Physical Review Letters

Relation of Random and Competing Nonrandom Couplings for Spin-Glasses

Fritz Haake, Maciej Lewenstein, Martin Wilkens

Published Dec 2, 1985• 9 citations

Abstract

By mapping the Edwards-Anderson model onto a nonrandom Ising model we argue that the lower critical dimension ${d}^{*}$ for the Edwards-Anderson spin-glass is close to 3 and may depend on the probability distribution of the bond strengths. For $d>{d}^{*}$, the approach to spin-glass order from the paramagnetic phase is characterized by Ising exponents. At $d={d}^{*}$, however, the critical exponents defined for $T>{T}_{c}$ are twice the corresponding Ising ones. These results are consistent with recent computer simulations.

DOI: 10.1103/physrevlett.55.2606

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Physical Review Letters

Relation of Random and Competing Nonrandom Couplings for Spin-Glasses

Fritz Haake, Maciej Lewenstein, Martin Wilkens

Published Dec 2, 1985• 9 citations

Publisher

Abstract

By mapping the Edwards-Anderson model onto a nonrandom Ising model we argue that the lower critical dimension

{d}^{*}

for the Edwards-Anderson spin-glass is close to 3 and may depend on the probability distribution of the bond strengths. For

d>{d}^{*}

, the approach to spin-glass order from the paramagnetic phase is characterized by Ising exponents. At

d={d}^{*}

, however, the critical exponents defined for

T>{T}_{c}

are twice the corresponding Ising ones. These results are consistent with recent computer simulations.

DOI: 10.1103/physrevlett.55.2606

Related papers

Stiffness of the Edwards-Anderson Model in all Dimensions

Stefan Boettcher

arXiv:cond-mat/0508061Physical Review LettersNov 4, 200591 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 $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.

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Enzo Marinari, C. Naitza, Francesco Zuliani

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Critical parameters of the three-dimensional Ising spin glass

Marco Baity‐Jesi, R. A. Baños, Andrés De la Cruz +21 more

We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to $L=40$ using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain ${T}_{\mathrm{c}}=1.1019(29)$ for the critical temperature, $\ensuremath{\nu}=2.562(42)$ for the thermal exponent, $\ensuremath{\eta}=\ensuremath{-}0.3900(36)$ for the anomalous dimension, and $\ensuremath{\omega}=1.12(10)$ for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield $\ensuremath{\alpha}=\ensuremath{-}5.69(13)$, $\ensuremath{\beta}=0.782(10)$, and $\ensuremath{\gamma}=6.13(11)$. We also compute several universal quantities at ${T}_{\mathrm{c}}$ .We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to $L=40$ using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain ${T}_{\mathrm{c}}=1.1019(29)$ for the critical temperature, $\ensuremath{\nu}=2.562(42)$ for the thermal exponent, $\ensuremath{\eta}=\ensuremath{-}0.3900(36)$ for the anomalous dimension, and $\ensuremath{\omega}=1.12(10)$ for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield $\ensuremath{\alpha}=\ensuremath{-}5.69(13)$, $\ensuremath{\beta}=0.782(10)$, and $\ensuremath{\gamma}=6.13(11)$. We also compute several universal quantities at ${T}_{\mathrm{c}}$.

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Phase diagram of the dilute Ising spin glass in general spatial dimension

Shye Shapira, Lior Klein, Joan Adler +2 more

We use high-temperature and low-concentration series to treat the dilute spin glass within a model with nearest-neighbor interactions which randomly assume the values +J, 0, -J with probabilities p/2, 1-p, p/2, respectively. Using the Harris no-free-end diagrams scheme in general spatial dimension, we obtained 15th-order series for ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ as a function of temperature for arbitrary dilution, 14th-order series for ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ as a function of dilution for selected temperatures, and 11th-order series for two higher derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ with respect to the ordering field, where ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ is the Edwards-Anderson spin-glass susceptibility. Analysis of these series yields values of ${\mathit{T}}_{\mathrm{SG}}$ (p), the critical temperature as a function of the dilution p or the analogous critical concentration ${\mathit{p}}_{\mathrm{SG}}$ (T). Thus we determine a critical line, separating the spin-glass phase from the paramagnetic phase in the T-p plane. We find values of the critical exponent \ensuremath{\gamma} and universal amplitude ratios along the critical line. Universal amplitude ratios and dominant exponents along the critical line are identical to those of the pure spin glass for a wide range of dilution, indicating the same critical behavior as that of the pure spin glass.We use high-temperature and low-concentration series to treat the dilute spin glass within a model with nearest-neighbor interactions which randomly assume the values +J, 0, -J with probabilities p/2, 1-p, p/2, respectively. Using the Harris no-free-end diagrams scheme in general spatial dimension, we obtained 15th-order series for ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ as a function of temperature for arbitrary dilution, 14th-order series for ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ as a function of dilution for selected temperatures, and 11th-order series for two higher derivatives of ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ with respect to the ordering field, where ${\mathrm{\ensuremath{\chi}}}^{\mathrm{EA}}$ is the Edwards-Anderson spin-glass susceptibility. Analysis of these series yields values of ${\mathit{T}}_{\mathrm{SG}}$(p), the critical temperature as a function of the dilution p or the analogous critical concentration ${\mathit{p}}_{\mathrm{SG}}$(T). Thus we determine a critical line, separating the spin-glass phase from the paramagnetic phase in the T-p plane. We find values of the critical exponent \ensuremath{\gamma} and universal amplitude ratios along the critical line. Universal amplitude ratios and dominant exponents along the critical line are identical to those of the pure spin glass for a wide range of dilution, indicating the same critical behavior as that of the pure spin glass.

Physical review. B, Condensed matterApr 1, 19946 citations

By mapping the Edwards-Anderson model onto a nonrandom Ising model we argue that the lower critical dimension

{d}^{*}

for the Edwards-Anderson spin-glass is close to 3 and may depend on the probability distribution of the bond strengths. For

d>{d}^{*}

, the approach to spin-glass order from the paramagnetic phase is characterized by Ising exponents. At

d={d}^{*}

, however, the critical exponents defined for

T>{T}_{c}

are twice the corresponding Ising ones. These results are consistent with recent computer simulations.

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 $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.

d=2, 3, \mathrm{and} 4

. The phase diagram, which is in qualitative agreement with mean-field results, exhibits paramagnetic, ferromagnetic, antiferromagnetic, and spin-glass phases. The spin-glass and paramagnetic phases meet along an extended second-order phase boundary, which terminates in two tricritical points. Critical and tricritical exponents are calculated. The spin-glass specific-heat exponent turns out to be large and negative, compatibly with recent experiments which show a rounded specific-heat anomaly.

We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to

L=40

using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain

{T}_{\mathrm{c}}=1.1019(29)

for the critical temperature, $\ensuremath{\nu}=2.562(42)$ for the thermal exponent, $\ensuremath{\eta}=\ensuremath{-}0.3900(36)$ for the anomalous dimension, and $\ensuremath{\omega}=1.12(10)$ for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield $\ensuremath{\alpha}=\ensuremath{-}5.69(13)$, $\ensuremath{\beta}=0.782(10)$, and $\ensuremath{\gamma}=6.13(11)$. We also compute several universal quantities at

{T}_{\mathrm{c}}

{\mathit{T}}_{\mathrm{SG}}

(p), the critical temperature as a function of the dilution p or the analogous critical concentration

{\mathit{p}}_{\mathrm{SG}}

(T). Thus we determine a critical line, separating the spin-glass phase from the paramagnetic phase in the T-p plane. We find values of the critical exponent \ensuremath{\gamma} and universal amplitude ratios along the critical line. Universal amplitude ratios and dominant exponents along the critical line are identical to those of the pure spin glass for a wide range of dilution, indicating the same critical behavior as that of the pure spin glass.