Two different generalization schemes for the molecular field approximation are described. Both contain the possibility to incorporate generalized effective fields that induce correlations between the degrees of freedom at the boundary of a chosen cluster. Applications are presented for low-dimensional spin-1/2 systems. For the Ising models the critical couplings and critical exponents as well as the spontaneous magnetization curve are determined. An application of the method to the quantum
S=
21 one-dimensional Heisenberg model is presented and reliable low-temperature estimates of the specific heat are evaluated. The data are combined with the quantum transfer matrix predictions for large system sizes and this leads to the following prediction for the low-temperature specific heat: ${C=AT}^{\ensuremath{\alpha}}$ with $\ensuremath{\alpha}=0.35\ifmmode\pm\else\textpm\fi{}0.07$.
We calculate the scaled constant-magnetization and constant magnetic field specific heats $\frac{[{C}_{M}(\ensuremath{\epsilon},M)\ensuremath{-}{C}_{M}(\ensuremath{\epsilon},0)]}{{M}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}}}}$ and $\frac{[{C}_{H}(\ensuremath{\epsilon},H)\ensuremath{-}{C}_{H}(\ensuremath{\epsilon},0)]}{{H}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}\ensuremath{\delta}}}}$ as functions of the scaled variable $x\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}}$ and $y\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{H}^{\frac{1}{\ensuremath{\beta}\ensuremath{\delta}}}}$, respectively [where $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and
Tc is the critical temperature],for the spin-\textonehalf{} Ising model (bcc lattice) and the spin-\textonehalf{}, $\ensuremath{\infty}$ Heisenberg models (fcc lattice). Our calculations are based on previously calculated scaling functions for the
M(H,T) equation of state. We also calculate the amplitudes of the zero-field specific heat for
T>Tc and
T<Tc. Thus, we obtain the functions ${C}_{M}(\ensuremath{\epsilon},M)$ and ${C}_{H}(\ensuremath{\epsilon},H)$, though for the Heisenberg models we cannot obtain the finite nonzero constant
CM(0,0)=CH(0,0). We compare our calculated functions with the data of Connelly, Loomis, and Mapother on nickel.
By combining conventional finite-temperature many-body perturbation theory with cluster expansions, we develop a systematic method to carry out high order arbitrary temperature perturbative calculations on a computer. The method is well suited to studying the thermodynamic properties of quantum disordered and quantum critical phases at finite temperatures. As an application, we calculate the magnetic susceptibility, internal energy, and specific heat of the bilayer Heisenberg model. It is shown that for a wide range of coupling constants these expansions show excellent convergence at all temperatures. Comparing the direct series (without extrapolations) for the bulk susceptibility to quantum Monte Carlo simulations we find an almost perfect agreement between the two methods even at the quantum critical coupling separating the dimerized and antiferromagnetic phases. The convergence fails only at very low temperatures, which are also difficult to reach by quantum Monte Carlo simulations.
The self-consistency conditions inherent in the mean-field approximation are extended to take into account some pair correlations.Ii is shown that this extension is systematically improvable and applicable to the spin systems with short-range interactions in the entire temperature region.With two correlations included and the renormalization-group ideas implemented, the critical properties of the Ising model are satisfactorily revealed.Moreover, the improved estimates of the thermodynamical quantities for the quantum linear XY model from the finite-size calculations are found down to very low temperatures.
We review the estimates for the critical indices of the Ising model below
TC and conclude (for 3 dimensions): ${\ensuremath{\alpha}}^{\ensuremath{'}}=0.066_{\ensuremath{-}0.04}^{}{}_{}{}^{+0.16}$, $\ensuremath{\beta}=0.312_{\ensuremath{-}0.005}^{}{}_{}{}^{+0.002}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.310_{\ensuremath{-}0.05}^{}{}_{}{}^{+0.04}$ for the specific heat, magnetization, and magnetic susceptibility, respectively. For 2 dimensions, we estimate ${\ensuremath{\gamma}}^{\ensuremath{'}}=1.75_{\ensuremath{-}0.00}^{}{}_{}{}^{+0.01}$. In order to explain previous estimates of ${\ensuremath{\alpha}}^{\ensuremath{'}}=0$, we point out that a low power can, in practice, look deceptively like a logarithm. Finally, we discuss the behavior of the specific heat at constant magnetization.