Physical Review Letters

Location of the Ising Spin-Glass Multicritical Point on Nishimori's Line

Pierre Le Doussal, A. Brooks Harris

Published Aug 1, 1988• 109 citations

Abstract

We present arguments, based on local gauge invariance, that the multicritical point of Ising spin-glasses should be located on a particular line of the phase diagram known as Nishimori's line [$tanh(\ensuremath{\beta}J)=2p\ensuremath{-}1$ for the $\ifmmode\pm\else\textpm\fi{}J$ distribution]. One scaling axis is along the line, and the other is along the temperature direction. This scenario is generic for any random Ising model with a Nishimori line, in any number of dimensions, if the transitions are second order. The renormalization-group fixed point located inside Nishimori's manifold is expected to control multicriticality for a wider class of models.

DOI: 10.1103/physrevlett.61.625

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Physical Review Letters

Location of the Ising Spin-Glass Multicritical Point on Nishimori's Line

Pierre Le Doussal, A. Brooks Harris

Published Aug 1, 1988• 109 citations

Publisher

Abstract

We present arguments, based on local gauge invariance, that the multicritical point of Ising spin-glasses should be located on a particular line of the phase diagram known as Nishimori's line [

tanh(\ensuremath{\beta}J)=2p\ensuremath{-}1

for the

\ifmmode\pm\else\textpm\fi{}J

distribution]. One scaling axis is along the line, and the other is along the temperature direction. This scenario is generic for any random Ising model with a Nishimori line, in any number of dimensions, if the transitions are second order. The renormalization-group fixed point located inside Nishimori's manifold is expected to control multicriticality for a wider class of models.

DOI: 10.1103/physrevlett.61.625

Related papers

ε expansion for the Nishimori multicritical point of spin glasses

Pierre Le Doussal, A. Brooks Harris

The renormalization-group recursion relations obtained by Chen and Lubensky for the multicritical point associated with simultaneous critical fluctuations in both the spin-glass and ferromagnetic order are reanalyzed. To first order in \ensuremath{\epsilon}==6-d we find that the multicritical fixed point is located inside the Nishimori manifold and that the scaling axes agree with those obtained recently from general arguments. It is confirmed that the scaling along the Nishimori line and at the paramagnetic-spin-glass transition are related. We also point out some universal properties of the multicritical point of possible experimental interest.The renormalization-group recursion relations obtained by Chen and Lubensky for the multicritical point associated with simultaneous critical fluctuations in both the spin-glass and ferromagnetic order are reanalyzed. To first order in \ensuremath{\epsilon}==6-d we find that the multicritical fixed point is located inside the Nishimori manifold and that the scaling axes agree with those obtained recently from general arguments. It is confirmed that the scaling along the Nishimori line and at the paramagnetic-spin-glass transition are related. We also point out some universal properties of the multicritical point of possible experimental interest.

Physical review. B, Condensed matterNov 1, 198966 citations

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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 $\ensuremath{\phi}\ensuremath{\equiv}{y}_{1}/{y}_{2}=2.62(6)$.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:0803.0444Physical Review EMay 15, 200845 citations

Spin-glass–ferromagnetic–paramagnetic multicritical point

Rajiv R. P. Singh

We study the criticality at the spin-glass-ferromagnetic-paramagnetic multicritical point in the d=3, \ifmmode\pm\else\textpm\fi{}J distribution, random-bond Ising model. Using high-temperature expansions to order ${\mathit{T}}^{\mathrm{\ensuremath{-}}34}$, we estimate that the multicritical point N lies on the Nishimori line at ${\mathit{T}}_{\mathit{c}}$ /J=1.690\ifmmode\pm\else\textpm\fi{}0.016. Along this line the critical exponents are found to be \ensuremath{\gamma}=1.80\ifmmode\pm\else\textpm\fi{}0.15 and \ensuremath{\nu}=0.85\ifmmode\pm\else\textpm\fi{}0.08. The latter is clearly consistent with the rigorous exponent inequality \ensuremath{\nu}\ensuremath{\ge}2/d. We also calculate the crossover exponent \ensuremath{\varphi} and show that the scaling axes at N are in agreement with the recent predictions of Le Doussal and Harris.We study the criticality at the spin-glass-ferromagnetic-paramagnetic multicritical point in the d=3, \ifmmode\pm\else\textpm\fi{}J distribution, random-bond Ising model. Using high-temperature expansions to order ${\mathit{T}}^{\mathrm{\ensuremath{-}}34}$, we estimate that the multicritical point N lies on the Nishimori line at ${\mathit{T}}_{\mathit{c}}$/J=1.690\ifmmode\pm\else\textpm\fi{}0.016. Along this line the critical exponents are found to be \ensuremath{\gamma}=1.80\ifmmode\pm\else\textpm\fi{}0.15 and \ensuremath{\nu}=0.85\ifmmode\pm\else\textpm\fi{}0.08. The latter is clearly consistent with the rigorous exponent inequality \ensuremath{\nu}\ensuremath{\ge}2/d. We also calculate the crossover exponent \ensuremath{\varphi} and show that the scaling axes at N are in agreement with the recent predictions of Le Doussal and Harris.

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Ising spin-glass critical and multicritical fixed distributions from a renormalization-group calculation with quenched randomness

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The Ising model with a quenched random distribution of ferromagnetic and antiferromagnetic interactions has been investigated by tracking the probability distribution of interactions under rescaling. As the initial probability of a ferromagnetic interaction is increased, the phase diagram in three dimensions shows three ordered phases: antiferromagnetic, spin glass and ferromagnetic. We find that, within the spin-glass region, effective interactions between pairs of spins at successive length scales occur in a chaotic sequence, while the average magnitudes of the ferromagnetic and the antiferromagnetic interactions both flow to infinity under iteration. Our calculated spin-glass multicritical points are on the Nishimori line in cases of both bimodal (±J) and diluted (trimodal) distributions, and these points flow under rescaling to a common fixed distribution with a specific heat exponent α of approximately −3.7.The Ising model with a quenched random distribution of ferromagnetic and antiferromagnetic interactions has been investigated by tracking the probability distribution of interactions under rescaling. As the initial probability of a ferromagnetic interaction is increased, the phase diagram in three dimensions shows three ordered phases: antiferromagnetic, spin glass and ferromagnetic. We find that, within the spin-glass region, effective interactions between pairs of spins at successive length scales occur in a chaotic sequence, while the average magnitudes of the ferromagnetic and the antiferromagnetic interactions both flow to infinity under iteration. Our calculated spin-glass multicritical points are on the Nishimori line in cases of both bimodal (±J) and diluted (trimodal) distributions, and these points flow under rescaling to a common fixed distribution with a specific heat exponent α of approximately −3.7.

Journal of Applied PhysicsNov 15, 199129 citations

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Michael Hinczewski, A. Nihat Berker

The Ising spin glasses are investigated on three dual pairs of hierarchical lattices, using exact renormalization-group transformation of the quenched bond probability distribution. The goal is to investigate a recent conjecture that relates, on such pairs of dual lattices, the locations of the multicritical points, which occur on the Nishimori symmetry line. Toward this end we precisely determine the global phase diagrams for these six hierarchical spin glasses, using up to $2.5\ifmmode\times\else\texttimes\fi{}{10}^{9}$ probability bins to represent the quenched distribution subjected to an exact renormalization-group transformation. We find in all three cases that the conjecture is realized to a very good approximation, even when the mutually dual models belong to different spatial dimensionalities $d$ and have different phase diagram topologies at the multicritical points of the conjecture and even though the contributions to the conjecture from each lattice of the dual pair are strongly asymmetric. In all six phase diagrams, we find reentrance near the multicritical point. In the models with $d=2$ or 1.5, the spin glass phase does not occur and the phase boundary between the ferromagnetic and paramagnetic phases is second order with a strong violation of universality.The Ising spin glasses are investigated on three dual pairs of hierarchical lattices, using exact renormalization-group transformation of the quenched bond probability distribution. The goal is to investigate a recent conjecture that relates, on such pairs of dual lattices, the locations of the multicritical points, which occur on the Nishimori symmetry line. Toward this end we precisely determine the global phase diagrams for these six hierarchical spin glasses, using up to $2.5\ifmmode\times\else\texttimes\fi{}{10}^{9}$ probability bins to represent the quenched distribution subjected to an exact renormalization-group transformation. We find in all three cases that the conjecture is realized to a very good approximation, even when the mutually dual models belong to different spatial dimensionalities $d$ and have different phase diagram topologies at the multicritical points of the conjecture and even though the contributions to the conjecture from each lattice of the dual pair are strongly asymmetric. In all six phase diagrams, we find reentrance near the multicritical point. In the models with $d=2$ or 1.5, the spin glass phase does not occur and the phase boundary between the ferromagnetic and paramagnetic phases is second order with a strong violation of universality.

arXiv:cond-mat/0507293Physical Review BOct 3, 200529 citations

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

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

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 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.

arXiv:0810.0685Journal of Physics Conference SeriesJan 1, 20094 citations

We present arguments, based on local gauge invariance, that the multicritical point of Ising spin-glasses should be located on a particular line of the phase diagram known as Nishimori's line [

tanh(\ensuremath{\beta}J)=2p\ensuremath{-}1

for the

\ifmmode\pm\else\textpm\fi{}J

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 $\ensuremath{\phi}\ensuremath{\equiv}{y}_{1}/{y}_{2}=2.62(6)$.

{\mathit{T}}_{\mathit{c}}

/J=1.690\ifmmode\pm\else\textpm\fi{}0.016. Along this line the critical exponents are found to be \ensuremath{\gamma}=1.80\ifmmode\pm\else\textpm\fi{}0.15 and \ensuremath{\nu}=0.85\ifmmode\pm\else\textpm\fi{}0.08. The latter is clearly consistent with the rigorous exponent inequality \ensuremath{\nu}\ensuremath{\ge}2/d. We also calculate the crossover exponent \ensuremath{\varphi} and show that the scaling axes at N are in agreement with the recent predictions of Le Doussal and Harris.

d

and have different phase diagram topologies at the multicritical points of the conjecture and even though the contributions to the conjecture from each lattice of the dual pair are strongly asymmetric. In all six phase diagrams, we find reentrance near the multicritical point. In the models with

d=2

or 1.5, the spin glass phase does not occur and the phase boundary between the ferromagnetic and paramagnetic phases is second order with a strong violation of universality.