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Physical review. B, Condensed matter

Phase transition in the Ising ferromagnetic model with fixed spins

A. Labarta, Joaquín Marro, Benjamín Martínez, Javier Tejada

Published Jul 1, 1988• 3 citations
Publisher

Abstract

The ferromagnetic Ising spin-(1/2 model in a finite simple-cubic lattice \ensuremath{\Lambda} is studied by Monte Carlo methods when two subsets of the lattice sites in \ensuremath{\Lambda}, say ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$, contain (the same number of) spins fixed at \ifmmode\pm\else\textpm\fi{}1, respectively, the global defect concentration being x\ensuremath{\le}0.25. We study the thermodynamic properties of the model for different choices of \ensuremath{\Omega}=${\ensuremath{\Omega}}^{+}$\ensuremath{\cup}${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$. 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 \ensuremath{\Omega}. Any small randomness in \ensuremath{\Omega}, 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 ${\ensuremath{\Omega}}^{+}$ and ${\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}$ differ from each other. We also discuss our findings in relation to the random-field and frustration problems.

DOI
10.1103/physrevb.38.500

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The ferromagnetic Ising spin-(1/2 model in a finite simple-cubic lattice \ensuremath{\Lambda} is studied by Monte Carlo methods when two subsets of the lattice sites in \ensuremath{\Lambda}, say \ensuremathΩ+{\ensuremath{\Omega}}^{+}\ensuremathΩ+ and \ensuremathΩ\ensuremath−{\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}\ensuremathΩ\ensuremath−, contain (the same number of) spins fixed at \ifmmode\pm\else\textpm\fi{}1, respectively, the global defect concentration being x\ensuremath{\le}0.25. We study the thermodynamic properties of the model for different choices of \ensuremath{\Omega}=\ensuremathΩ+{\ensuremath{\Omega}}^{+}\ensuremathΩ+\ensuremath{\cup}\ensuremathΩ\ensuremath−{\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}\ensuremathΩ\ensuremath−. 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 \ensuremath{\Omega}. Any small randomness in \ensuremath{\Omega}, 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 \ensuremathΩ+{\ensuremath{\Omega}}^{+}\ensuremathΩ+ and \ensuremathΩ\ensuremath−{\ensuremath{\Omega}}^{\mathrm{\ensuremath{-}}}\ensuremathΩ\ensuremath− differ from each other. We also discuss our findings in relation to the random-field and frustration problems.
In this paper we report a study of the influence of the symmetry which characterizes the quenched defects (namely, 1/2 spins fixed either ‘‘up’’ or ‘‘down’’) distributions into the nature of the involved phase transitions. We have performed a Monte Carlo simulation involving L3 cubic lattices with L≤40 and, in order to conclude above the phase transition nature, a detailed finite size scaling analysis has been done. It is concluded that the transition remains second order with pure critical exponents for regularly spaced defects. Any small randomness in the location of the defects, however, makes the transition first order.
A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.
We study a short-range site-random Ising model in three dimensions (3D) using a Monte Carlo (MC) method and find a phase transition to a spin glass (SG) phase at a very high temperature ( T c ≃2.83 J). Values of the critical exponents estimated from a finite-size scaling analysis are different from those in both the 3D∓J\mp J∓J Ising model and the 3D Ising ferromagnet. We suggest that the SG phase in this model is a new SG phase which is characterized by a nonzero SG order parameter q ≠0.
We have performed Monte Carlo simulations of the ferromagnetic order-disorder transition in an off-lattice version of the Heisenberg model. Varying the temperature at a fixed density, we located the transition for three different densities by means of a finite-size scaling analysis. The obtained critical exponents differ from those for the lattice Heisenberg model. Although the limited size of our systems warrants caution, the results question whether the lattice and the off-lattice model are in the same universality class.
We consider the random-bond ±J Ising model on a square lattice as a function of the temperature T and of the disorder parameter p (p=1 corresponds to the pure Ising model). We investigate the critical behavior along the paramagnetic-ferromagnetic transition line at low temperatures, below the temperature of the multicritical Nishimori point at T *=0.9527(1), p *=0.89083(3). We present finite-size scaling analyses of Monte Carlo results at two temperature values, T≈0.645 and T=0.5. The results show that the paramagnetic-ferromagnetic transition line is reentrant for T<T *, that the transitions are continuous and controlled by a strong-disorder fixed point with critical exponents ν=1.50(4), η=0.128(8), and β=0.095(5). This fixed point is definitely different from the Ising fixed point controlling the paramagnetic-ferromagnetic transitions for T>T *. Our results for the critical exponents are consistent with the hyperscaling relation 2β/ν−η=d−2=0.