Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0106554 · Physical review. B, Condensed matter

Statistics of lowest droplets in two-dimensional Gaussian Ising spin glasses

Marco Picco, Félix Ritort, Marta Sales‐Pardo

Published May 30, 2003• 11 citations
arXivHTMLPDF

Abstract

An approach to determine the value of the zero-temperature thermal exponent $\ensuremath{\theta}$ in spin glasses is presented. It consists in describing the energy level spectrum in spin glasses only in terms of the properties of the lowest energy droplets and the lowest droplet exponents (LDE's) ${\ensuremath{\lambda}}_{l},{\ensuremath{\theta}}_{l}$ that describe the statistics of their sizes and gaps. We show how these LDE's yield the standard thermal exponent of droplet theory $\ensuremath{\theta}$ through the relation $\ensuremath{\theta}={\ensuremath{\theta}}_{l}+d{\ensuremath{\lambda}}_{l}.$ The present approach provides a new way to measure the thermal exponent $\ensuremath{\theta}$ without any assumption about the correct procedure to generate typical low-lying excitations as is commonly done in many perturbation methods including domain wall calculations. To illustrate the usefulness of the method we present a detailed investigation of the properties of the lowest energy droplets in two-dimensional Gaussian Ising spin glasses. By independent measurements of both LDE's and an aspect-ratio analysis, we find $\ensuremath{\theta}(2d)\ensuremath{\simeq}\ensuremath{-}0.46(1)<{\ensuremath{\theta}}_{\mathrm{DW}}(2d)\ensuremath{\simeq}\ensuremath{-}0.287$ where ${\ensuremath{\theta}}_{\mathrm{DW}}$ is the thermal exponent obtained in domain-wall theory. We also discuss the origin of finite-volume corrections in the behavior of the LDE ${\ensuremath{\theta}}_{l}$ and relate them to the finite-volume corrections in the statistics of extreme values. Finally, we analyze some geometrical properties of the lowest energy droplets, finding results in agreement with those recently reported by Kawashima and Aoki [J. Phys. Soc. Jpn. $69,$ 169 (2000)]. All in all, we show that typical large-scale droplets are not probed by most of the present perturbation methods, since they probably do not have a compact structure as has been recently suggested. We speculate that a multifractal scenario could be at the roots of the reported discrepancies on the value of the thermal exponent $\ensuremath{\theta}$ in the two-dimensional Gaussian Ising spin glass.

DOI
10.1103/physrevb.67.184421

Related papers

Large-scale low-energy excitations in the two-dimensional Ising spin glass

Alexander K. Hartmann, A. P. Young

We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, ${d}_{s},$ and stiffness exponent, ${\ensuremath{\theta}}^{\ensuremath{'}},$ for ``droplet'' excitations, are in reasonable agreement with estimates from ``domain wall'' calculations, and so support the predictions of the ``droplet theory.'' Restricting our analysis to small lattices, we do not find an effective value of ${\ensuremath{\theta}}^{\ensuremath{'}}$ close to $\ensuremath{-}0.47$ as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give ${\ensuremath{\theta}}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.47$ for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the ``TNT'' scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.

arXiv:cond-mat/0205659•Physical review. B, Condensed matter•Sep 16, 2002•45 citations

Low-temperature behavior of two-dimensional Gaussian Ising spin glasses

J. Houdayer, Alexander K. Hartmann

We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\ensuremath{\beta}=1∕T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state, and the specific heat. We confirm that ${T}_{c}=0$. All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.

arXiv:cond-mat/0402036•Physical Review B•Jul 1, 2004•133 citations

Droplets in the two-dimensional ±J Ising spin glass

Alexander K. Hartmann

By using problem-dependent mappings to computer-science problems and by applying sophisticated algorithms, one can numerically study some important problems much better compared to applying standard approaches, such as Monte Carlo simulations. Here, by using calculations of ground states of suitable perturbed systems, droplets are obtained in two-dimensional $\ifmmode\pm\else\textpm\fi{}J$ spin glasses, which are in the focus of a currently very lively debate. By using a sophisticated matching algorithm, exact ground states of large systems up to ${L}^{2}={256}^{2}$ spins can be generated. Furthermore, no equilibration or extrapolation to $T=0$ is necessary. Three different $\ifmmode\pm\else\textpm\fi{}J$ models are studied here: (a) with free boundary conditions, (b) with fixed boundary conditions, and (c) a diluted system wherein a fraction $p=0.125$ of all bonds is zero. For large systems, the droplet energy shows for all three models a power-law behavior ${E}_{\text{dp}}\ensuremath{\sim}{L}^{{\ensuremath{\theta}}_{\text{dp}}^{\ensuremath{'}}}$ with ${\ensuremath{\theta}}_{\text{dp}}^{\ensuremath{'}}<0$. This is different from the previous studies of domain walls, wherein a convergence to a constant nonzero value $({\ensuremath{\theta}}_{\text{DW}}=0)$ was found. After correcting for the noncompactness of the droplets, the results are likely to be compatible with ${\ensuremath{\theta}}_{\text{dp}}\ensuremath{\approx}\ensuremath{-}0.29$ for all three models. This is in accordance with the Gaussian system wherein ${\ensuremath{\theta}}_{\text{dp}}=\ensuremath{-}0.287(4)$ ($\ensuremath{\nu}\ensuremath{\approx}3.5$ via $\ensuremath{\nu}=\ensuremath{-}1/{\ensuremath{\theta}}_{\text{dp}}$). Nevertheless, the disorder-averaged spin-spin correlation exponent $\ensuremath{\eta}$ is determined here via the probability to have a nonzero-energy droplet, and $\ensuremath{\eta}\ensuremath{\approx}0.22$ is found for all three models, in contrast to the model with the Gaussian interactions, where $\ensuremath{\eta}=0$ is exact.

Physical Review B•Apr 18, 2008•20 citations

Droplets in the two-dimensional +-J spin glass: evidence for (non-) universality

Alexander K. Hartmann

Using mappings to computer-science problems and by applying sophisticated algorithms, one can study numerically many problems much better compared to applying standard approaches like Monte Carlo simulations. Here, using calculations of ground states of suitable perturbed systems, droplets are obtained in two-dimensional +-J spin glasses, which are in the focus of a currently very lifely debate. Since a sophisticated matching algorithm is applied here, exact ground states of large systems up to L^2=256^2 spins can be generated. Furthermore, no equilibration or extrapolation to T=0 is necessary. Three different +-J models are studied here: a) with open boundary conditions, b) with fixed boundary conditions and c) a diluted system where a fraction p=0.125 of all bonds is zero. For large systems, the droplet energy shows for all three models a power-law behavior E_D L^θ'_D with θ'_D<0. This is different from previous studies of domain walls, where a convergence to a constant non-zero value (θ_dw=0) has been found for such models. After correcting for the non-compactness of the droplets, the results are likely to be compatible with θ_D= -0.29 for all three models. This is in accordance with the Gaussian system where θ_D=-0.287(4) (ν=3.5 via ν=-1/θ_D). Nevertheless, the disorder-averaged spin-spin correlation exponent ηis determined here via the probability to have a non-zero-energy droplet, and η~0.22$ is found for all three models, this being in contrast to the behavior of the model with Gaussian interactions, where exactly η=0.

arXiv:0704.2748•Apr 20, 2007•0 citations

Energy size effects of two-dimensional Ising spin glasses

Ian A. Campbell, Alexander K. Hartmann, Helmut G. Katzgraber

We analyze exact ground-state energies of two-dimensional Ising spin glasses with either Gaussian or bimodal nearest-neighbor interactions for large system sizes and for three types of boundary conditions: free on both axes, periodic on both axes, and free on one axis and periodic on the other. We find accurate values for bulk-, edge-, and corner-site energies. Fits for the system with Gaussian bonds are excellent for all types of boundary conditions over the whole range of system sizes. In particular, the leading behavior for nonfree boundary conditions is governed by the stiffness exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.282$ describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.

arXiv:cond-mat/0402627•Physical Review B•Aug 31, 2004•35 citations

Temperature Chaos and Bond Chaos in Edwards-Anderson Ising Spin Glasses: Domain-Wall Free-Energy Measurements

M. Sasaki, Koji Hukushima, Hajime Yoshino +1 more

Domain-wall free energy $\ensuremath{\delta}F$, entropy $\ensuremath{\delta}S$, and the correlation function ${C}_{\mathrm{temp}}$ of $\ensuremath{\delta}F$ are measured independently in the four-dimensional $\ifmmode\pm\else\textpm\fi{}J$ Edwards-Anderson (EA) Ising spin glass. The stiffness exponent $\ensuremath{\theta}$, the fractal dimension of domain walls ${d}_{s}$, and the chaos exponent $\ensuremath{\zeta}$ are extracted from the finite-size scaling analysis of $\ensuremath{\delta}F$, $\ensuremath{\delta}S$, and ${C}_{\mathrm{temp}}$, respectively, well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation $\ensuremath{\zeta}={d}_{s}/2\ensuremath{-}\ensuremath{\theta}$ derived by the droplet theory within our numerical accuracy. We also study bond chaos induced by random variation of bonds, and find that the bond and temperature perturbations yield the universal chaos effects described by a common scaling function and the chaos exponent. These results strongly support the appropriateness of the droplet theory for the description of chaos effect in the EA Ising spin glasses.

arXiv:cond-mat/0411138•Physical Review Letters•Dec 20, 2005•44 citations
An approach to determine the value of the zero-temperature thermal exponent \ensuremathθ\ensuremath{\theta}\ensuremathθ in spin glasses is presented. It consists in describing the energy level spectrum in spin glasses only in terms of the properties of the lowest energy droplets and the lowest droplet exponents (LDE's) \ensuremathλl,\ensuremathθl{\ensuremath{\lambda}}_{l},{\ensuremath{\theta}}_{l}\ensuremathλl​,\ensuremathθl​ that describe the statistics of their sizes and gaps. We show how these LDE's yield the standard thermal exponent of droplet theory \ensuremathθ\ensuremath{\theta}\ensuremathθ through the relation \ensuremathθ=\ensuremathθl+d\ensuremathλl.\ensuremath{\theta}={\ensuremath{\theta}}_{l}+d{\ensuremath{\lambda}}_{l}.\ensuremathθ=\ensuremathθl​+d\ensuremathλl​. The present approach provides a new way to measure the thermal exponent \ensuremathθ\ensuremath{\theta}\ensuremathθ without any assumption about the correct procedure to generate typical low-lying excitations as is commonly done in many perturbation methods including domain wall calculations. To illustrate the usefulness of the method we present a detailed investigation of the properties of the lowest energy droplets in two-dimensional Gaussian Ising spin glasses. By independent measurements of both LDE's and an aspect-ratio analysis, we find \ensuremathθ(2d)\ensuremath≃\ensuremath−0.46(1)<\ensuremathθDW(2d)\ensuremath≃\ensuremath−0.287\ensuremath{\theta}(2d)\ensuremath{\simeq}\ensuremath{-}0.46(1)<{\ensuremath{\theta}}_{\mathrm{DW}}(2d)\ensuremath{\simeq}\ensuremath{-}0.287\ensuremathθ(2d)\ensuremath≃\ensuremath−0.46(1)<\ensuremathθDW​(2d)\ensuremath≃\ensuremath−0.287 where \ensuremathθDW{\ensuremath{\theta}}_{\mathrm{DW}}\ensuremathθDW​ is the thermal exponent obtained in domain-wall theory. We also discuss the origin of finite-volume corrections in the behavior of the LDE \ensuremathθl{\ensuremath{\theta}}_{l}\ensuremathθl​ and relate them to the finite-volume corrections in the statistics of extreme values. Finally, we analyze some geometrical properties of the lowest energy droplets, finding results in agreement with those recently reported by Kawashima and Aoki [J. Phys. Soc. Jpn. 69,69,69, 169 (2000)]. All in all, we show that typical large-scale droplets are not probed by most of the present perturbation methods, since they probably do not have a compact structure as has been recently suggested. We speculate that a multifractal scenario could be at the roots of the reported discrepancies on the value of the thermal exponent \ensuremathθ\ensuremath{\theta}\ensuremathθ in the two-dimensional Gaussian Ising spin glass.
We study large-scale, low-energy excitations in the Ising spin glass with Gaussian interactions in two dimensions at zero temperature, using an optimization algorithm to determine exact ground states. Periodic boundary conditions are applied. Our results for the fractal dimension of the surface, ds,{d}_{s},ds​, and stiffness exponent, ${\ensuremath{\theta}}^{\ensuremath{'}},$ for ``droplet'' excitations, are in reasonable agreement with estimates from ``domain wall'' calculations, and so support the predictions of the ``droplet theory.'' Restricting our analysis to small lattices, we do not find an effective value of ${\ensuremath{\theta}}^{\ensuremath{'}}$ close to $\ensuremath{-}0.47$ as has been recently proposed. The effects of averaging over droplets of different sizes are studied and are also found to be too small to give ${\ensuremath{\theta}}^{\ensuremath{'}}\ensuremath{\approx}\ensuremath{-}0.47$ for smaller sizes. Larger corrections to finite-size scaling would be needed in three and four dimensions in order for the numerical data used to support the ``TNT'' scenario to be compatible with the droplet theory prediction that the stiffness exponent is the same for droplets and domain walls.
We perform Monte Carlo simulations of large two-dimensional Gaussian Ising spin glasses down to very low temperatures $\ensuremath{\beta}=1∕T=50$. Equilibration is ensured by using a cluster algorithm including Monte Carlo moves consisting of flipping fundamental excitations. We study the thermodynamic behavior using the Binder cumulant, the spin-glass susceptibility, the distribution of overlaps, the overlap with the ground state, and the specific heat. We confirm that Tc=0{T}_{c}=0Tc​=0. All results are compatible with an algebraic divergence of the correlation length with an exponent $\ensuremath{\nu}$. We find $\ensuremath{-}1∕\ensuremath{\nu}=\ensuremath{-}0.295(40)$, which is compatible with the value for the domain-wall and droplet exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.29$ found previously in ground-state studies. Hence the thermodynamic behavior of this model seems to be governed by one single exponent.
By using problem-dependent mappings to computer-science problems and by applying sophisticated algorithms, one can numerically study some important problems much better compared to applying standard approaches, such as Monte Carlo simulations. Here, by using calculations of ground states of suitable perturbed systems, droplets are obtained in two-dimensional $\ifmmode\pm\else\textpm\fi{}J$ spin glasses, which are in the focus of a currently very lively debate. By using a sophisticated matching algorithm, exact ground states of large systems up to L2=2562{L}^{2}={256}^{2}L2=2562 spins can be generated. Furthermore, no equilibration or extrapolation to T=0T=0T=0 is necessary. Three different $\ifmmode\pm\else\textpm\fi{}J$ models are studied here: (a) with free boundary conditions, (b) with fixed boundary conditions, and (c) a diluted system wherein a fraction p=0.125p=0.125p=0.125 of all bonds is zero. For large systems, the droplet energy shows for all three models a power-law behavior ${E}_{\text{dp}}\ensuremath{\sim}{L}^{{\ensuremath{\theta}}_{\text{dp}}^{\ensuremath{'}}}$ with ${\ensuremath{\theta}}_{\text{dp}}^{\ensuremath{'}}<0$. This is different from the previous studies of domain walls, wherein a convergence to a constant nonzero value $({\ensuremath{\theta}}_{\text{DW}}=0)$ was found. After correcting for the noncompactness of the droplets, the results are likely to be compatible with ${\ensuremath{\theta}}_{\text{dp}}\ensuremath{\approx}\ensuremath{-}0.29$ for all three models. This is in accordance with the Gaussian system wherein ${\ensuremath{\theta}}_{\text{dp}}=\ensuremath{-}0.287(4)$ ($\ensuremath{\nu}\ensuremath{\approx}3.5$ via $\ensuremath{\nu}=\ensuremath{-}1/{\ensuremath{\theta}}_{\text{dp}}$). Nevertheless, the disorder-averaged spin-spin correlation exponent $\ensuremath{\eta}$ is determined here via the probability to have a nonzero-energy droplet, and $\ensuremath{\eta}\ensuremath{\approx}0.22$ is found for all three models, in contrast to the model with the Gaussian interactions, where $\ensuremath{\eta}=0$ is exact.
Using mappings to computer-science problems and by applying sophisticated algorithms, one can study numerically many problems much better compared to applying standard approaches like Monte Carlo simulations. Here, using calculations of ground states of suitable perturbed systems, droplets are obtained in two-dimensional +-J spin glasses, which are in the focus of a currently very lifely debate. Since a sophisticated matching algorithm is applied here, exact ground states of large systems up to L^2=256^2 spins can be generated. Furthermore, no equilibration or extrapolation to T=0 is necessary. Three different +-J models are studied here: a) with open boundary conditions, b) with fixed boundary conditions and c) a diluted system where a fraction p=0.125 of all bonds is zero. For large systems, the droplet energy shows for all three models a power-law behavior E_D L^θ'_D with θ'_D<0. This is different from previous studies of domain walls, where a convergence to a constant non-zero value (θ_dw=0) has been found for such models. After correcting for the non-compactness of the droplets, the results are likely to be compatible with θ_D= -0.29 for all three models. This is in accordance with the Gaussian system where θ_D=-0.287(4) (ν=3.5 via ν=-1/θ_D). Nevertheless, the disorder-averaged spin-spin correlation exponent ηis determined here via the probability to have a non-zero-energy droplet, and η~0.22$ is found for all three models, this being in contrast to the behavior of the model with Gaussian interactions, where exactly η=0.
We analyze exact ground-state energies of two-dimensional Ising spin glasses with either Gaussian or bimodal nearest-neighbor interactions for large system sizes and for three types of boundary conditions: free on both axes, periodic on both axes, and free on one axis and periodic on the other. We find accurate values for bulk-, edge-, and corner-site energies. Fits for the system with Gaussian bonds are excellent for all types of boundary conditions over the whole range of system sizes. In particular, the leading behavior for nonfree boundary conditions is governed by the stiffness exponent $\ensuremath{\theta}\ensuremath{\approx}\ensuremath{-}0.282$ describing the scaling of domain-wall and droplet excitations. For the system with a bimodal distribution of bonds the fit is good for free boundary conditions but worse for other geometries, particularly for periodic-free boundary conditions where there appear to be unorthodox corrections to scaling up to large sizes. Finally, by introducing hard bonds we test explicitly for the Gaussian case the relationship between domain walls and the standard scaling behavior.
Domain-wall free energy $\ensuremath{\delta}F$, entropy $\ensuremath{\delta}S$, and the correlation function Ctemp{C}_{\mathrm{temp}}Ctemp​ of $\ensuremath{\delta}F$ are measured independently in the four-dimensional $\ifmmode\pm\else\textpm\fi{}J$ Edwards-Anderson (EA) Ising spin glass. The stiffness exponent $\ensuremath{\theta}$, the fractal dimension of domain walls ds{d}_{s}ds​, and the chaos exponent $\ensuremath{\zeta}$ are extracted from the finite-size scaling analysis of $\ensuremath{\delta}F$, $\ensuremath{\delta}S$, and Ctemp{C}_{\mathrm{temp}}Ctemp​, respectively, well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation $\ensuremath{\zeta}={d}_{s}/2\ensuremath{-}\ensuremath{\theta}$ derived by the droplet theory within our numerical accuracy. We also study bond chaos induced by random variation of bonds, and find that the bond and temperature perturbations yield the universal chaos effects described by a common scaling function and the chaos exponent. These results strongly support the appropriateness of the droplet theory for the description of chaos effect in the EA Ising spin glasses.