Physical review. B, Condensed matter

Evidence against a spin-glass transition in three dimensions

Gyan V Bhanot, K.J.M. Moriarty, Román Salvador

Published Jan 1, 1989• 1 citation

Abstract

We study the Ising spin-glass model in two and three dimensions on small lattices by exactly computing the partition function for a large number of bond configurations. An analysis of the zeros of these partition functions suggests that there is no nonanalyticity in the free energy of the spin glass in either two or three dimensions.

DOI: 10.1103/physrevb.39.356

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Phase Transition of Ising Spin Glass Models in Two Dimensions

Michinori Shiomi, Fumitaka Matsubara, Takayuki Shirakura

We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy $\overline{\Delta F(T)}$ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, $\overline{\Delta F(T)}$ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, $\overline{\Delta F(T)}$ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .

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Renormalization-Group Approach to Spin Glass Transition of a Random Bond Ising Model in Two- and Three-Dimensions

T. Tatsumi

Progress of Theoretical PhysicsFeb 1, 197810 citations

Numerical studies of Ising spin glasses in two, three, and four dimensions

R. N. Bhatt, A. P. Young

Physical review. B, Condensed matterApr 1, 1988301 citations

Domain-Wall Renormalization-Group Study of Spin-Glass Models

Koji Hukushima

An efficient Monte Carlo method is extended to evaluate directly domain-wall free energy for randomly frustrated spin systems. Using the method, critical phenomena of Ising and Heisenberg spin glasses in three dimensions are investigated. Clear evidence of a finite-temperature spin-glass phase transition is presented in the Ising spin glass, while there is no such tendency in Heisenberg spin glass.

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Critical behavior of three-dimensional Ising spin glass models

Martin Hasenbusch B, Andrea Pelissetto C, Ettore Vicari

arXiv:0809.3329Physical Review BDec 17, 2008119 citations

Spin glass behavior in a random Coulomb antiferromagnet

J. Rehn, Roderich Moessner, A. P. Young

arXiv:1607.04587Physical review. ESep 21, 20165 citations

Physical review. B, Condensed matter

Evidence against a spin-glass transition in three dimensions

Gyan V Bhanot, K.J.M. Moriarty, Román Salvador

Published Jan 1, 1989• 1 citation

Publisher

Abstract

DOI: 10.1103/physrevb.39.356

Related papers

Phase Transition of Ising Spin Glass Models in Two Dimensions

Michinori Shiomi, Fumitaka Matsubara, Takayuki Shirakura

We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy $\overline{\Delta F(T)}$ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, $\overline{\Delta F(T)}$ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, $\overline{\Delta F(T)}$ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy $\overline{\Delta F(T)}$ and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution, $\overline{\Delta F(T)}$ at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution, $\overline{\Delta F(T)}$ decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .

Journal of the Physical Society of JapanSep 1, 200010 citations

Renormalization-Group Approach to Spin Glass Transition of a Random Bond Ising Model in Two- and Three-Dimensions

T. Tatsumi

An Ising spin glass of random bonds with the Gaussian and two-peaked distributions in two- and three-dimensions is investigated with the cumulant expansion and two-step decimation transformations. It is found that the spin glass state exists in two- and three-dimensions and that the ferromagnetic phase shrinks to that of the pure system in two-dimension for the two-peaked distribution. The specific heat exponent at the spin glass transition point is found to be -1.08 and -0.471 for the square and simple cubic lattices, respectively. Also the effect of bond correlations on the spin glass state is considered.An Ising spin glass of random bonds with the Gaussian and two-peaked distributions in two- and three-dimensions is investigated with the cumulant expansion and two-step decimation transformations. It is found that the spin glass state exists in two- and three-dimensions and that the ferromagnetic phase shrinks to that of the pure system in two-dimension for the two-peaked distribution. The specific heat exponent at the spin glass transition point is found to be -1.08 and -0.471 for the square and simple cubic lattices, respectively. Also the effect of bond correlations on the spin glass state is considered.

Progress of Theoretical PhysicsFeb 1, 197810 citations

Numerical studies of Ising spin glasses in two, three, and four dimensions

R. N. Bhatt, A. P. Young

We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at ${T}_{c}$ =0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero ${T}_{c}$ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that ${T}_{c}$ is nonzero with a long-range-ordered phase below ${T}_{c}$ . Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.We present the results of numerical simulations on Ising spin glasses in zero magnetic field with nearest-neighbor interactions on hyper-cubic lattices in two, three, and four dimensions with both Gaussian and \ifmmode\pm\else\textpm\fi{}J bond distributions. Finite-size scaling is used to analyze the results. In two dimensions (d=2) we agree with earlier work that the transition temperature is at ${T}_{c}$=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero ${T}_{c}$ but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that ${T}_{c}$ is nonzero with a long-range-ordered phase below ${T}_{c}$. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.

Physical review. B, Condensed matterApr 1, 1988301 citations

Domain-Wall Renormalization-Group Study of Spin-Glass Models

Koji Hukushima

Progress of Theoretical Physics SupplementJan 1, 20000 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

Spin glass behavior in a random Coulomb antiferromagnet

J. Rehn, Roderich Moessner, A. P. Young

We study spin glass behavior in a random Ising Coulomb antiferromagnet in two and three dimensions using Monte Carlo simulations. In two dimensions, we find a transition at zero temperature with critical exponents consistent with those of the Edwards-Anderson model, though with large uncertainties. In three dimensions, evidence for a finite-temperature transition, as occurs in the Edwards-Anderson model, is rather weak. This may indicate that the sizes are too small to probe the asymptotic critical behavior, or possibly that the universality class is different from that of the Edwards-Anderson model and has a lower critical dimension equal to three.We study spin glass behavior in a random Ising Coulomb antiferromagnet in two and three dimensions using Monte Carlo simulations. In two dimensions, we find a transition at zero temperature with critical exponents consistent with those of the Edwards-Anderson model, though with large uncertainties. In three dimensions, evidence for a finite-temperature transition, as occurs in the Edwards-Anderson model, is rather weak. This may indicate that the sizes are too small to probe the asymptotic critical behavior, or possibly that the universality class is different from that of the Edwards-Anderson model and has a lower critical dimension equal to three.

arXiv:1607.04587Physical review. ESep 21, 20165 citations

We examine the phase transition of Ising spin glass models in two dimensions (2D), calculating complementary two quantities, i.e., the interface free energy

\overline{\Delta F(T)}

and the Binder parameter g L , on finite L × L ( L ≤24) lattices at very low temperatures T . We find that these quantities exhibit quite different features depending on the distribution of bonds. For the ± J distribution,

\overline{\Delta F(T)}

at very low temperatures slightly increases with L and g L intersect at T ∼0.25 J . These results suggest a non-zero temperature phase transition, T c ≠0. On the other hand, for the Gaussian distribution,

\overline{\Delta F(T)}

decreases as L increases even at T = 0 and g L for different L converges to unity at T = 0. These results confirm the assumption of the zero temperature phase transition, T c = 0. Finite-size scaling analyses support those results. Thus we suggest that, in 2D, the existence of a finite-temperature phase transition depends on the distribution of bonds and it exists when the bond distribution is ± J .

{T}_{c}

=0, and obtain the correlation-length exponent \ensuremath{\nu}, and the exponent \ensuremath{\eta}, at the zero-temperature transition for the \ifmmode\pm\else\textpm\fi{}J model. In d=3 dimensions we concentrate on results for the Gaussian distribution, since our results for the \ifmmode\pm\else\textpm\fi{}J distribution have been presented earlier. As expected, we find similar results for the two distributions, namely a nonzero

{T}_{c}

but evidence that d=3 is close to the lower critical dimension. In a four-dimensional spin glass with Gaussian bonds we find that only a modest amount of computer time is required to show that

{T}_{c}

is nonzero with a long-range-ordered phase below

{T}_{c}

. Our estimates for critical exponents in d=4 dimensions agree well with results from recent high-temperature-series expansions.

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