Investigation of Finite-Size 2D Ising Model with a Noisy Matrix of Spin-Spin Interactions

• Main Authors: Boris Kryzhanovsky, Magomed Malsagov, Iakov Karandashev
• Format: Article
• Language: English
• Published: MDPI AG 2018-08-01
• Series: Entropy
• Subjects: Ising model; noisy connections; ground state; free energy; internal energy; magnetization; specific heat; entropy; critical temperature
• Online Access: http://www.mdpi.com/1099-4300/20/8/585

We analyze changes in the thermodynamic properties of a spin system when it passes from the classical two-dimensional Ising model to the spin glass model, where spin-spin interactions are random in their values and signs. Formally, the transition reduces to a gradual change in the amplitude of the multiplicative noise (distributed uniformly with a mean equal to one) superimposed over the initial Ising matrix of interacting spins. Considering the noise, we obtain analytical expressions that are valid for lattices of finite sizes. We compare our results with the results of computer simulations performed for square N = L × L lattices with linear dimensions L = 50 ÷ 1000. We find experimentally the dependencies of the critical values (the critical temperature, the internal energy, entropy and the specific heat) as well as the dependencies of the energy of the ground state and its magnetization on the amplitude of the noise. We show that when the variance of the noise reaches one, there is a jump of the ground state from the fully correlated state to an uncorrelated state and its magnetization jumps from 1 to 0. In the same time, a phase transition that is present at a lower level of the noise disappears.