Ising model

About: Ising model is a research topic. Over the lifetime, 25508 publications have been published within this topic receiving 555000 citations.

Scaling laws for ising models near T c

Abstract: A model for describing the behavior of Ising models very near ${T}_{c}$ is introduced. The description is based upon dividing the Ising model into cells which are microscopically large but much smaller than the coherence length and then using the total magnetization within each cell as a collective variable. The resulting calculation serves as a partial justification for Ifidom's conjecture about the homogeneity of the free energy and at the same time gives his result $s\ensuremath{ u}\ensuremath{'}=\mathrm{\ensuremath{\gamma}}\ensuremath{'}+2\mathrm{\ensuremath{\beta}}$.

Statistics of the Two-Dimensional Ferromagnet. Part II

Abstract: In an effort to make statistical methods available for the treatment of cooperational phenomena, the Ising model of ferromagnetism is treated by rigorous Boltzmann statistics. A method is developed which yields the partition function as the largest eigenvalue of some finite matrix, as long as the manifold is only one dimensionally infinite. The method is carried out fully for the linear chain of spins which has no ferromagnetic properties. Then a sequence of finite matrices is found whose largest eigenvalue approaches the partition function of the two-dimensional square net as the matrix order gets large. It is shown that these matrices possess a symmetry property which permits location of the Curie temperature if it exists and is unique. It lies at $\frac{J}{k{T}_{c}}=0.8814$ if we denote by $J$ the coupling energy between neighboring spins. The symmetry relation also excludes certain forms of singularities at ${T}_{c}$, as, e.g., a jump in the specific heat. However, the information thus gathered by rigorous analytic methods remains incomplete.

Effect of random defects on the critical behaviour of Ising models

Abstract: A cumulant expansion is used to calculate the transition temperature of Ising models with random-bond defects. For a concentration, x, of missing interactions in the simple-square Ising model the author finds -Tc-1 dTc/dx mod x=0=1.329 compared with the mean-field value of one. If the interactions are independent random variable with a width delta J/J identical to epsilon , the result is -Tc-1 dTc/d epsilon 2 mod epsilon =0=0.312 compared with the mean-field results of zero. An approximation yields the specific heat in the critical regime as C approximately C0/(1+x gamma 2C0), where gamma is a constant and C0 is the unperturbed specific heat at a renormalized temperature. Thus, the specific heat divergence is broadened over a temperature interval Delta T, with Delta T/Tc approximately x(1 alpha )/, where alpha is the critical exponent for the specific heat, and a maximum value of order x-1 is attained. Heuristic arguments show that this smoothing effect occurs if alpha >0.