The mean-field renormalisation group of Indekeu et al. (1982) is applied to the antiferromagnetic nearest-neighbour Ising model on the triangular lattice. The resulting phase diagram in the temperature-field plane is in good agreement with other calculations, while the predicted specific-heat index, alpha , is negative and thus qualitatively wrong. The study indicates that the method may be a useful approach, at least for determining phase diagrams, for frustrated systems and systems with competing interactions where conventional mean-field theory is dubious.
Phase diagrams of two inhomogeneous annealed Ising models on the triangular lattice are constructed in an exact way. The interimpurity exchange constants (bonds), of the same but arbitrary value and sign, can be located along the horizontal edges of the triangles. In the first model the remaining non-random couplings are ferromagnetic and in the second they are antiferromagnetic. The dependence of the correlation functions and magnetisation on the temperature is calculated and discussed. It is shown that, for certain values of the impurity bond couplings and a range of their concentrations, there is no net magnetic moment at T=0 and low temperatures. The magnetisation appears at some finite temperature and then vanishes at a higher one. This increase of the magnetisation with the temperature is connected with the role of entropy in determining the minimum of the free energy.
The partition function of the Ising model with
N spins, written as a functional integral, is approximated by replacing the field ${\ensuremath{\psi}}_{l}$ at the lattice site
l by the root-mean-square value of these fields, and evaluating the integral by a saddle-point method. A phase transition, whose mechanism is mathematically similar to that of the spherical model, is predicted at a temperature lower than the Weiss critical temperature. After parametrizing the mean-square field, one may apply a general variational principle to choose the parameters which minimize the difference between the approximate free energy and the exact Ising-model free energy. This optimum choice leads to the Horwitz-Callen theory.
The authors study a two-dimensional Ising model on a triangular lattice which is divided into two sublattices. One consists of a honeycomb of sites and the other of the remaining interstitial sites. The interaction energies within the honeycomb and between the honeycomb and the interstitial sites differ from each other. The method used consists of a combination of transfer matrix techniques and the phenomenological renormalisation procedure. Results are presented for the critical temperature and exponents in zero field and the phase diagram in the temperature-field plane is obtained.
We study a triangular Ising model subject to nearest-neighbor pair interactions
K, an external magnetic field
H, and triangular three-spin interactions
P.
K>0(<0) corresponds to ferromagnetic (antiferromagnetic) coupling. The phase diagram and thermodynamics are invariant under $(K,H,P)\ensuremath{\leftrightarrow}(K,\ensuremath{-}H,\ensuremath{-}P)$. A simple position-space renormalization group is formulated, based on three interpenetrating three-spin cells, which preserves the three-sublattice symmetry of the antiferromagnetic phase. The complete (
K,H,P) phase diagram consists of five interconnecting transition surfaces (two second-order and three first-order), which bound the four single-phase regions (\ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\uparrow}, \ensuremath{\uparrow}\ensuremath{\uparrow}\ensuremath{\downarrow}, \ensuremath{\uparrow}\ensuremath{\downarrow}\ensuremath{\downarrow}, and \ensuremath{\downarrow}\ensuremath{\downarrow}\ensuremath{\downarrow}). The edges of these surfaces form loci of second-order, critical-endpoint, and Baxter-Wu transitions. When
P=0, this system is known to provide a useful model of the order-disorder transition in monolayer films adsorbed on a hexagonal substrate. Calculation of the magnetization
M(K,H) and heat capacity
cM(K,H) of the Ising model allows quantitative comparison with experimental data for helium on graphite. The observed dependence
Tc(n) of the transition temperature on coverage is well fit with a single effective antiferromagnetic pair interaction. The corresponding calculated heat capacities (no additional adjustable parameters) are in remarkable agreement with experimental data near and below
Tc.