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arXiv:cond-mat/0411138 · Physical Review Letters

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

M. Sasaki, Koji Hukushima, Hajime Yoshino, H. Takayama

Published Dec 20, 2005• 44 citations
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Abstract

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.

DOI
10.1103/physrevlett.95.267203

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Domain-wall free energy \ensuremathδF\ensuremath{\delta}F\ensuremathδF, entropy \ensuremathδS\ensuremath{\delta}S\ensuremathδS, and the correlation function Ctemp{C}_{\mathrm{temp}}Ctemp​ of \ensuremathδF\ensuremath{\delta}F\ensuremathδF are measured independently in the four-dimensional \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ Edwards-Anderson (EA) Ising spin glass. The stiffness exponent \ensuremathθ\ensuremath{\theta}\ensuremathθ, the fractal dimension of domain walls ds{d}_{s}ds​, and the chaos exponent \ensuremathζ\ensuremath{\zeta}\ensuremathζ are extracted from the finite-size scaling analysis of \ensuremathδF\ensuremath{\delta}F\ensuremathδF, \ensuremathδS\ensuremath{\delta}S\ensuremathδS, and Ctemp{C}_{\mathrm{temp}}Ctemp​, respectively, well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation \ensuremathζ=ds/2\ensuremath−\ensuremathθ\ensuremath{\zeta}={d}_{s}/2\ensuremath{-}\ensuremath{\theta}\ensuremathζ=ds​/2\ensuremath−\ensuremathθ 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.
Domain-wall free-energy δF\delta FδF, entropy δS\delta SδS, and the correlation function, CtempC_{\rm temp}Ctemp​, of δF\delta FδF are measured independently in the four-dimensional ±J\pm J±J Edwards-Anderson (EA) Ising spin glass. The stiffness exponent θ\thetaθ, the fractal dimension of domain walls dsd_{\rm s}ds​ and the chaos exponent ζ\zetaζ are extracted from the finite-size scaling analysis of δF\delta FδF, δS\delta SδS and CtempC_{\rm temp}Ctemp​ respectively well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation ζ=ds/2−θ\zeta=d_{\rm s}/2-\thetaζ=ds​/2−θ 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.
Domain-wall free-energy, the difference in the free-energy between the system with the periodic boundary condition (BC) and that with the anti-periodic BC, is measured in the 4 dimensional ±J Ising spin glass. Temperature dependence of the domain-wall free-energy for each sample shows that the favorable BC changes with temperature, suggesting the existence of temperature chaos. We also investigate fragility of the equilibrium state against small changes of couplings { Jij } (bond chaos) by measuring the domain-wall free-energy. As a result, we find scaling behavior which suggests that the chaos exponent and the scaling function for temperature perturbation are the same as those for bond perturbation, meaning that temperature chaos and bond chaos belong to the same universarity class on fragility.
Spin glasses have competing interactions that lead to a rough energy landscape which is highly susceptible to small perturbations. These chaotic effects strongly affect numerical simulations and, as such, gaining a deeper understanding of chaos in spin glasses is of much importance. The use of thermal boundary conditions is an effective approach to study chaotic phenomena. Here we generalize population annealing Monte Carlo, combined with thermal boundary conditions, to study bond chaos due to small perturbations in the spin-spin couplings of the three-dimensional Edwards-Anderson Ising spin glass. We show that bond and temperature-induced chaos share the same scaling exponents and that bond chaos is stronger than temperature chaos.
For the long-range one-dimensional Ising spin-glass with random couplings decaying as J ( r ) ∝ r − σ , the scaling of the effective coupling defined as the difference between the free energies corresponding to periodic and antiperiodic boundary conditions J R ( N ) ≡ F (P) ( N ) − F (AP) ( N ) ∼ N θ ( σ ) defines the droplet exponent θ ( σ ). Here we study numerically the instability of the renormalization flow of the effective coupling J R ( N ) with respect to magnetic, disorder and temperature perturbations respectively, in order to extract the corresponding chaos exponents ζ H ( σ ), ζ J ( σ ) and ζ T ( σ ) as a function of σ . Our results for ζ T ( σ ) are interpreted in terms of the entropy exponent θ S ( σ ) ≃ 1/3 which governs the scaling of the entropy difference S (P) ( N ) − S (AP) ( N ) ∼ N θ S ( σ ) . We also study the instability of the ground state configuration with respect to perturbations, as measured by the spin overlap between the unperturbed and the perturbed ground states, in order to extract the corresponding chaos exponents and .
The statistics of the ground-state and domain-wall energies for the two-dimensional random-bond Ising model on square lattices with independent, identically distributed bonds having probability ppp of ${J}_{ij}=\ensuremath{-}1$ and $(1\ensuremath{-}p)$ of Jij=+1{J}_{ij}=+1Jij​=+1 are studied. We are able to consider large samples of up to 3202{320}^{2}3202 spins by using sophisticated matching algorithms. We study $L\ifmmode\times\else\texttimes\fi{}L$ systems, but we also consider $L\ifmmode\times\else\texttimes\fi{}M$ samples, for different aspect ratios R=L∕MR=L∕MR=L∕M. We find that the scaling behavior of the ground-state energy and its sample-to-sample fluctuations inside the spin-glass region $({p}_{c}\ensuremath{\leqslant}p\ensuremath{\leqslant}1\ensuremath{-}{p}_{c})$ are characterized by simple scaling functions. In particular, the fluctuations exhibit a cusplike singularity at pc{p}_{c}pc​. Inside the spin-glass region, the average domain-wall energy converges to a finite nonzero value as the sample size becomes infinite, holding RRR fixed. Here, large finite-size effects are visible, which can be explained for all ppp by a single exponent $\ensuremath{\omega}\ensuremath{\approx}2∕3$, provided higher-order corrections to scaling are included. Finally, we confirm the validity of aspect-ratio scaling for $R\ensuremath{\rightarrow}0$: the distribution of the domain-wall energies converges to a Gaussian for $R\ensuremath{\rightarrow}0$, although the domain walls of neighboring subsystems of size $L\ifmmode\times\else\texttimes\fi{}L$ are not independent.
We study the fragility of spin glasses to small temperature perturbations numerically using population annealing Monte Carlo. We apply thermal boundary conditions to a three-dimensional Edwards-Anderson Ising spin glass. In thermal boundary conditions all eight combinations of periodic versus antiperiodic boundary conditions in the three spatial directions are present, each appearing in the ensemble with its respective statistical weight determined by its free energy. We show that temperature chaos is revealed in the statistics of crossings in the free energy for different boundary conditions. By studying the energy difference between boundary conditions at free-energy crossings, we determine the domain-wall fractal dimension. Similarly, by studying the number of crossings, we determine the chaos exponent. Our results also show that computational hardness in spin glasses and the presence of chaos are closely related.