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Journal of the Physical Society of Japan

Correlation Function <SziSzj> in Spin-1/2 Alternating Heisenberg-XYAntiferromagnets in One Dimension

Kiyomi Okamoto

Published Mar 15, 1987• 18 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.56.912

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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 spin-1/2 alternating Heisenberg model with X Y -like anisotropy is studied by numerical methods. This model exhibits critical behavior in the ground state as the bond-alternation parameter δ approaches 0. The critical exponents characterizing this ground-state critical point are estimated. The results agree well with those of the analytical calculation of Nakano and Fukuyama. A logarithmic correction of the type predicted by Black and Emery is found to be consistent with numerical data in the isotropic Heisenberg model but not in anisotropic models.
We study the ground-state and finite-temperature properties of the one-dimensional isotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions which compete with each other. Extrapolating the exact results for finite-size systems of up to 20 spins, we estimate the ground-state energy and the ground-state two-spin correlation function in the infinite-size limit. Using the cluster transfer matrix method, we also calculate the temperature dependence of the internal energy and the specific heat and that of the inverse correlation length and the wave number, by both of which the asymptotic behavior of the two-spin correlation function in the long-distance limit is characterized. We compare the results of these calculations with those obtained previously for the case where the nearest-neighbor interaction is antiferromagnetic.
The ground-state properties of the one-dimensional spin-1/2 alternating Heisenberg-Ising antiferromagnet are studied both analytically and numerically. The phase diagram is qualitatively predicted by the use of the phase Hamiltonian: The ground state is either the Néel or the effective singlet state according to the values of the alternation and anisotropy parameters. The phase transition between these states is of the first order. The ground state energy, its derivative with respect to the alternation parameter, excitation gap, and the various long-range order parameters are exactly calculated by the numerical method for finite systems ( N ≦20). The numerical calculations support the analytically predicted phase diagram. The correlation length is also numerically calculated at finite temperatures by the use of the quantum transfer-matrix method.
We compute the ground-state spin-spin correlation function for the one-dimensional antiferromagnetic Heisenberg model of spin 1/2, 1, and 3/2. The spin-1 chain has a disordered ground state with a correlation length of 6.2 lattice spacings whereas the spin-1/2 and the spin-3/2 chains both have (ln(r)${)}^{\mathrm{\ensuremath{\sigma}}}$/r decaying correlation functions. The logarithmic corrections are different for the spin-1/2 and the spin-3/2 chains. Finite-size scaling is used to analyze the data for chains of up to 128 spins.