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Journal of Applied Physics

Phase transition in Ising ferromagnetic lattices with fixed spins (abstract)

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

Published Apr 15, 1988• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.340888

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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.
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.
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 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.
Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from +Msp+{M}_{\mathrm{sp}}+Msp​ to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from 0+{0}^{+}0+ to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature Tc{T}_{c}Tc​ and at Tc{T}_{c}Tc​. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.
Monte Carlo simulations are performed to analyze phase transitions in three-dimensional systems described by the 3-state Potts model with nonmagnetic impurities. Numerical results are presented for systems with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 on lattices of size L varying between 20 and 44. Binder’s cumulant analysis shows that the introduction of quenched disorder in the form of non-magnetic impurities induces a crossover from first-order to second-order phase transition. The finite-size scaling method is used to calculate the static critical exponents α, γ, β, and ν for specific heat, susceptibility, magnetization, and correlation length, respectively.