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Progress of Theoretical Physics Supplement

Fragility of the Equilibrium State in the Four-Dimensional ±JIsing Spin Glass

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

Published Jan 1, 2005• 0 citations
Publisher

Abstract

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.

DOI
10.1143/ptps.157.64

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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.
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.
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.
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.
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 .
An efficient Monte Carlo method is extended to evaluate directly domain-wall free energy for randomly frustrated spin systems. Using the method, critical phenomena of spin-glass phase transition are investigated in the 4d+/-J Ising model under the replica boundary condition. Our values of the critical temperature and exponent, obtained by finite-size scaling, are in good agreement with those of the standard Monte Carlo and the series expansion studies. In addition, two exponents, the stiffness exponent and the fractal dimension of the domain wall, which characterize the ordered phase, are obtained. The latter value is larger than d-1, indicating that the domain wall is really rough in the 4d Ising spin-glass phase.