arXiv:2207.08193 · Physical review. B./Physical review. B

Ground-state interface exponents of the diluted Sherrington-Kirkpatrick spin glass

Wenlong Wang

Published Oct 27, 2022• 2 citations

Abstract

We present a large-scale simulation of the ground state interface properties of the diluted Sherrington-Kirkpatrick spin glass of Gaussian disorder for a broad range of the bond occupation probability $p$ using the strong disorder renormalization group and the population annealing Monte Carlo methods. The model is studied in the framework of the diluted one-dimensional Kotliar-Anderson-Stein spin glass with power-law interactions in the mean-field regime. We find that the interface is space-filling independent of $p$, i.e., the fractal dimension ${d}_{s}=1$. The stiffness exponent $\ensuremath{\theta}$ is likely also independent of $p$, despite that the energy finite-size correction exponent $\ensuremath{\omega}$ varies with $p$ as recently found. The energy finite-size scaling is also analyzed and compared with that of the $\ifmmode\pm\else\textpm\fi{}J$ disorder, finding that the thermodynamic energy is universal in both $p$ and the disorder, and the exponent $\ensuremath{\omega}$ varies with $p$ but is universal in the disorder.

DOI: 10.1103/physrevb.106.134209

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Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

Stefan Boettcher

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arXiv:2207.08193 · Physical review. B./Physical review. B

Ground-state interface exponents of the diluted Sherrington-Kirkpatrick spin glass

Wenlong Wang

Published Oct 27, 2022• 2 citations

arXiv HTML PDF

Abstract

p

using the strong disorder renormalization group and the population annealing Monte Carlo methods. The model is studied in the framework of the diluted one-dimensional Kotliar-Anderson-Stein spin glass with power-law interactions in the mean-field regime. We find that the interface is space-filling independent of

p

, i.e., the fractal dimension

{d}_{s}=1

. The stiffness exponent

\ensuremath{\theta}

is likely also independent of

p

, despite that the energy finite-size correction exponent

\ensuremath{\omega}

varies with

p

as recently found. The energy finite-size scaling is also analyzed and compared with that of the

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

disorder, finding that the thermodynamic energy is universal in both

p

and the disorder, and the exponent

\ensuremath{\omega}

varies with

p

but is universal in the disorder.

DOI: 10.1103/physrevb.106.134209

Related papers

Ground State Properties of the Diluted Sherrington-Kirkpatrick Spin Glass

Stefan Boettcher

We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond diluted to an overall density p, such that the average degree diverges as ∼pN with the system size N. Ground state energies are obtained with high accuracy for random instances over a wide range of fixed p. Since this is an NP-hard combinatorial problem, we employ the extremal optimization heuristic to that end. We find that the exponent describing the finite-size corrections ω varies continuously with p, a somewhat surprising result, as one would not expect that gradual bond dilution would change the T=0 universality class of a statistical model. For p→1, the familiar result of ω(p=1)≈2/3 for the SK model is obtained.We present a numerical study of ground states of the dilute versions of the Sherrington-Kirkpatrick (SK) mean-field spin glass. In contrast to so-called "sparse" mean-field spin glasses that have been studied widely on random networks of finite (average or regular) degree, the networks studied here are randomly bond diluted to an overall density p, such that the average degree diverges as ∼pN with the system size N. Ground state energies are obtained with high accuracy for random instances over a wide range of fixed p. Since this is an NP-hard combinatorial problem, we employ the extremal optimization heuristic to that end. We find that the exponent describing the finite-size corrections ω varies continuously with p, a somewhat surprising result, as one would not expect that gradual bond dilution would change the T=0 universality class of a statistical model. For p→1, the familiar result of ω(p=1)≈2/3 for the SK model is obtained.

arXiv:1912.04974Physical Review LettersApr 28, 202011 citations

Finite-size corrections for ground states of Edwards-Anderson spin glasses

Stefan Boettcher, Stefan Falkner

Extensive computations of ground-state energies of the Edwards-Anderson spin glass on bond-diluted, hypercubic lattices are conducted in dimensions d =3, ..., 7. Results are presented for bond densities exactly at the percolation threshold, p = p c , and deep within the glassy regime, p > p c , where finding ground states is one of the hardest combinatorial optimization problems. Finite-size corrections of the form 1/ N ω are shown to be consistent throughout with the prediction ω=1− y / d , where y refers to the "stiffness" exponent that controls the formation of domain wall excitations at low temperatures. At p = p c , an extrapolation for d → ∞ appears to match our mean-field results for these corrections. In the glassy phase, however, ω does not approach its anticipated mean-field value of 2/3, obtained from simulations of the Sherrington-Kirkpatrick spin glass on an N -clique graph. Instead, the value of ω reached at the upper critical dimension matches another type of mean-field spin glass models, namely those on sparse random networks of regular degree called Bethe lattices.Extensive computations of ground-state energies of the Edwards-Anderson spin glass on bond-diluted, hypercubic lattices are conducted in dimensions d =3, ..., 7. Results are presented for bond densities exactly at the percolation threshold, p = p c , and deep within the glassy regime, p > p c , where finding ground states is one of the hardest combinatorial optimization problems. Finite-size corrections of the form 1/ N ω are shown to be consistent throughout with the prediction ω=1− y / d , where y refers to the "stiffness" exponent that controls the formation of domain wall excitations at low temperatures. At p = p c , an extrapolation for d → ∞ appears to match our mean-field results for these corrections. In the glassy phase, however, ω does not approach its anticipated mean-field value of 2/3, obtained from simulations of the Sherrington-Kirkpatrick spin glass on an N -clique graph. Instead, the value of ω reached at the upper critical dimension matches another type of mean-field spin glass models, namely those on sparse random networks of regular degree called Bethe lattices.

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The fractal dimension of excitations in glassy systems gives information on the critical dimension at which the droplet picture of spin glasses changes to a description based on replica symmetry breaking where the interfaces are space filling. Here, the fractal dimension of domain-wall interfaces is studied using the strong-disorder renormalization group method pioneered by Monthus [Fractals 23, 1550042 (2015)FRACEG0218-348X10.1142/S0218348X15500425] both for the Edwards-Anderson spin-glass model in up to 8 space dimensions, as well as for the one-dimensional long-ranged Ising spin-glass with power-law interactions. Analyzing the fractal dimension of domain walls, we find that replica symmetry is broken in high-enough space dimensions. Because our results for high-dimensional hypercubic lattices are limited by their small size, we have also studied the behavior of the one-dimensional long-range Ising spin-glass with power-law interactions. For the regime where the power of the decay of the spin-spin interactions with their separation distance corresponds to 6 and higher effective space dimensions, we find again the broken replica symmetry result of space filling excitations. This is not the case for smaller effective space dimensions. These results show that the dimensionality of the spin glass determines which theoretical description is appropriate. Our results will also be of relevance to the Gardner transition of structural glasses.The fractal dimension of excitations in glassy systems gives information on the critical dimension at which the droplet picture of spin glasses changes to a description based on replica symmetry breaking where the interfaces are space filling. Here, the fractal dimension of domain-wall interfaces is studied using the strong-disorder renormalization group method pioneered by Monthus [Fractals 23, 1550042 (2015)FRACEG0218-348X10.1142/S0218348X15500425] both for the Edwards-Anderson spin-glass model in up to 8 space dimensions, as well as for the one-dimensional long-ranged Ising spin-glass with power-law interactions. Analyzing the fractal dimension of domain walls, we find that replica symmetry is broken in high-enough space dimensions. Because our results for high-dimensional hypercubic lattices are limited by their small size, we have also studied the behavior of the one-dimensional long-range Ising spin-glass with power-law interactions. For the regime where the power of the decay of the spin-spin interactions with their separation distance corresponds to 6 and higher effective space dimensions, we find again the broken replica symmetry result of space filling excitations. This is not the case for smaller effective space dimensions. These results show that the dimensionality of the spin glass determines which theoretical description is appropriate. Our results will also be of relevance to the Gardner transition of structural glasses.

arXiv:1703.08679Physical Review LettersSep 7, 201718 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.

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arXiv:1509.05372Physical review. EMar 14, 201621 citations

Nature of spin glass order in physical dimensions

Bharadwaj Vedula, M A Moore, Auditya Sharma

We have studied the diluted Heisenberg spin glass model in a 3-component random field for the commonly-used one-dimensional long-range model where the probability that two spins separated by a distance $r$ interact with one another falls as $1/r^{2 σ}$ , for two values of $σ$ , $0.75$ and $0.85$ . No de Almeida-Thouless line is expected at these $σ$ values. The spin glass correlation length $ξ_{\text{SG}}$ varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when $ξ_{\text{SG}}$ becomes comparable to the system size $L$ , there are departures which we attribute to the features deriving from the TNT picture of spin glasses. For the case $σ=0.85$ these features go away for system sizes with $L >L^*$ , where $L^*$ is large ( $\approx 4000-8000$ lattice spacings). In the case of $σ= 0.75$ we have been unable to study large enough systems to determine its value of $L^*$ . We sketch a renormalization group scenario to explain how these features could arise. On this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce TNT features and some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.We have studied the diluted Heisenberg spin glass model in a 3-component random field for the commonly-used one-dimensional long-range model where the probability that two spins separated by a distance $r$ interact with one another falls as $1/r^{2 σ}$, for two values of $σ$, $0.75$ and $0.85$. No de Almeida-Thouless line is expected at these $σ$ values. The spin glass correlation length $ξ_{\text{SG}}$ varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when $ξ_{\text{SG}}$ becomes comparable to the system size $L$, there are departures which we attribute to the features deriving from the TNT picture of spin glasses. For the case $σ=0.85$ these features go away for system sizes with $L >L^*$, where $L^*$ is large ($\approx 4000-8000$ lattice spacings). In the case of $σ= 0.75$ we have been unable to study large enough systems to determine its value of $L^*$. We sketch a renormalization group scenario to explain how these features could arise. On this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce TNT features and some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.

arXiv:2410.19069Oct 24, 20240 citations

p

p

, i.e., the fractal dimension

{d}_{s}=1

. The stiffness exponent

\ensuremath{\theta}

is likely also independent of

p

, despite that the energy finite-size correction exponent

\ensuremath{\omega}

varies with

p

as recently found. The energy finite-size scaling is also analyzed and compared with that of the

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

disorder, finding that the thermodynamic energy is universal in both

p

and the disorder, and the exponent

\ensuremath{\omega}

varies with

p

but is universal in the disorder.

r

interact with one another falls as

1/r^{2 σ}

, for two values of

σ

0.75

and

0.85

. No de Almeida-Thouless line is expected at these

σ

values. The spin glass correlation length

ξ_{\text{SG}}

varies with the random field as expected from the Imry-Ma argument and the droplet scaling picture of spin glasses. However, when

ξ_{\text{SG}}

becomes comparable to the system size

L

, there are departures which we attribute to the features deriving from the TNT picture of spin glasses. For the case

σ=0.85

these features go away for system sizes with

L >L^*

, where

L^*

is large (

\approx 4000-8000

lattice spacings). In the case of

σ= 0.75

we have been unable to study large enough systems to determine its value of

L^*

. We sketch a renormalization group scenario to explain how these features could arise. On this scenario finite size effects on the droplet scaling picture in low-dimensional spin glasses produce TNT features and some aspects of Parisi's replica symmetry breaking theory of the Sherrington-Kirkpatrick model.