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各向異性海森堡模型的臨界性質
周東川
ABSTRACT
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We consider the spin-（圖表省略）
anisotropic Heisenberg model described by the Hamiltonian
（圖表省略）
This hamiltonian corresponds to the Ising, Heisenberg and XY models
when（圖表省略）
respectively. Exact high temperature series expansions are derived to
order T-7'' for the second-order fluctuations and to order T-6 for the
fourth-order fluctuations in the (ferromagnetic order parameters) for an
arbitrary lattice, and are derived to T-7 for the secondorder fluctuations
in the antiferromagnetic order parameters for open lattices. These series
are analysed by various methods to give the critical temperatures and the
critical exponents for all values of J∥and j┴.
For a given lattice, when the anisotropy varies from the Ising limit
(J┴=0) to the isotropic limit (│J∥│=│J┴│), the critical temperature
decreases slowly (near zero slop) in the Ising limit and rapidly in the
isotropic limit. Similar behavior is found when the system varies from the
XY limit to the isotropic limit.
For the cubic lattices our estimates of the critical exponents are
consistent with the universality hypothesis. The susceptibility exponents
γ for the cubic lattices and the staggered susceptibility exponents γN
for the bcc and sc lattices have the values γ（圖表省略）γN（圖表省略）
1.38, 1.25, and 1.31 when the system is isotropic (J∥=│J┴│),
Ising-like (│J┴│<│J∥│) and XY-like (│J∥│<│J┴│), respectively.
For the cubic lattices we also find that the gap exponent △4=1.81, 1.56,
ans 1.66 when 0<J┴=J∥,│J┴│<J∥, and │J∥│<J┴ respectively.
For two-dimensional lattices, it is difficult to obtain reliable estimates
of the critical temperature from the series expansions for the
second-order fluctuation, since the series coefficients are irregular in
the isotropic parameters, we find an evidence in favor of a phase
transition in the twodimensional lattices for all positive values of J∥
and J1.
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