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.
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.
Real space renormalisation group techniques and the replica method of Edwards and Anderson (1975), combined with successive variational approximations going beyond mean field, are used to calculate phase diagrams of various Ising systems. For a bond dilute 2D Ising model with nearest-neighbour interactions, real-space renormalisation group techniques provide very precise results which are used to test the reliability of the replica method. Combined with variational approximations going beyond mean field, this last method is applied to 3D Ising models with competing nearest-neighbour and next-nearest-neighbour interactions.
Positions of critical temperatures of antiferromagnetic Ising model with ferromagnetic next nearest neighbour interactions on the triangular lattice are investigated by making use of their properties of the Kosterlitz-Thouless type of phase transition. In particular, finite-size scaling method and Monte Carlo Renormalization Group methods (MCRG) with flow diagrams of the anomalous dimension η show the fixed line structure where the correlation length diverges. Another MCRG with flow diagram of the stiffness constant also shows a critical property clearly. The temperature dependence of the long range order is also discussed in order to locate the lower critical point.
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.
An antiferromagnetic Ising system on a triangular lattice is studied by means of a transfer matrix and the phenomenological scaling method. System consists of a semi-infinite strip and helical boundary conditions are imposed at the end of each row. At external magnetic fields h =0 and 6 J , where J is a nearest neighbor coupling constant, a unite residual entropy is obtained at the zero temperature. With a small field, specific heat is found to have two peaks; a high-temperature peak corresponding to the enhanced short-range order, and a low-temperature peak to the appearance of a long-range order. A phase diagram in h - T space is obtained by means of the phenomenological scaling, and agrees fairly well with the one obtained by Monte Carlo simulation. The main deficiency is the finite transition temperature at h =0, which may be due to a crossover to the XY-like phase transition.
The renormalisation group method of Kadanoff is applied to the spin- 1 / 2 Ising model in two dimensions using a transformation applicable to both the square and triangular lattices. Two types of cluster are studied for the triangular lattice.The first cluster yields a new fixed point but the results for the critical temperature and critical exponents are in poor agreement with the known exact values. The second cluster maps directly onto the fixed point found by Kadanoff for the square lattice yielding excellent values for both the critical exponents and the critical temperature.