A relatively simple method for computing thermodynamic quantities in the Ising model with impurities in thermodynamic equilibrium is proposed. The specific heat, the susceptibility, and the short-range order of spins localized on bonds of different types are investigated for the plane square lattice.
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.
PACS. 05.50 – Lattice theory and statistics; Ising problems PACS. 75.10 – General theory and models of magnetic ordering We describe results of the cluster algorithm Special Purpose Processor simulations of the 2D Ising model with impurity bonds. Use of large lattices, with the number of spins up to 10 6, permitted to define critical region of temperatures, where both finite size corrections and corrections to scaling are small. High accuracy data unambiguously show increase of magnetization and magnetic susceptibility effective exponents β and γ, caused by impurities. The M and χ singularities became more sharp, while the specific heat singularity is smoothed. The specific heat is found to be in a good agreement with Dotsenko-Dotsenko theoretical predictions in the The problem of inhomogenities influence on phase transitions has a long history. The first model with some kind of impurities to be studied, is, due to its simplicity, the Ising model. Effect of randomness on the critical behaviour of different Ising
Within a new type of correlated-effective-field theory we calculate the effective-field parameters (including their response to external field), critical temperatures, susceptibility, short-range order parameters, and specific heat associated with an anisotropic, square-lattice, spin-\textonehalf{} Ising ferromagnet. The present formalism, which is shown to recover the Bethe-Peierls thermodynamics for the pure Z-coordinated-lattice, spin-\textonehalf{} Ising model, yields satisfactory results; in particular, the crossover from a pure Ising ferromagnet on a square lattice to that on a linear chain is reasonably exhibited in the behavior of the susceptibility and the specific heat.
The problem of the effect of inhomogeneous impurities in a lattice structure on the critical behaviour of systems which undergo a second-order phase transition is studied on a model system, the 2D Ising Model with impurity bonds randomly distributed over the lattice. It is known that in the critical region the 2D Ising Model is equivalent to the model of free fermions. We show that the effect of impurities is to add a four-fermion interaction with the corresponding charge proportional to the concentration of impurities. The resulting fermion model is simple enough and can be studied exactly by renormalization group methods. We show that for any small concentration of impurity bonds a new critical regime is established if we go sufficiently close to the phase-transition point. In particular we find that the specific heat singularity changes from C ˜ ln 1/|τ|(τ=(Τ − Τc/Τc to C ˜ ln ln 1/|τ| for τ≪τi ˜ exp(−const/c i), c i being the concentration of impurities, while the spin-spin correlation function in the critical point changes more seriously: from <σ R σΟ> ˜ 1/R1/4 to <σ R σΟ> ˜ exp { − (const/c i)(ln ln R2 )} for R≫R i˜ exp (const/c i), which corresponds to a change of the critical exponent η from 1/4 to 0.
We consider the thermodynamics of a square Ising lattice which is constructed from unit cells of height n lattice spacings and width m lattice spacings. In a particular unit cell the values of the 2mn exchange energies can be chosen at will. Once specified this basic unit cell is repeated to make up the entire planar lattice. For arbitrary n and for m=2 and m=3 we obtain the exact equations for the critical point. Furthermore, we show that when n is finite and m=1 or m=2 the specific heat has a singularity of the form $A\mathrm{ln}|1\ensuremath{-}\frac{T}{{T}_{c}}|$ and we derive an explicit formula for A. Our method can be extended to values of m>2. A numerical evaluation of our results shows that the average of A over impurity configurations is smaller for m=1 than for m=2.
We consider thermodynamic properties, e.g. specific heat and magnetic susceptibility, of alternating Heisenberg spin chains. Due to a hidden Ising symmetry, these chains can be decomposed into a set of finite chain fragments. The problem of finding the thermodynamic quantities is effectively separated into two parts. First we deal with finite objects; secondly we can incorporate the fragments into a statistical ensemble. As functions of the coupling constants, the models exhibit special features in the thermodynamic quantities, e.g. the specific heat displays double peaks at low enough temperatures. These features stem from first-order quantum phase transitions at zero temperature, which have been investigated in the first part of this work.