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arXiv:0811.2101 · Journal of Statistical Physics

Strong-Disorder Paramagnetic-Ferromagnetic Fixed Point in the Square-Lattice ±J Ising Model

Francesco Parisen Toldin, Andrea Pelissetto C, Ettore Vicari

Published Mar 13, 2009• 44 citations
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Abstract

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.

DOI
10.1007/s10955-009-9705-5

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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.
We investigate the critical behavior of the random-bond $\ifmmode\pm\else\textpm\fi{}J$ Ising model on a square lattice at the multicritical Nishimori point in the $T\text{\ensuremath{-}}p$ phase diagram, where TTT is the temperature and ppp is the disorder parameter (p=1p=1p=1 corresponds to the pure Ising model). We perform a finite-size scaling analysis of high-statistics Monte Carlo simulations along the Nishimori line defined by $2p\ensuremath{-}1=\text{tanh}(1/T)$, along which the multicritical point lies. The multicritical Nishimori point is located at ${p}^{\ensuremath{\ast}}=0.890\text{ }81(7)$, ${T}^{\ensuremath{\ast}}=0.9528(4)$, and the renormalization-group dimensions of the operators that control the multicritical behavior are y1=0.655(15){y}_{1}=0.655(15)y1​=0.655(15) and y2=0.250(2){y}_{2}=0.250(2)y2​=0.250(2); they correspond to the thermal exponent $\ensuremath{\nu}\ensuremath{\equiv}1/{y}_{2}=4.00(3)$ and to the crossover exponent $\ensuremath{\phi}\ensuremath{\equiv}{y}_{1}/{y}_{2}=2.62(6)$.
We consider the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ model defined on a simple cubic lattice and study its behavior close to the multicritical Nishimori point, where the paramagnetic-ferromagnetic, the paramagnetic-glassy, and the ferromagnetic-glassy transition lines meet in the $T\text{\ensuremath{-}}p$ phase diagram (ppp characterizes the disorder distribution and gives the fraction of ferromagnetic bonds). For this purpose, we perform Monte Carlo simulations on cubic lattices of size $L\ensuremath{\leqslant}32$ and a finite-size-scaling analysis of the numerical results. The magnetic-glassy multicritical point is found at p∗=0.76820(4){p}^{*}=0.768\phantom{\rule{0.2em}{0ex}}20(4)p∗=0.76820(4), along the Nishimori line given by $2p\ensuremath{-}1=\mathrm{tanh}(J∕T)$. We determine the renormalization-group dimensions of the operators that control the renormalization-group flow close to the multicritical point, y1=1.02(5){y}_{1}=1.02(5)y1​=1.02(5), y2=0.61(2){y}_{2}=0.61(2)y2​=0.61(2), and the susceptibility exponent $\ensuremath{\eta}=\ensuremath{-}0.114(3)$. The temperature and crossover exponents are $\ensuremath{\nu}=1∕{y}_{2}=1.64(5)$ and $\ensuremath{\phi}={y}_{1}∕{y}_{2}=1.67(10)$, respectively. We also investigate the model-AAA dynamics, obtaining the dynamic critical exponent z=5.0(5)z=5.0(5)z=5.0(5).
We report our Monte Carlo results on the critical and multicritical behavior of the ± J Ising model [with a random-exchange probability P ( J xy ) = p δ( J xy - J )+(1- p )δ( J xy + J )], in two and three dimensions. We study the transition line between the paramagnetic and ferromagnetic phase, which extends from p = 1 to a multicritical (Nishimori) point. By a finite-size scaling analysis, we provide strong numerical evidence that in three dimensions the critical behavior along this line belongs to the same universality class as that of the critical transition in the randomly dilute Ising model. In two dimensions we confirm that the critical behavior is controlled by the pure Ising fixed point and that disorder is marginally irrelevant, giving rise to universal logarithmic corrections. In both two and three dimensions, we also determine the location of the multicritical Nishimori point, as well as the renormalization-group dimensions of the operators that control the renormalization-group flow close to it.
The random-bond Ising model on the square lattice has several disordered critical points, depending on the probability distribution of the bonds. There are a finite-temperature multicritical point, called the Nishimori point, and a zero-temperature fixed point, for both a binary distribution where the coupling constants take the values ± J and a Gaussian disorder distribution. Inclusion of dilution in the ± J distribution ( J = 0 for some bonds) gives rise to another zero-temperature fixed point which can be identified with percolation in the non-frustrated case ( J ≥ 0). We study these fixed points using numerical (transfer matrix) methods. We determine the location, critical exponents, and central charge of the different fixed points and study the spin–spin correlation functions. Our main findings are the following. (1) We confirm that the Nishimori point is universal with respect to the type of disorder, i.e. we obtain the same central charge and critical exponents for the ± J and Gaussian distributions of disorder. (2) The Nishimori point, the zero-temperature fixed point for the ± J and Gaussian distributions of disorder, and the percolation point in the diluted case all belong to mutually distinct universality classes. (3) The paramagnetic phase is re-entrant below the Nishimori point, i.e. the zero-temperature fixed points are not located exactly below the Nishimori point, either for the ± J distribution, or for the Gaussian distribution.
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 study the critical behavior of the three-dimensional $\ifmmode\pm\else\textpm\fi{}J$ Ising model [with random-exchange probability $P({J}_{xy})=p\ensuremath{\delta}({J}_{xy}\ensuremath{-}J)+(1\ensuremath{-}p)\ensuremath{\delta}({J}_{xy}+J)$] at the transition line between the paramagnetic and ferromagnetic phases, which extends from p=1p=1p=1 to a magnetic-glassy multicritical point at $p={p}_{N}\ensuremath{\approx}0.768$. By a finite-size scaling analysis of Monte Carlo simulations at various values of ppp in the region pN<p<1{p}_{N}<p<1pN​<p<1, we provide strong numerical evidence that the critical behavior along the ferromagnetic transition line belongs to the same universality class as the three-dimensional randomly dilute Ising model. We obtain the results $\ensuremath{\nu}=0.682(3)$ and $\ensuremath{\eta}=0.036(2)$ for the critical exponents, which are consistent with the estimates $\ensuremath{\nu}=0.683(2)$ and $\ensuremath{\eta}=0.036(1)$ at the transition of randomly dilute Ising models.