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Physical Review B

Low-temperature phase boundary of dilute-lattice spin glasses

Stefan Boettcher, Emiliano Marchetti

Published Mar 27, 2008• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.77.100405

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The thermal-to-percolative crossover exponent \ensuremathϕ\ensuremath{\phi}\ensuremathϕ, 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 pc{p}_{c}pc​ using an improved reduction algorithm. Simulations extend to system sizes above N=108N={10}^{8}N=108 in dimensions d=2,…,7d=2,\dots{},7d=2,…,7. The results can be related to the behavior of the transition temperature Tg\ensuremath∼(p\ensuremath−pc)\ensuremathϕ{T}_{g}\ensuremath{\sim}{(p\ensuremath{-}{p}_{c})}^{\ensuremath{\phi}}Tg​\ensuremath∼(p\ensuremath−pc​)\ensuremathϕ between the paramagnetic and the glassy regime for p\ensuremath↘pcp\ensuremath{\searrow}{p}_{c}p\ensuremath↘pc​. In three dimensions, where our simulations predict \ensuremathϕ=1.127(5)\ensuremath{\phi}=1.127(5)\ensuremathϕ=1.127(5), this scaling form for Tg{T}_{g}Tg​ 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 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.
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.
We investigate the ferromagnetic-glassy transitions that separate the low-temperature ferromagnetic and spin-glass phases in the temperature-disorder phase diagram of three-dimensional Ising spin-glass models. For this purpose, we consider the cubic-lattice $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model with bond distribution $P(J)=p\ensuremath{\delta}(J\ensuremath{-}1)+(1\ensuremath{-}p)\ensuremath{\delta}(J+1)$, and present a numerical Monte Carlo study of the critical behavior along the line that marks the onset of ferromagnetism. The finite-size scaling analysis of the Monte Carlo data shows that the ferromagnetic-glassy transition line is slightly reentrant. As a consequence, for an interval of the disorder parameter ppp, around $p\ensuremath{\approx}0.77$, the system presents a low-temperature glassy phase, an intermediate ferromagnetic phase, and a high-temperature paramagnetic phase. Along the ferromagnetic-glassy transition line, magnetic correlations show a universal critical behavior with critical exponents $\ensuremath{\nu}=0.96(2)$ and $\ensuremath{\eta}=\ensuremath{-}0.39(2)$. The hyperscaling relation $\ensuremath{\beta}/\ensuremath{\nu}=(1+\ensuremath{\eta})/2$ is satisfied at the transitions, so that $\ensuremath{\beta}/\ensuremath{\nu}=0.305(10)$. This magnetic critical behavior represents a new universality class for ferromagnetic transitions in Ising-type disordered systems. Overlap correlations are apparently not critical and show a smooth behavior across the transition.
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.
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, dc=6{d}_{c}=6dc​=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=3d=3d=3 with Gaussian bonds.
We study in Ising spin glasses the finite-size effects near the spin-glass transition in zero field and at the de Almeida-Thouless transition in a field by Monte Carlo methods and by analytical approximations. In zero field, the finite-size scaling function associated with the spin-glass susceptibility of the Sherrington-Kirkpatrick mean-field spin-glass model is of the same form as that of one-dimensional spin-glass models with power-law long-range interactions in the regime where they can be a proxy for the Edwards-Anderson short-range spin-glass model above the upper critical dimension. We also calculate a simple analytical approximation for the spin-glass susceptibility crossover function. The behavior of the spin-glass susceptibility near the de Almeida-Thouless transition line has also been studied, but here we have only been able to obtain analytically its behavior in the asymptotic limit above and below the transition. We have also simulated the one-dimensional system in a field in the non-mean-field regime to illustrate that when the Imry-Ma droplet length scale exceeds the system size one can then be erroneously lead to conclude that there is a de Almeida-Thouless transition even though it is absent.