arXiv:0802.1941 · Physical Review B

Reduction of dilute Ising spin glasses

Stefan Boettcher, James E. Davidheiser

Published Jun 25, 2008• 6 citations

Abstract

The recently proposed reduction method for diluted spin glasses is investigated in depth. In particular, the Edwards-Anderson model with $\ifmmode\pm\else\textpm\fi{}J$ and Gaussian bond disorder on hypercubic lattices in $d=2$, 3, and 4 is studied for a range of bond dilutions. The results demonstrate the effectiveness of using bond dilution to elucidate low-temperature properties of Ising spin glasses and provide a starting point to enhance the methods used in reduction. Based on that, a greedy heuristic called ``dominant bond reduction'' is introduced and explored.

DOI: 10.1103/physrevb.77.214432

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Reduction of two-dimensional dilute Ising spin glasses

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Low-temperature phase boundary of dilute-lattice spin glasses

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arXiv:0802.1941 · Physical Review B

Reduction of dilute Ising spin glasses

Stefan Boettcher, James E. Davidheiser

Published Jun 25, 2008• 6 citations

arXiv HTML PDF

Abstract

The recently proposed reduction method for diluted spin glasses is investigated in depth. In particular, the Edwards-Anderson model with

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

and Gaussian bond disorder on hypercubic lattices in

d=2

, 3, and 4 is studied for a range of bond dilutions. The results demonstrate the effectiveness of using bond dilution to elucidate low-temperature properties of Ising spin glasses and provide a starting point to enhance the methods used in reduction. Based on that, a greedy heuristic called ``dominant bond reduction'' is introduced and explored.

DOI: 10.1103/physrevb.77.214432

Related papers

Reduction of two-dimensional dilute Ising spin glasses

Stefan Boettcher, Alexander K. Hartmann

The recently proposed reduction method is applied to the Edwards-Anderson model on bond-diluted square lattices. In combination with a graph-theoretical matching algorithm, this allows us to calculate numerically exact ground states of large systems. Low-temperature domain-wall excitations are studied to determine the stiffness exponent ${y}_{2}$ . A value of ${y}_{2}=\ensuremath{-}0.281(3)$ is found, consistent with previous results obtained on undiluted lattices. This comparison demonstrates the validity of the reduction method for bond-diluted spin systems and provides strong support for similar studies proclaiming accurate results for stiffness exponents in dimensions $d=3,\dots{},7$ .The recently proposed reduction method is applied to the Edwards-Anderson model on bond-diluted square lattices. In combination with a graph-theoretical matching algorithm, this allows us to calculate numerically exact ground states of large systems. Low-temperature domain-wall excitations are studied to determine the stiffness exponent ${y}_{2}$. A value of ${y}_{2}=\ensuremath{-}0.281(3)$ is found, consistent with previous results obtained on undiluted lattices. This comparison demonstrates the validity of the reduction method for bond-diluted spin systems and provides strong support for similar studies proclaiming accurate results for stiffness exponents in dimensions $d=3,\dots{},7$.

arXiv:cond-mat/0503486Physical Review BJul 14, 20055 citations

Obtaining stiffness exponents from bond-diluted lattice spin glasses

Stefan Boettcher, S. E. Cooke

Recently, a method has been proposed to obtain accurate predictions for low-temperature properties of lattice spin glasses that is practical even above the upper critical dimension, ${d}_{c}=6$ . This method is based on the observation that bond-dilution enables the numerical treatment of larger lattices, and that the subsequent combination of such data at various bond densities into a finite-size scaling ansatz produces more robust scaling behavior. In the present study we test the potential of such a procedure, in particular, to obtain the stiffness exponent for the hierarchical Migdal-Kadanoff lattice. Critical exponents for this model are known with great accuracy and any simulations can be executed to very large lattice sizes at almost any bond density, effecting an insightful comparison that highlights the advantages---as well as the weaknesses---of this method. These insights are applied to the Edwards-Anderson model in $d=3$ with Gaussian bonds.Recently, a method has been proposed to obtain accurate predictions for low-temperature properties of lattice spin glasses that is practical even above the upper critical dimension, ${d}_{c}=6$. This method is based on the observation that bond-dilution enables the numerical treatment of larger lattices, and that the subsequent combination of such data at various bond densities into a finite-size scaling ansatz produces more robust scaling behavior. In the present study we test the potential of such a procedure, in particular, to obtain the stiffness exponent for the hierarchical Migdal-Kadanoff lattice. Critical exponents for this model are known with great accuracy and any simulations can be executed to very large lattice sizes at almost any bond density, effecting an insightful comparison that highlights the advantages---as well as the weaknesses---of this method. These insights are applied to the Edwards-Anderson model in $d=3$ with Gaussian bonds.

arXiv:cond-mat/0501547Physical Review BJun 9, 20053 citations

Low-temperature phase boundary of dilute-lattice spin glasses

Stefan Boettcher, Emiliano Marchetti

The thermal-to-percolative crossover exponent $\ensuremath{\phi}$, well known for ferromagnetic systems, is studied extensively for Edwards--Anderson spin glasses. The scaling of defect energies are determined at the bond percolation threshold ${p}_{c}$ using an improved reduction algorithm. Simulations extend to system sizes above $N={10}^{8}$ in dimensions $d=2,\dots{},7$ . The results can be related to the behavior of the transition temperature ${T}_{g}\ensuremath{\sim}{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\phi}}$ between the paramagnetic and the glassy regime for $p\ensuremath{\searrow}{p}_{c}$. In three dimensions, where our simulations predict $\ensuremath{\phi}=1.127(5)$, this scaling form for ${T}_{g}$ provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.The thermal-to-percolative crossover exponent $\ensuremath{\phi}$, well known for ferromagnetic systems, is studied extensively for Edwards--Anderson spin glasses. The scaling of defect energies are determined at the bond percolation threshold ${p}_{c}$ using an improved reduction algorithm. Simulations extend to system sizes above $N={10}^{8}$ in dimensions $d=2,\dots{},7$. The results can be related to the behavior of the transition temperature ${T}_{g}\ensuremath{\sim}{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\phi}}$ between the paramagnetic and the glassy regime for $p\ensuremath{\searrow}{p}_{c}$. In three dimensions, where our simulations predict $\ensuremath{\phi}=1.127(5)$, this scaling form for ${T}_{g}$ provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.

Physical Review BMar 27, 20083 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

Low-temperature excitations of dilute lattice spin glasses

Stefan Boettcher

A new approach to exploring low-temperature excitations in finite-dimensional lattice spin glasses is proposed. By focusing on bond-diluted lattices just above the percolation threshold, large system sizes L can be obtained which lead to enhanced scaling regimes and more accurate exponents. Furthermore, this method in principle remains practical for any dimension, yielding exponents that so far have been elusive. This approach is demonstrated by determining the stiffness exponent for dimensions d = 3, d = 6 (the upper critical dimension), and d = 7. Key is the application of an exact reduction algorithm, which eliminates a large fraction of spins, so that the reduced lattices never exceed ∼ 10 3 variables for sizes as large as L = 30 in d = 3, L = 9 in d = 6, or L = 8 in d = 7. Finite-size scaling analysis gives y 3 = 0.24(1) for d = 3, significantly improving on previous work. The results for d = 6 and d = 7, y 6 = 1.1(1) and y 7 = 1.24(5), are entirely new and are compared with mean-field predictions made for d ⩾ 6.A new approach to exploring low-temperature excitations in finite-dimensional lattice spin glasses is proposed. By focusing on bond-diluted lattices just above the percolation threshold, large system sizes L can be obtained which lead to enhanced scaling regimes and more accurate exponents. Furthermore, this method in principle remains practical for any dimension, yielding exponents that so far have been elusive. This approach is demonstrated by determining the stiffness exponent for dimensions d = 3, d = 6 (the upper critical dimension), and d = 7. Key is the application of an exact reduction algorithm, which eliminates a large fraction of spins, so that the reduced lattices never exceed ∼ 10 3 variables for sizes as large as L = 30 in d = 3, L = 9 in d = 6, or L = 8 in d = 7. Finite-size scaling analysis gives y 3 = 0.24(1) for d = 3, significantly improving on previous work. The results for d = 6 and d = 7, y 6 = 1.1(1) and y 7 = 1.24(5), are entirely new and are compared with mean-field predictions made for d ⩾ 6.

arXiv:cond-mat/0303431Europhysics Letters (EPL)Aug 1, 200438 citations

Physics of the Edwards–Anderson spin glass in dimensions d = 3, … ,8 from heuristic ground state optimization

Stefan Boettcher

We present a collection of simulations of the Edwards–Anderson lattice spin glass at T=0 to elucidate the nature of low-energy excitations over a range of dimensions that reach from physically realizable systems to the mean-field limit. Using heuristic methods, we sample the ground states of instances to determine their energies while eliciting excitations through manipulating boundary conditions. We exploit the universality of the phase diagram of bond-diluted lattices to make such a study in higher dimensions computationally feasible. As a result, we obtain a variety of accurate exponents for domain wall stiffness and finite-size corrections, which allow us to examine their dimensional behavior and their connection with predictions from mean-field theory. We also provide an experimentally testable prediction for the thermal-to-percolative crossover exponent in dilute lattice Ising spin glasses.We present a collection of simulations of the Edwards–Anderson lattice spin glass at T=0 to elucidate the nature of low-energy excitations over a range of dimensions that reach from physically realizable systems to the mean-field limit. Using heuristic methods, we sample the ground states of instances to determine their energies while eliciting excitations through manipulating boundary conditions. We exploit the universality of the phase diagram of bond-diluted lattices to make such a study in higher dimensions computationally feasible. As a result, we obtain a variety of accurate exponents for domain wall stiffness and finite-size corrections, which allow us to examine their dimensional behavior and their connection with predictions from mean-field theory. We also provide an experimentally testable prediction for the thermal-to-percolative crossover exponent in dilute lattice Ising spin glasses.

arXiv:2407.14646Frontiers in PhysicsSep 20, 20242 citations

The recently proposed reduction method for diluted spin glasses is investigated in depth. In particular, the Edwards-Anderson model with

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

and Gaussian bond disorder on hypercubic lattices in

d=2

{y}_{2}

. A value of ${y}_{2}=\ensuremath{-}0.281(3)$ is found, consistent with previous results obtained on undiluted lattices. This comparison demonstrates the validity of the reduction method for bond-diluted spin systems and provides strong support for similar studies proclaiming accurate results for stiffness exponents in dimensions

d=3,\dots{},7

Recently, a method has been proposed to obtain accurate predictions for low-temperature properties of lattice spin glasses that is practical even above the upper critical dimension,

{d}_{c}=6

. This method is based on the observation that bond-dilution enables the numerical treatment of larger lattices, and that the subsequent combination of such data at various bond densities into a finite-size scaling ansatz produces more robust scaling behavior. In the present study we test the potential of such a procedure, in particular, to obtain the stiffness exponent for the hierarchical Migdal-Kadanoff lattice. Critical exponents for this model are known with great accuracy and any simulations can be executed to very large lattice sizes at almost any bond density, effecting an insightful comparison that highlights the advantages---as well as the weaknesses---of this method. These insights are applied to the Edwards-Anderson model in

d=3

with Gaussian bonds.

{p}_{c}

using an improved reduction algorithm. Simulations extend to system sizes above

N={10}^{8}

in dimensions

d=2,\dots{},7

. The results can be related to the behavior of the transition temperature ${T}_{g}\ensuremath{\sim}{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\phi}}$ between the paramagnetic and the glassy regime for $p\ensuremath{\searrow}{p}_{c}$. In three dimensions, where our simulations predict $\ensuremath{\phi}=1.127(5)$, this scaling form for

{T}_{g}

provides a rare experimental test of predictions arising from the equilibrium theory of low-temperature spin glasses. For dimensions near and above the upper critical dimension, the results provide a challenge to reconcile mean-field theory with finite-dimensional properties.

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