Physical Review Letters

Integer Optimization and Zero-Temperature Fixed Point in Ising Random-Field Systems

Andrew T. Ogielski

Published Sep 8, 1986• 200 citations

Abstract

Phase transition in $d=3$ ferromagnetic Ising models with random fields is analyzed directly at the zero-temperature critical point. Critical behavior is extracted from correlation functions averaged over an ensemble of exact ground states obtained with a new integer optimization algorithm. For Gaussian distribution of random fields finite-size scaling demonstrates a continuous phase transition with effective disconnected susceptibility exponent $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}\ensuremath{\approx}\ensuremath{-}0.9$, correlation-length exponent $\ensuremath{\nu}\ensuremath{\approx}1.0$, and magnetization exponent $\ensuremath{\beta}\ensuremath{\approx}0.05$.

DOI: 10.1103/physrevlett.57.1251

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Physical Review Letters

Integer Optimization and Zero-Temperature Fixed Point in Ising Random-Field Systems

Andrew T. Ogielski

Published Sep 8, 1986• 200 citations

Publisher

Abstract

Phase transition in

d=3

ferromagnetic Ising models with random fields is analyzed directly at the zero-temperature critical point. Critical behavior is extracted from correlation functions averaged over an ensemble of exact ground states obtained with a new integer optimization algorithm. For Gaussian distribution of random fields finite-size scaling demonstrates a continuous phase transition with effective disconnected susceptibility exponent

\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}\ensuremath{\approx}\ensuremath{-}0.9

, correlation-length exponent

\ensuremath{\nu}\ensuremath{\approx}1.0

, and magnetization exponent

\ensuremath{\beta}\ensuremath{\approx}0.05

DOI: 10.1103/physrevlett.57.1251

Related papers

Critical exponents of four-dimensional random-field Ising systems

Alexander K. Hartmann

The ferromagnet-to-paramagnet transition of the four-dimensional random-field Ising model with Gaussian distribution of the random fields is studied. Exact ground states of systems with sizes up to ${32}^{4}$ are obtained using graph theoretical algorithms. The magnetization, the disconnected susceptibility, the susceptibility, and a specific-heat-like quantity are calculated. Using finite-size scaling techniques, the corresponding critical exponents are obtained: $\ensuremath{\beta}=0.13(5),$ $\overline{\ensuremath{\gamma}}=2.83(50),$ $\ensuremath{\gamma}=1.42(20).$ For the specific heat both $\ensuremath{\alpha}=0$ (logarithmic divergence) and an algebraic divergence with $\ensuremath{\alpha}=0.26(5)$ are compatible with the data. Furthermore, values for the critical randomness ${h}_{c}=4.18(1)$ and the correlation-length exponent $\ensuremath{\nu}=0.78(10)$ were found. These independently obtained exponents are compatible with all (hyper)scaling relations and support the two-exponent scenario $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$The ferromagnet-to-paramagnet transition of the four-dimensional random-field Ising model with Gaussian distribution of the random fields is studied. Exact ground states of systems with sizes up to ${32}^{4}$ are obtained using graph theoretical algorithms. The magnetization, the disconnected susceptibility, the susceptibility, and a specific-heat-like quantity are calculated. Using finite-size scaling techniques, the corresponding critical exponents are obtained: $\ensuremath{\beta}=0.13(5),$ $\overline{\ensuremath{\gamma}}=2.83(50),$ $\ensuremath{\gamma}=1.42(20).$ For the specific heat both $\ensuremath{\alpha}=0$ (logarithmic divergence) and an algebraic divergence with $\ensuremath{\alpha}=0.26(5)$ are compatible with the data. Furthermore, values for the critical randomness ${h}_{c}=4.18(1)$ and the correlation-length exponent $\ensuremath{\nu}=0.78(10)$ were found. These independently obtained exponents are compatible with all (hyper)scaling relations and support the two-exponent scenario $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$

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Marco Picco, Nicolas Sourlas

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Fluctuations and criticality in the random-field Ising model

Panagiotis E. Theodorakis, Ioannis Georgiou, Nikolaos G. Fytas

We investigate the critical properties of the $d=3$ random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength $h$ and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength ${h}_{\mathrm{c}}$ and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.We investigate the critical properties of the $d=3$ random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength $h$ and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength ${h}_{\mathrm{c}}$ and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.

Physical Review EMar 8, 201311 citations

Phase transition in

d=3

\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\eta}}\ensuremath{\approx}\ensuremath{-}0.9

, correlation-length exponent

\ensuremath{\nu}\ensuremath{\approx}1.0

, and magnetization exponent

\ensuremath{\beta}\ensuremath{\approx}0.05

The ferromagnet-to-paramagnet transition of the four-dimensional random-field Ising model with Gaussian distribution of the random fields is studied. Exact ground states of systems with sizes up to

{32}^{4}

are obtained using graph theoretical algorithms. The magnetization, the disconnected susceptibility, the susceptibility, and a specific-heat-like quantity are calculated. Using finite-size scaling techniques, the corresponding critical exponents are obtained: $\ensuremath{\beta}=0.13(5),$ $\overline{\ensuremath{\gamma}}=2.83(50),$ $\ensuremath{\gamma}=1.42(20).$ For the specific heat both $\ensuremath{\alpha}=0$ (logarithmic divergence) and an algebraic divergence with $\ensuremath{\alpha}=0.26(5)$ are compatible with the data. Furthermore, values for the critical randomness

{h}_{c}=4.18(1)

and the correlation-length exponent $\ensuremath{\nu}=0.78(10)$ were found. These independently obtained exponents are compatible with all (hyper)scaling relations and support the two-exponent scenario $(\overline{\ensuremath{\gamma}}=2\ensuremath{\gamma}).$

{\mathit{h}}_{\mathit{r}}

{)}^{2}

being close to one this gives further support for a two-exponent scaling scenario implying \ensuremath{\eta}\ifmmode\bar\else\textasciimacron\fi{}=2\ensuremath{\eta}. The magnetization behaves discontinuously at the transition, i.e., \ensuremath{\beta}=0. However, no divergence for the specific heat and in particular no latent heat is found. Also the probability distribution of the magnetization does not show a multipeak structure that would be characteristic for the phase-coexistence at first-order phase-transition points.

The disorder-driven phase transition of the random-field Ising model is observed using exact ground-state computer simulations for hypercubic lattices in

d=5,6,7

dimensions. Finite-size scaling analyses are used to calculate the critical point and the critical exponents of the specific heat, magnetization, susceptibility, and of the correlation length. For dimensions

d=6,7

which are larger or equal to the assumed upper critical dimension,

{d}_{u}=6

, mean-field behavior is found, i.e., $\ensuremath{\alpha}=0, \ensuremath{\beta}=1/2, \ensuremath{\gamma}=1, \ensuremath{\nu}=1/2$. For the analysis of the numerical data, it appears to be necessary to include recently proposed corrections to scaling at and beyond the upper critical dimension.

We investigate the critical properties of the

d=3

random-field Ising model with a Gaussian field distribution at zero temperature. By implementing suitable graph-theoretical algorithms, we perform a large-scale numerical simulation of the model for a vast range of values of the disorder strength

h

and system sizes $\mathcal{V}=L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$, with $L\ensuremath{\le}156$. Using the sample-to-sample fluctuations of various quantities and proper finite-size scaling techniques we estimate with high accuracy the critical disorder strength

{h}_{\mathrm{c}}

and the correlation length exponent $\ensuremath{\nu}$. Additional simulations in the area of the estimated critical-field strength and relevant scaling analysis of the bond energy suggest bounds for the specific heat critical exponent $\ensuremath{\alpha}$ and the violation of the hyperscaling exponent $\ensuremath{\theta}$. Finally, a data collapse analysis of the order parameter and disconnected susceptibility provides accurate estimates for the critical exponent ratios $\ensuremath{\beta}/\ensuremath{\nu}$ and $\overline{\ensuremath{\gamma}}/\ensuremath{\nu}$, respectively.