation function < S z i S z j > in the ground state of one dimensional spin-1/2 alternating antiferromagnetic Heisenberg model, taking the X Y -like anisotropy into consideration. We use the phase Hamiltonian method developed by Nakano and Fukuyama and apply the self-consistent harmonic approximation to the non-harmonic terms in the phase Hamiltonian. The long-distance behavior of the correlation function is< S z i S z j >∼( x /ξ) -1/2 exp (- x /ξ), where x is the distance between the i-th and j-th spins. The correlation length ξ is proportional to the inverse of the energy gap between the ground state and the first excited state. The results are cornpared with those of the exact numerical analysis of finite size systems and the Monte Carlo simulations.
The ground state properties of the one-dimensional spin-1/2 alternating Heisenberg- XY ferromagnet are analytically investigated by the use of the phase Hamiltonian representation. To treat the non-harmonic terms in the phase Hamiltonian, both the self-consistent harmonic approximation (SCHA) and the renormalization group method are applied. The ground state is either the effective singlet (ES) state or the so-called spin-fluid state. The phase diagram is depicted on the Δ -δ plane, where Δ and δ are the anisotropy and alternation parameters, respectively. The longitudinal and transverse spin correlation functions (< S i z S j z > and < S i + S j - >) in the ES state are also discussed in the framework of the SCHA.
The thermal evolution of equal-time spin correlations in the antiferromagnetic isotropic spin-1/2 Heisenberg chain is examined by full exact diagonalization of a ten-site periodic cluster. The calculation retains the complete spectrum and evaluates canonical ensemble averages without decoupling approximations. Correlations through the maximum independent separation are reported over 0.10 ≤ kBT/J ≤ 5. The low-temperature state displays the expected alternating antiferromagnetic pattern and appreciable correlations across the finite ring. Heating suppresses the magnitude of every nonlocal correlator, with longer-range terms disappearing first. The nearest-neighbour correlation approaches the high-temperature limit continuously, while an effective finite-cluster correlation length falls rapidly around kBT/J of order unity. Low- and high-temperature consistency checks, SU (2) symmetry, and comparison with Bethe-ansatz expectations are used to validate the computation. The results provide a compact reproducible reference for interpreting thermal loss of short-range order, while clearly separating finite-size numerical evidence from thermodynamic-limit conclusions.
We investigate the thermal properties of s=1/2 and 1 antiferromagnetic Heisenberg spin chains by an expansion in finite lattices. The expansion is based on expressing the quantity of interest, in the thermodynamic limit, as a sum over contributions from different length scales. The first few terms in the expansion, obtained by diagonalizing chains of length up to 13 for s=1/2 and length up to 9 for s=1, provide an accurate estimate for the internal energy and specific heat for T/J\ensuremath{\ge}0.25 and 0.35, respectively. At lower temperatures, by studying the exponential decay of the various terms in the expansion, we estimate the correlation length \ensuremath{\xi}. We get good estimates for \ensuremath{\xi} down to T/J=0.06 for s=1/2 and T/J=0.02 for s=1. We find that for s=1 \ensuremath{\xi} increases much more slowly than for s=1/2 and is consistent with a value of \ensuremath{\xi}\ensuremath{\approxeq}7 at T=0.