Phase transition in the Ising ferromagnetic model with fixed spins

The ferromagnetic Ising spin-(1/2 model in a finite simple-cubic lattice Λ is studied by Monte Carlo methods when two subsets of the lattice sites in Λ, say ${\Omega }^{+}$ and ${\Omega }^{\mathrm{-}}$, contain (the same number of) spins fixed at ±1, respectively, the global defect concentration being x≤0.25. We study the thermodynamic properties of the model for different choices of Ω=${\Omega }^{+}$∪${\Omega }^{\mathrm{-}}$. A finite-size-scaling analysis reveals that the transition remains second order with pure critical exponents for regularly spaced defects, the critical temperature varying with the symmetry of Ω. Any small randomness in Ω, however, makes the transition weakly first order; the transition becomes more abrupt for defects located fully at random, and the long-range order is suppressed when the numbers of defects in ${\Omega }^{+}$ and ${\Omega }^{\mathrm{-}}$ differ from each other. We also discuss our findings in relation to the random-field and frustration problems.