The antiferromagnetic spin-
1∕2 chain with power-law decaying interactions $\ensuremath{\mid}{J}_{n}\ensuremath{\mid}\ensuremath{\propto}1∕{n}^{p}$ is studied by a Green's function method. The dispersion relation ${\ensuremath{\omega}}_{k}$ and an expression for the critical temperature
TN as a function of
p are presented. By analyzing the small-
k behavior of ${\ensuremath{\omega}}_{k}$, it is shown that there will be no long-range order at finite temperature when $p\ensuremath{\geqslant}2$, which is in good agreement with Bruno's rigorous proof. Our work refurbishes a previous work also by Green's function method, and validates the Tyablikov decoupling approximation to long-range interactions with
1<p<2.
By the Green’s function method, where spin operators are transformed to fermion operators, an antiferromagnetic Heisenberg chain system has been investigated within the Hartree-Fock approximation. The static correlation ≪SzkSz-k> cannot be reduced to the Ornstein-Zernike form and has no singularity at k=π. The absorption function χzz’’(k,ω) cannot be expressed by the δ-function-like spectrum even at 0°K, but rather by a narrow diffused one, which is due to the two-magnon process. The present dispersion relation has a gap at k=π for T=0°K. The second order phase transition, then, may not occur in the linear chain. The results obtained are discussed comparing with those in the X-Y model and experimental data on TMMC.
The one-dimensional spin-1 Heisenberg antiferromagnetic chains are studied by a generalized two-time Green's-function method at finite temperature. 〈
Siz〉 is set to zero at each site to guarantee the lack of the long-range order in the one-dimensional Heisenberg system. The Green's functions are decoupled in terms of the correlation functions of spin operators. A set of self-consistent equations of the correlation functions are derived and solved numerically. Correlation functions of spin operators, thermodynamic properties such as the internal energy, and specific heat in the entire temperature region are obtained. The Haldane gap in the excitation spectrum appears naturally in the analytical result. When k is zero, there is a gap 2\ensuremath{\Delta}, which is 1.0J and close to the theoretical result of White and Huse. And there is a broad maximum in the temperature dependence of the specific heat. The properties of the thermodynamic quantities are consistent with the numerical results on the finite chains. \textcopyright{} 1996 The American Physical Society.