Physical review. B, Condensed matter

Edge critical exponent for the three-dimensional Ising model by Monte Carlo simulations

K. K. Mon, J. L. Vallés

Published Aug 1, 1989• 3 citations

Abstract

We present the first Monte Carlo estimate of the edge critical exponent ${\ensuremath{\Delta}}_{2}$ for the three-dimensional nearest-neighbor ferromagnetic Ising model at the ordinary edge transition. The finite-size dependence of the spin-spin correlation function is used to obtain the edge-bulk exponent ${\ensuremath{\eta}}_{0}$,2=1.704\ifmmode\pm\else\textpm\fi{}0.1, and this gives ${\ensuremath{\Delta}}_{2}$/${\ensuremath{\nu}}_{b}$=-1.185\ifmmode\pm\else\textpm\fi{}0.1. This last value is in agreement with the \ensuremath{\epsilon}-expansion estimate of -29/24=-1.208. . . .

DOI: 10.1103/physrevb.40.2419

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Physical review. B, Condensed matter

Edge critical exponent for the three-dimensional Ising model by Monte Carlo simulations

K. K. Mon, J. L. Vallés

Published Aug 1, 1989• 3 citations

Publisher

Abstract

We present the first Monte Carlo estimate of the edge critical exponent

{\ensuremath{\Delta}}_{2}

for the three-dimensional nearest-neighbor ferromagnetic Ising model at the ordinary edge transition. The finite-size dependence of the spin-spin correlation function is used to obtain the edge-bulk exponent

{\ensuremath{\eta}}_{0}

,2=1.704\ifmmode\pm\else\textpm\fi{}0.1, and this gives

{\ensuremath{\Delta}}_{2}

{\ensuremath{\nu}}_{b}

=-1.185\ifmmode\pm\else\textpm\fi{}0.1. This last value is in agreement with the \ensuremath{\epsilon}-expansion estimate of -29/24=-1.208. . . .

DOI: 10.1103/physrevb.40.2419

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Monte Carlo estimates of the corner critical exponents of the three-dimensional Ising model

Pik-Yin Lai, K. K. Mon

Journal of Physics A Mathematical and GeneralDec 7, 19892 citations

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Michel Pleimling, Walter Selke

arXiv:cond-mat/9806175Physical review. B, Condensed matterJan 1, 19999 citations

Finite-size-scaling estimation of critical exponents in the two-dimensional axial next-nearest-neighbor Ising model

Marcelo D. Grynberg, Horacio Ceva

The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.The magnetic critical exponent of the two-dimensional axial next-nearest-neighbor Ising model is estimated by means of the finite-size scaling of the order parameter. For a finite system without ordering field, this magnitude is computed from the two leading eigenvectors of the transfer matrix. The ferromagnetic-paramagnetic transition temperatures estimated with this method exhibit a small lattice-size dependence. Moreover, they are in good agreement with those obtained using phenomenological renormalization, and also with analytical approximations. The correlation-length critical exponent is calculated by means of phenomenological renormalization. The results found with these numerical techniques indicate that the ferromagnetic-paramagnetic transition of the model seems to be continuous, in contradiction with a previous result obtained using the persistence-length criterion, which indicated that the transition is always first order. We found that the phase transition is first order only when the ratio of the next-nearest-neighbor and nearest-neighbor interactions equals one-half.

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

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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.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.

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Physical review. B, Condensed matterApr 1, 1984318 citations

Critical behavior of three-dimensional Ising spin glass models

Martin Hasenbusch B, Andrea Pelissetto C, Ettore Vicari

We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size $L$ : the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter $p$ , $p=0.5$ and $p=0.7$ (up to $L=28$ and $L=20$ , respectively), and the bond-diluted bimodal model for bond-occupation probability ${p}_{b}=0.45$ (up to $L=16$ ). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at ${p}_{b}=0.7$ and ${p}_{b}=0.35$ (up to $L=10$ ) and of the Ising spin glass model with Gaussian bond distribution (up to $L=8$ ) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size $L$: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter $p$, $p=0.5$ and $p=0.7$ (up to $L=28$ and $L=20$, respectively), and the bond-diluted bimodal model for bond-occupation probability ${p}_{b}=0.45$ (up to $L=16$). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at ${p}_{b}=0.7$ and ${p}_{b}=0.35$ (up to $L=10$) and of the Ising spin glass model with Gaussian bond distribution (up to $L=8$) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.

arXiv:0809.3329Physical Review BDec 17, 2008119 citations

We present the first Monte Carlo estimate of the edge critical exponent

{\ensuremath{\Delta}}_{2}

{\ensuremath{\eta}}_{0}

,2=1.704\ifmmode\pm\else\textpm\fi{}0.1, and this gives

{\ensuremath{\Delta}}_{2}

{\ensuremath{\nu}}_{b}

=-1.185\ifmmode\pm\else\textpm\fi{}0.1. This last value is in agreement with the \ensuremath{\epsilon}-expansion estimate of -29/24=-1.208. . . .

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.

The Monte Carlo renormalization group is applied to the three-dimensional Ising model on simple cubic lattices with

{8}^{3}

{16}^{3}

{32}^{3}

, and

{64}^{3}

sites. The comparison of block-spin correlation functions from the largest lattices yields the nearest-neighbor critical coupling

{K}_{1}^{c}=0.221654(6)

. After allowing for (i) interpolation to this best estimate for

{K}_{1}^{c}

, (ii) an apparent finite-size effect in the renormalization-group transformation due to the measurement of correlation functions of too few (seven) operators, and (iii) the extrapolation for the effect of a slow transient towards the fixed point, the values $\ensuremath{\nu}=0.629(4)$ and $\ensuremath{\eta}=0.031(5)$ are obtained for the thermal and magnetic exponents. The correction-to-scaling exponent $\ensuremath{\omega}$ is estimated to be around 1; to obtain an accuracy competitive with other methods requires measurements with more than seven operators. We briefly review the problem of redundant operators and indicate the future prospects for this kind of calculation.

We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size

L

: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter

p

p=0.5

and

p=0.7

(up to

L=28

and

L=20

, respectively), and the bond-diluted bimodal model for bond-occupation probability

{p}_{b}=0.45

(up to

L=16

). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at

{p}_{b}=0.7

and

{p}_{b}=0.35

(up to

L=10

) and of the Ising spin glass model with Gaussian bond distribution (up to

L=8

) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.