Stiffness of the Edwards-Anderson Model in all Dimensions

Abstract

A comprehensive description in all dimensions is provided for the scaling exponent y of low-energy excitations in the Ising spin glass introduced by Edwards and Anderson. A combination of extensive numerical as well as theoretical results suggest that its lower critical dimension is exactly d l = 5/2. Such a result would be an essential feature of any complete model of low-temperature spin glass order and imposes a constraint that may help to distinguish between theories.PACS numbers: 05.50.+q , 75.10.Nr , 02.60.Pn Imagining physical systems in non-integer dimensions, such as through an ǫ-expansion near the upper [1] or lower critical dimension [2], has provided many important results for the understanding of the physics in realistic dimensions. Often, peculiarities found in unphysical dimensions may impact real-world physics and enhance our understanding [3]. Here, we will explore the variation in dimension of the low-temperature behavior in the Edwards-Anderson (EA) spin glass model [4].A quantity of fundamental importance for the modeling of amorphous magnetic materials through spin glasses [5] is the "stiffness" exponent y [6,7]. As Hooke's law describes the response in increasing elastic energy imparted to a system for increasing displacement L from its equilibrium position, the stiffness of a spin configuration describes the typical rise in magnetic energy ∆E due to an induced defect-interface of size L. But unlike uniform systems with a convex potential energy function over its configuration space (say, a parabola for the sole variable in Hooke's law), an amorphous many-body system exhibits a function more reminiscent of a highdimensional mountain landscape [8]. Any defect-induced displacement of size L in such a complicated energy landscape may move a system through many ups-and-downs in energy ∆E. Averaging over many incarnations of such a system results in a typical energy scaleThe importance of this exponent for small excitations in disordered spin systems has been discussed in many contexts [5,6,7,9,10,11,12]. Spin systems with y > 0 provide resistance ("stiffness") against the spontaneous formation of defects at sufficiently low temperatures T ; an indication that a phase transition T c > 0 to an ordered state exists. For instance, in an Ising ferromagnet, the energy ∆E is always proportional to the size of the interface, i. e. y = d − 1, consistent with the fact that T c > 0 only when d > 1. When y ≤ 0, a system is unstable (such as the d = 1 ferromagnet) to spontaneous fluctuations which proliferate, preventing any ordered state. Thus, determining the "lower critical dimension" d l , where y d l = 0, is of significant importance [6,13,14,15], and understanding the mechanism leading to d l , however un-natural, provides definite clues to the origin of order [2]. For instance, in homogeneous systems with a continuous symmetry [16], such as the Heisenberg ferromagnet, the possibility of "soft modes" (Goldstone bosons) perpendicular to the direction of magnetization have proved to weaken order furt...

Local field distributions in spin glasses

“…For the finite-range EA model there is a spin-glass transition above the lower critical dimension, d > d l , believed [14] to be d l = 5/2, but the existence of an ultrametric hierarchy in short-range systems is controversial, not proven and disbelieved by many practitioners.…”

Numerical study of the overlap Lee–Yang singularities in the three-dimensional Edwards–Anderson model

“…¶ The results, in Table 3, show a value of x 1 incompatible with x 1 = D = 3. We have also included corrections to scaling, using both ω = 1 (Goldstone-like correction) [34] and ω = 3 (Ising ordered correction) [34], ω = y = 0.255 (droplet) [35,36,37], ω = θ(0) = 0.39 (replicon and also 1/ν which controls the scaling correction of q EA (L) [21]), ω = 0.79 = 2θ(0) (twice the replicon [21]) and ω = 0.65 = θ(q EA ) [21] but in neither case is the asymptotic x 1 = D behavior recovered (see Table 3). In addition, we have forced the fits with x 1 = 3 and leaving free ω and the statistical quality of the fits was bad.…”

Spatial correlation functions and dynamical exponents in very large samples of four-dimensional spin glasses

“…(9). A closer inspection reveals the deviation of ξ 2,3 at larger times due to the badness of the estimation of the tail contribution, so that a secure range for a fit to obtain z is ξ 2,3 ∈ [3,5], though compatible results are obtained in a wider time window, as can be seen in Table III. …”

“…5 we plot the best estimate for the exponents z and α in a way that makes evident that Table III. Lines are best fits: T z(T ) = 9.7(2) and α = 1.025 (9). z(T ) 9.7(2)/T and α 1.025(9) for T < T c .…”

“…At this point, numerical simulations are very useful and we feel that studying the four-dimensional case is very important to interpolate between the field theoretical results above d u and the tridimensional case. The latter is the most explored case in large scale numerical simulations, which is indeed the crucial case, but also very close to the lower critical dimension (most probably d l 2.5 [8,9]). …”