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arXiv:cond-mat/9612001 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Critical behavior of the three-dimensional Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions

Emilio N. M. Cirillo, Giuseppe Gonnella, Alessandro Pelizzola

Published Jan 1, 1997• 22 citations
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Abstract

The critical and multicritical behavior of the simple cubic Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions is studied using the cube and star-cube approximations of the cluster variation method and the recently proposed cluster variation--Pad\'eapproximant method. Particular attention is paid to the line of critical end points of the ferromagnetic-paramagnetic phase transition: its (multi)critical exponents are calculated, and their values suggest that the transition belongs to a different universality class. A rough estimate of the crossover exponent is also given.

DOI
10.1103/physreve.55.r17

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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
The critical and multicritical behavior of the simple cubic Ising model with nearest-neighbor, next-nearest-neighbor, and plaquette interactions is studied using the cube and star-cube approximations of the cluster variation method and the recently proposed cluster variation--Pad'eapproximant method. Particular attention is paid to the line of critical end points of the ferromagnetic-paramagnetic phase transition: its (multi)critical exponents are calculated, and their values suggest that the transition belongs to a different universality class. A rough estimate of the crossover exponent is also given.
A three-dimensional Ising model with the plaquette-type (next-nearest-neighbor and four-spin) interactions is investigated numerically. This extended Ising model, the so-called gonihedric model, was introduced by Savvidy and Wegner as a discretized version of the interacting (closed) surfaces without surface tension. The gonihedric model is notorious for its slow relaxation to the thermal equilibrium (glassy behavior), which deteriorates the efficiency of the Monte Carlo sampling. We employ the transfer-matrix (TM) method, implementing Novotny's idea, which enables us to treat an arbitrary number of spins N for one TM slice even in three dimensions. This arbitrariness admits systematic finite-size-scaling analyses. Accepting the extended parameter space by Cirillo et al., we analyzed the (multi-) criticality of the gonihedric model for N</=13. Thereby, we found that, as first noted by Cirillo et al. analytically (cluster-variation method), the data are well described by the multicritical (crossover) scaling theory. That is, the previously reported nonstandard criticality for the gonihedric model is reconciled with a crossover exponent and the ordinary three-dimensional-Ising universality class. We estimate the crossover exponent and the correlation-length critical exponent at the multicritical point as phi=0.6(2) and nu; =0.45(15), respectively.
Phase transitions and critical characteristics of the layered antiferromagnetic Ising model in the case of a cubic lattice with next-nearest-neighbor intralayer interactions are studied in the framework of the Monte Carlo method implementing the replica algorithm. The characteristics of the phase transitions are analyzed within the histogram method and with the Binder cumulants. For the model under study, it is found that the transition from the superantiferromagnetic phase to the paramagnetic one is a second order phase transition. The static critical exponents for the specific heat α, susceptibility γ, order parameter β, correlation radius ν, and the Fisher exponent η are calculated using the finite-size scaling theory. It is shown that the three-dimensional Ising model for the cubic lattice with next-nearest-neighbor interactions belongs to the same universality class of critical behavior as the completely frustrated three-dimensional Ising model.
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.
The phase diagram of the Ising square ferromagnet with nearest-neighbor, next-nearest-neighbor, and four-spin (plaquette) interactions is investigated using the cluster variation method in the square approximation. A complete analysis in temperature is performed for the ferromagnetic, superantiferromagnetic, and frustrated region of the ground state. Particular attention is devoted to the role of the four-spin interaction. The most remarkable effect is the occurrence, in several regions of the interaction parameter space, of first-order transitions and sometimes reentrance phenomena. Critical and tricritical points are almost analytically determined.
Phase transitions and critical properties of the frustrated Ising model on a cubic lattice with next-nearest-neighbor intralayer interactions are investigated by the replica Monte Carlo method. Estimations are made for the magnitude relation of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J2/J1 in the value ranges of r ∈ [0.0, 1.0]. The phase diagram of the dependence of critical temperature on the next-nearest-neighbor interaction has been plotted. The static critical exponents of the heat capacity, the susceptibility, the ordering parameter and the correlation length as well as the Fisher exponent are calculated by means of the finite-size scaling theory. The universality class of the critical behavior of this model is revealed to remain within the limits of values r ∈ [0.0, 0.4]. It is found that the change in the next-nearest-neighbor interaction value in the range of r ∈ [0.9, 1.0] leads to another critical behavior.
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).