Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0810.0685 · Journal of Physics Conference Series

Critical and multicritical behavior of the ±J Ising model in two and three dimensions

Martin Hasenbusch B, Francesco Parisen Toldin, Andrea Pelissetto C, Ettore Vicari

Published Jan 1, 2009• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1742-6596/145/1/012055

Related papers

Critical behavior of the three-dimensional ±J Ising model at the paramagnetic-ferromagnetic transition line

Martin Hasenbusch B, Francesco Parisen Toldin, Andrea Pelissetto C +1 more

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 $p=1$ 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 $p$ in the region ${p}_{N}<p<1$, 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.

arXiv:0704.0427•Physical Review B•Sep 6, 2007•48 citations

Magnetic-glassy multicritical behavior of the three-dimensional ±J Ising model

Martin Hasenbusch B, Francesco Parisen Toldin, Andrea Pelissetto C +1 more

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 ($p$ 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 ${p}^{*}=0.768\phantom{\rule{0.2em}{0ex}}20(4)$, 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, ${y}_{1}=1.02(5)$, ${y}_{2}=0.61(2)$, 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-$A$ dynamics, obtaining the dynamic critical exponent $z=5.0(5)$.

arXiv:0707.2866•Physical Review B•Nov 8, 2007•38 citations

Multicritical Nishimori point in the phase diagram of the ± J Ising model on a square lattice

Martin Hasenbusch B, Francesco Parisen Toldin, Andrea Pelissetto C +1 more

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 $T$ is the temperature and $p$ is the disorder parameter ($p=1$ 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 ${y}_{1}=0.655(15)$ and ${y}_{2}=0.250(2)$; they correspond to the thermal exponent $\ensuremath{\nu}\ensuremath{\equiv}1/{y}_{2}=4.00(3)$ and to the crossover exponent $\ensuremath{\phi}\ensuremath{\equiv}{y}_{1}/{y}_{2}=2.62(6)$.

arXiv:0803.0444•Physical Review E•May 15, 2008•45 citations

Universal dependence on disorder of two-dimensional randomly diluted and random-bond ±J Ising models

Martin Hasenbusch B, Francesco Parisen Toldin, Andrea Pelissetto C +1 more

We consider the two-dimensional randomly site diluted Ising model and the random-bond +/-J Ising model (also called the Edwards-Anderson model), and study their critical behavior at the paramagnetic-ferromagnetic transition. The critical behavior of thermodynamic quantities can be derived from a set of renormalization-group equations, in which disorder is a marginally irrelevant perturbation at the two-dimensional Ising fixed point. We discuss their solutions, focusing in particular on the universality of the logarithmic corrections arising from the presence of disorder. Then, we present a finite-size scaling analysis of high-statistics Monte Carlo simulations. The numerical results confirm the renormalization-group predictions, and in particular the universality of the logarithmic corrections to the Ising behavior due to quenched dilution.

arXiv:0804.2788•Physical Review E•Jul 11, 2008•53 citations

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

Francesco Parisen Toldin, Andrea Pelissetto C, Ettore Vicari

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.

arXiv:0811.2101•Journal of Statistical Physics•Mar 13, 2009•44 citations

Nonequilibrium critical dynamics of the two-dimensional ±J Ising model

Ramgopal Agrawal, Leticia F. Cugliandolo, Lara Faoro +2 more

The ±J Ising model is a simple frustrated spin model, where the exchange couplings independently take the discrete value -J with probability p and +J with probability 1-p. It is especially appealing due to its connection to quantum error correcting codes. Here, we investigate the nonequilibrium critical behavior of the two-dimensional ±J Ising model, after a quench from different initial conditions to a critical point T_{c}(p) on the paramagnetic-ferromagnetic (PF) transition line, especially above, below, and at the multicritical Nishimori point (NP). The dynamical critical exponent z_{c} seems to exhibit nonuniversal behavior for quenches above and below the NP, which is identified as a preasymptotic feature due to the repulsive fixed point at the NP, whereas for a quench directly to the NP, the dynamics reaches the asymptotic regime with z_{c}≃6.02(6). We also consider the geometrical spin clusters (of like spin signs) during the critical dynamics. Each universality class on the PF line is uniquely characterized by the stochastic Loewner evolution with corresponding parameter κ. Moreover, for the critical quenches from the paramagnetic phase, the model, irrespective of the frustration, exhibits an emergent critical percolation topology at the large length scales.

arXiv:2304.11997•Physical review. E•Dec 21, 2023•6 citations
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.
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.
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 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 two-dimensional randomly site diluted Ising model and the random-bond +/-J Ising model (also called the Edwards-Anderson model), and study their critical behavior at the paramagnetic-ferromagnetic transition. The critical behavior of thermodynamic quantities can be derived from a set of renormalization-group equations, in which disorder is a marginally irrelevant perturbation at the two-dimensional Ising fixed point. We discuss their solutions, focusing in particular on the universality of the logarithmic corrections arising from the presence of disorder. Then, we present a finite-size scaling analysis of high-statistics Monte Carlo simulations. The numerical results confirm the renormalization-group predictions, and in particular the universality of the logarithmic corrections to the Ising behavior due to quenched dilution.
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.
The ±J Ising model is a simple frustrated spin model, where the exchange couplings independently take the discrete value -J with probability p and +J with probability 1-p. It is especially appealing due to its connection to quantum error correcting codes. Here, we investigate the nonequilibrium critical behavior of the two-dimensional ±J Ising model, after a quench from different initial conditions to a critical point T_{c}(p) on the paramagnetic-ferromagnetic (PF) transition line, especially above, below, and at the multicritical Nishimori point (NP). The dynamical critical exponent z_{c} seems to exhibit nonuniversal behavior for quenches above and below the NP, which is identified as a preasymptotic feature due to the repulsive fixed point at the NP, whereas for a quench directly to the NP, the dynamics reaches the asymptotic regime with z_{c}≃6.02(6). We also consider the geometrical spin clusters (of like spin signs) during the critical dynamics. Each universality class on the PF line is uniquely characterized by the stochastic Loewner evolution with corresponding parameter κ. Moreover, for the critical quenches from the paramagnetic phase, the model, irrespective of the frustration, exhibits an emergent critical percolation topology at the large length scales.