A scaling transformation is developed for the pure square-lattice-spin- 1 / 2 Ising model with different exchange couplings J 1 , J 2 in the two lattice directions. The transformation preserves Onsager's exact duality relationship and so can treat cases of strong lattice anisotropy. It is generalised to the dilute case and used to obtain critical exponents and critical curves (transition temperature T c as a function of concentration p of magnetic bonds for various exchange ratios gamma =J 2 /J 1 ). For gamma small, corresponding to weakly coupled chains, the transition temperature falls rapidly with dilution, then levels off, finally vanishing at the two-dimensional percolation concentration. Extensions are made to three-dimensional dilute magnets, including weakly coupled layer and chain magnets as special cases. Critical curves and exponents are calculated, and a quantitative description is obtained of various crosscovers between low- and high-dimensional behaviour which occur in the quasi-low-dimensional magnets as the transition temperature falls with dilution and the weak exchanges become significant.
Real space renormalisation group scaling techniques are used to investigate the static critical behaviour of the pure and dilute, classical, anisotropic Heisenberg model. Transfer matrix methods are employed to obtain asymptotically exact expressions for the correlation lengths and susceptibilities of the one-dimensional system. The resulting scaling relationships are combined with an approximate bond moving scheme to treat pure and dilute models in higher dimensionalities. Detailed discussions are given for the dependence of correlation lengths and susceptibilities on temperature, anisotropy and concentration, and fcr the critical temperature on anisotropy and concentration. Particular emphasis is given to the weakly anisotropic system near percolation threshold and comparisons are made between the results of the present analysis and those of neutron-scattering experiments on dilute quasi-two- and three-dimensional systems.
We study the effects of dilution to the critical properties of site-diluted Ising model in two dimensions using Monte Carlo simulations. Quenched disorder from the dilution is incorporated into the Ising model via random empty sites on the square lattice of Ising spins. Thermodynamic quantities such as the magnetization
M per spin, energy
E per spin, magnetic susceptibility
χ per spin, and specific heat
C per spin are then calculated after the system has equilibrated. At small dilution concentrations
d<0.1, we find that the value of the critical exponent
β does not deviate from its pure Ising value. At higher dilution concentrations
d>0.1, however, we find
β to strongly depend on the value of
d. We are able to locate a critical temperature
Tc and a critical dilution concentration
dc where the phase transition occurs. We find
Tc to depend linearly on
d. In the phase diagrams of
M,
E,
χ, and
C, we find that the phase transition line eventually disappears at high dilutions. Our results suggest that there is a transition from Strong Universality at low dilution to Weak Universality at high dilution. Lastly, we find a wide and nonsingular increase in the magnetic susceptibility
χ at the low temperature and high dilution region.
We have performed Monte Carlo calculations of the thermodynamic properties of the bond-dilute Ising model on the square lattice. The results are obtained for the magnetization, internal energy, specific heat, and susceptibility for several concentrations of missing bonds. The dependence of the Curie temperature on dilution is in good agreement with the results obtained by other methods. The phase transition is found to remain sharp with the same value of the susceptibility exponent as for the pure Ising model.
A two-step renormalisation group approach-a Migdal-Kadanoff (or decimation) transformation (MK) followed by a mean-field renormalisation group (MFRG)-has been applied to random bond Ising models on d-dimensional hypercubic lattices. Calculations of the critical couplings and percolation concentrations for two and three dimensions show much improvement, even for small size clusters, compared with the results of separate MK or MFRG calculations. Critical exponents nu T and nu p are also improved in some cases. Estimates of critical curves in the concentration-temperature plane are also obtained for the two- and three-dimensional models.
The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.
Exact analytic results are presented for the critical properties of the dilute Ising model on a Bethe lattice, where a fraction p of the sites are occupied at random by spins. It is found that the critical behaviour is the same as that of the pure system (p=1) except for transitions which occur by varying p at zero temperature, where the critical behaviour is different and corresponds to the percolation problem. For T to 0 and p to p c , the percolation concentration, the zero-field magnetization shows crossover behaviour of the form expected from scaling theories. Outside the critical region, a rapidly convergent numerical method is developed and used to calculate the spontaneous magnetization.