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Journal of the Physical Society of Japan

Ferromagnetic Transition of ±JIsing Spin Glass Model on Square Lattice

Nobuyasu Ito, Takashi Matsuhisa, Hidetsugu Kitatani

Published Apr 15, 1998• 27 citations
Publisher

Abstract

The ferromagnetic transition of ± J -Ising spin-glass model on square lattice is studied by using Monte Carlo simulation. At ferromagnetic-bond density p =0.96 where the system shows ferromagnetic transition, the spontaneous magnetization, the ratio < m 4 >/ < m 2 > 2 and the non-equilibrium relaxation (NER) are studied. The transition is estimated to be at K c =0.5100 (1). The analysis by NER gives the most reliable estimation with the least CPU time for simulations among the three methods. The value of magnetization exponent β is estimated to be 0.130(10) which coincides with that of the purely ferromagnetic Ising model ( p =1). The dynamical critical exponent z is estimated to be 2.418(23) from the NER analysis which is different from that of the pure model. The size of simulated lattices is up to 2001×2000, and the number of totally updated Ising spins is 5.5×10 15 .

DOI
10.1143/jpsj.67.1188

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The ferromagnetic transition of ± J -Ising spin-glass model on square lattice is studied by using Monte Carlo simulation. At ferromagnetic-bond density p =0.96 where the system shows ferromagnetic transition, the spontaneous magnetization, the ratio < m 4 >/ < m 2 > 2 and the non-equilibrium relaxation (NER) are studied. The transition is estimated to be at K c =0.5100 (1). The analysis by NER gives the most reliable estimation with the least CPU time for simulations among the three methods. The value of magnetization exponent β is estimated to be 0.130(10) which coincides with that of the purely ferromagnetic Ising model ( p =1). The dynamical critical exponent z is estimated to be 2.418(23) from the NER analysis which is different from that of the pure model. The size of simulated lattices is up to 2001×2000, and the number of totally updated Ising spins is 5.5×10 15 .
Ferromagnetic (FM) transition of the ± J Ising model is studied for square and simple-cubic lattices by Monte Carlo simulations. Non-equilibrium relaxation (NER) process of magnetization from the all-aligned spin state is analyzed. The transition temperature is estimated at several values of FM bond concentrations to provide the boundary of the FM phase. The estimated NER exponent of magnetization shows non-universal behavior and it becomes larger smoothly when the transition temperature becomes higher.
We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
We investigate a spin-3/2 Ising model on a square lattice by using Monte Carlo (MC) simulations and a density matrix renormalization group (DMRG) method. We obtain phase diagrams of the ground state. We carried out the MC simulations in order to calculate the critical temperature and the critical exponent η. We estimate the critical exponents ν, α and γ by using a finite size scaling analysis. We also calculate the critical temperature, the central charge c and the critical exponent ν by using the DMRG method. From the results obtained by the MC simulations and by the DMRG method, we discuss a universality class of the system. It is found that the values of critical exponents are determined by means of the degeneracy of the ground state. This result suggests that the universality does not hold for the spin-3/2 Ising model. Furthermore we find a possibility of a breakdown of the hyperscaling law. We also confirm that the ferrimagnetic phase exists in a parameter region where the ground state has a fourfold degeneracy, and that a transition between the ferrimagnetic phase and the ferromagnetic phase exists in addition to that between the ferromagnetic phase and the paramagnetic phase.
The ±J Ising model in three dimensions is investigated using dynamical scaling analysis for a nonequilibrium relaxation process. The spin-glass (SG) transition is analyzed under an assumed dynamical exponent z proposed for the scaling analysis of the SG transition. To estimate dynamical exponent z precisely, we introduce an extrapolation scheme for the asymptotic value of the logarithmic derivative of the dynamical Binder parameter. We obtain the phase diagram in the temperature–concentration (p–T) space, as well as the p-dependence of critical exponents γ, ν, and z. For the static exponents of the SG transition, the universality is strongly indicated numerically along the whole region of the paramagnetic (PM)–SG phase boundary. A remarkable crossover behavior is observed for estimations of critical exponents at the boundary close to the multicritical point (MCP) of PM–ferromagnetic (FM)–SG phases. In addition, the dynamical assumption for z resulting from the slow relaxation in the SG region disappears in the FM region, and the crossover between them occurs around the Nishimori Line.
Exact dynamical properties are discussed for the ± J Ising model around the multicritical point (MCP). It is found that the relation of relaxations between the non-equilibrium remanent magnetization m ( t ) and the equilibrium autocorrelation function q ( t ) at the MCP is different from that at the pure critical point. The dynamic critical exponent for the ferromagnetic ordering defined by and that for the spin glass ordering defined by become identical at the MCP. Accurate numerical calculations for them are performed in two and three dimensions using the non-equilibrium relaxation analysis. The MCP is located at and the exponent is estimated as for the square lattice. They are estimated as and for the simple cubic lattice.
We describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.