Ising model classical spin systems

Monte Carlo simulations of classical spin systems such as the Ising model can be performed simply and very efficiently using local update algorithms such as the Metropolis or heat-bath algorithms. However, in the vicinity of a continuous phase transition, these algorithms display dramatic critical... 

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

Applying superposition approximations to the Ising model, one finds an evidence for the phase transition existence but the critical parameter to is systematically underestimated (To is overestimated respectively). Errors in calculation of to are greater for low dimensions d. Therefore, the superposition approximation is effective, first of all, for the qualitative description of the phase transition in a spin system. In the vicinity of phase transition a number of critical exponents a, /3,7,..., could be introduced, which characterize the critical point, like oc f-for . M oc (i-io), or xt oc i—io for the magnetic permeability. Superposition approximations give only classical values of the critical exponents a = ao, 0 = f o, j — jo, ., obtained earlier in the classical molecular field theory [13, 14], say fio = 1/2, 7o = 1, whereas exact magnitudes of the critical exponents depend on the space dimension d. To describe the intermediate order in a spin system in terms of the superposition approximation, an additional correlation length is introduced, 0 = which does not coincide with the true In the phase... 

In dimensions d > 1, an exact solution is not available because the strong disorder renormalization group can be implemented only numerically. Moreover, mapping the spin system onto free fermions is restricted to one dimension. Therefore, simulation studies have mostly used Monte Carlo methods. The quantum-to-classical mapping for the Hamiltonian in Eq. [30] can be performed analogously to the clean case. The result is a disordered classical Ising model in d + 1 dimensions with the disorder perfectly correlated in one dimension (in 1-1-1 dimensions, this is the famous McCoy-Wu modeF ). The classical Hamiltonian reads... 

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