In the framework of the spin flip dynamics algorithm with an original expression for the energy, an influence of the interaction of second, third neighbors, four-particle interaction, temperature and size of a 1D Ising ferromagnet on the heat capacity critical exponent is studied. Periodic border conditions are used. The critical index α is calculated by the finite size scaling method. It is established that the growth of energy parameters of the multi-spin interaction significantly affects the heat capacity exponent. Interaction energy of third neighbors decreases the value of α to a greater extent than the second-neighbor interaction J2 and four-particle one J1 – 4. Different characters of the influence of pair and four-particle interactions on α are revealed. Energy parameters are established for which this critical exponent attains negative values. For instance, taking into account the interaction energy of third neighbors, with an approach to the critical region, the heat capacity critical exponent has a minimum near negative value of α = –0.0043 ± 0.0005. With increasing number of nodes in the 1D Ising chain, heat capacity critical exponent increases. Extrapolation according to the inverse size of the system in the thermodynamic limit gives the values α = 0.19 ± 0.02, α = 0.17 ± 0.02, α = 0.20 ± 0.02, when one takes into account the interaction energy of second neighbors J2 = 0.1, third neighbors J3 = 0.1 and four-particle interaction J1 – 4 = 0.1, respectively. Note that the second neighbor and four-particle interaction energies do not change the sign of the critical exponent for N > 6, and when approaching the critical region the critical exponent α tends to zero. The results obtained are compared with the values for Ising model with "broken ends" boundary conditions, as well as with other known results for two-dimensional and three-dimensional cases.
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.
A three-dimensional Ising model with the plaquette-type (next-nearest-neighbor and four-spin) interactions is investigated numerically. This extended Ising model, the so-called gonihedric model, was introduced by Savvidy and Wegner as a discretized version of the interacting (closed) surfaces without surface tension. The gonihedric model is notorious for its slow relaxation to the thermal equilibrium (glassy behavior), which deteriorates the efficiency of the Monte Carlo sampling. We employ the transfer-matrix (TM) method, implementing Novotny's idea, which enables us to treat an arbitrary number of spins N for one TM slice even in three dimensions. This arbitrariness admits systematic finite-size-scaling analyses. Accepting the extended parameter space by Cirillo et al., we analyzed the (multi-) criticality of the gonihedric model for N</=13. Thereby, we found that, as first noted by Cirillo et al. analytically (cluster-variation method), the data are well described by the multicritical (crossover) scaling theory. That is, the previously reported nonstandard criticality for the gonihedric model is reconciled with a crossover exponent and the ordinary three-dimensional-Ising universality class. We estimate the crossover exponent and the correlation-length critical exponent at the multicritical point as phi=0.6(2) and nu; =0.45(15), respectively.