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arXiv:1412.2448 · Physical Review B

Critical point scaling of Ising spin glasses in a magnetic field

Joonhyun Yeo, M A Moore

Published Mar 30, 2015• 11 citations
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Abstract

Critical point scaling in a field $H$ applies for the limits $t\ensuremath{\rightarrow}0$ (where $t=T/{T}_{c}\ensuremath{-}1)$ and $H\ensuremath{\rightarrow}0$ but with the ratio $R=t/{H}^{2/\ensuremath{\Delta}}$ finite. $\ensuremath{\Delta}$ is a critical exponent of the zero-field transition. We study the replicon correlation length $\ensuremath{\xi}$ and from it the crossover scaling function $f(R)$ defined via $1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)$. We have calculated analytically $f(R)$ for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension $d=3$, we have determined the exponents and the critical scaling function $f(R)$ within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for $f(R)$ in $d=3$ in reasonable agreement with those of the Monte Carlo simulations at the values of $R$ for which they can be compared. If there were a de Almeida-Thouless (AT) line for $d\ensuremath{\le}6$, it would appear as a zero of the function $f(R)$ at some negative value of $R$, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for $d\ensuremath{\le}6$, which we review.

DOI
10.1103/physrevb.91.104432

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Critical point scaling in a field HHH applies for the limits t\ensuremath→0t\ensuremath{\rightarrow}0t\ensuremath→0 (where t=T/Tc\ensuremath−1)t=T/{T}_{c}\ensuremath{-}1)t=T/Tc​\ensuremath−1) and H\ensuremath→0H\ensuremath{\rightarrow}0H\ensuremath→0 but with the ratio R=t/H2/\ensuremathΔR=t/{H}^{2/\ensuremath{\Delta}}R=t/H2/\ensuremathΔ finite. \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ is a critical exponent of the zero-field transition. We study the replicon correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ and from it the crossover scaling function f(R)f(R)f(R) defined via 1/(\ensuremathξH4/(d+2\ensuremath−\ensuremathη))\ensuremath∼f(R)1/(\ensuremath{\xi}{H}^{4/(d+2\ensuremath{-}\ensuremath{\eta})})\ensuremath{\sim}f(R)1/(\ensuremathξH4/(d+2\ensuremath−\ensuremathη))\ensuremath∼f(R). We have calculated analytically f(R)f(R)f(R) for the mean-field limit of the Sherrington-Kirkpatrick model. In dimension d=3d=3d=3, we have determined the exponents and the critical scaling function f(R)f(R)f(R) within two versions of the Migdal-Kadanoff (MK) renormalization group procedure. One of the MK versions gives results for f(R)f(R)f(R) in d=3d=3d=3 in reasonable agreement with those of the Monte Carlo simulations at the values of RRR for which they can be compared. If there were a de Almeida-Thouless (AT) line for d\ensuremath≤6d\ensuremath{\le}6d\ensuremath≤6, it would appear as a zero of the function f(R)f(R)f(R) at some negative value of RRR, but there is no evidence for such behavior. This is consistent with the arguments that there should be no AT line for d\ensuremath≤6d\ensuremath{\le}6d\ensuremath≤6, which we review.
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.
We study the transition of short-range Ising spin glasses in a magnetic field, within a general replica symmetric field theory, which contains three masses and eight cubic couplings, that is defined in terms of the fields representing the replicon, anomalous, and longitudinal modes. We discuss the symmetry of the theory in the limit of replica number $\stackrel{\ensuremath{\rightarrow}}{n}0,$ and consider the regular case where the longitudinal and anomalous masses remain degenerate. The spin-glass transitions in zero and nonzero fields are analyzed in a common framework. The mean-field treatment shows the usual results, that is a transition in zero field, where all the modes become critical, and a transition in nonzero field, at the de Almeida--Thouless (AT) line, with only the replicon mode critical. Renormalization group methods are used to study the critical behavior, to order $\ensuremath{\varepsilon}=6\ensuremath{-}d.$ In the general theory we find a stable fixed point associated with a spin-glass transition in a zero field. This fixed point becomes unstable in the presence of a small magnetic field, and we calculate crossover exponents, which we relate to zero-field critical exponents. In a finite magnetic field, we find no physical stable fixed point to describe the AT transition, in agreement with previous results of other authors.
The asymptotical behavior of physical quantities, like the order parameter, the replicon, and longitudinal masses, is studied around the zero-field spin-glass transition point when a small external magnetic field is applied. An effective field theory to model this asymptotics contains a small perturbation in its Lagrangian which breaks the zero-field symmetry. A first-order renormalization group supplemented by perturbational results provides the scaling functions. The perturbative zero of the scaling function for the replicon mass defines a generic Almeida-Thouless surface stemming from the zero-field fixed point.
For the Ising model with Gaussian random coupling of average J0{J}_{0}J0​ and unit variance, the zero-temperature spin-glass--ferromagnetic transition as a function of the control parameter J0{J}_{0}J0​ can be studied via the size-LLL-dependent renormalized coupling defined as the domain-wall energy ${J}^{R}(L)\ensuremath{\equiv}{E}_{\text{GS}}^{(\text{AF})}(L)\ensuremath{-}{E}_{\text{GS}}^{(\text{F})}(L)$ (i.e., the difference between the ground-state energies corresponding to antiferromagnetic and ferromagnetic boundary conditions in one direction). We study numerically the critical exponents of this zero-temperature transition within the Migdal-Kadanoff approximation as a function of the dimension d=2,3,4,5,6d=2,3,4,5,6d=2,3,4,5,6. We then compare with the mean-field spherical model. Our main conclusion is that in low dimensions, the critical stiffness exponent ${\ensuremath{\theta}}^{c}$ is clearly bigger than the spin-glass stiffness exponent ${\ensuremath{\theta}}^{\text{SG}}$, but that they turn out to coincide in high enough dimension and in the mean-field spherical model. We also discuss the finite-size scaling properties of the averaged value and of the width of the distribution of the renormalized couplings.
A transfer matrix method is used to study the variation with length scale L of the distribution P L (J eff ) of effective couplings in Ising spin glasses at zero temperature. For a gaussian initial distribution the authors find in two dimensions J(L) identical to ( mod J eff mod ) varies as L -1 nu / for 2<or=L<or=12, with nu =3.4+or-0.1, implying a zero-temperature phase transition with correlation length exponent nu . In three dimensions the effective couplings initially increase with length scale, J(L) varies as L -1 nu / for 2<or=L<or=4, with -1/ nu approximately=0.2, suggesting a phase transition at finite temperature. The results are described surprisingly well by the zero-temperature version of the Migdal-Kadanoff renormalisation group scheme.
Abstract We study the critical behavior of the Ising spin glass in five spatial dimensions through large-scale Monte Carlo simulations and finite-size scaling analysis. Numerical evidence for a phase transition is found both with and without an externally applied magnetic field. The critical exponents are computed in both cases. We compute with a 10% accuracy the lower critical dimension at zero magnetic field, finding a result consistent with estimates obtained with entirely different methods, by combining our estimates of critical exponents in five dimensions with previous results for other spatial dimensions. When the results in a magnetic field are compared with previous results in six spatial dimensions, qualitative differences emerge in the scaling behavior of the correlation functions at zero external momentum. This anomalous scaling does not extend to other wavevectors. We do not find indications of a quasi first-order phase transition in a magnetic field.