Double Criticality of the Sherrington-Kirkpatrick Model atT=0

Abstract

Numerical results up to 42nd order of replica symmetry breaking (RSB) are used to predict the singular structure of the SK spin glass at T = 0. We confirm predominant single parameter scaling and derive corrections for the T = 0 order function q(a), related to a Langevin equation with pseudotime 1/a. a = 0 and a = ∞ are shown to be two critical points for ∞-RSB, associated with two discrete spectra of Parisi block size ratios, attached to a continuous spectrum. Finite-RSB-sizescaling, associated exponents, and T = 0-energy are obtained with unprecedented accuracy.PACS numbers: 75.10. Hk,75.10.Nr,75.40.Cx The low temperature limit usually simplifies considerably the properties of magnetically ordered phases. Research in recent decades has however shown that frustrated systems can have rich behaviour even at T = 0. Spin glasses [1] are an extreme example from condensed matter, while others are a feature of computer and information science in problems such as hard satisfiability and error-correcting codes. In particular, even the potentially soluble infinite-range Ising spin glass model of Sherrington and Kirkpatrick [2] has left open many puzzling questions. Parisi devised an ansatz [3] for the order parameter of the SK-model, based on an infinite hierarchy of so-called replica symmetry breakings and related hierarchically to the distribution of overlaps of metastable solutions [10]. The determining equations for this ansatz have recently been rigorously proven to be exact [4], but its explicit solution remains elusive. Also only recently has the T = 0 SK problem been recognized as a critical one-dimensional theory [5,6].In view of the paradigmic role that the SK-model has played in the understanding and development of the statistical physics of complex systems, together with the potential that further comprehension of its subtleties has for extensions to other more-complicated systems in many fields of science, especially those involving zero-(or effectively zero-)temperature replica-symmetry-breaking [17] transitions, it seems important to pursue the better understanding of T = 0 RSB in the SK model. This letter is concerned with such a study and the exposure of several novel features, including new critical spectra, invariance points and quasi-dynamics.Parisi's order parameter is a function q(x, T ) on an interval 0 ≤ x ≤ 1, the limit of a stepwise function q i (T ), x i (T ) determined by extremization of a free energy. It provides the hierarchical distribution of pure state overlaps P (q) through P (q) = dx/dq [10]. Parisi's original work considered numerically an approximation with a small finite number of steps, but most recent studies of the SK model have been based on self-consistent solutions for his later non-trivial continuous order function, typically perturbatively in the deviation from the finite-temperature phase transition. Here the analysis is considered explicitly at T = 0 using very accurate studies of a very large sequence of RSB orders.In the limit of zero temperature Parisi's order functi...

Local field distributions in spin glasses

“…2 and Refs. [8], [10], and [29]). Through accurate studies of a large sequence of finite-replica-symmetry breakings Refs.…”

“…Through accurate studies of a large sequence of finite-replica-symmetry breakings Refs. [9] and [10] have demonstrated that within replica theory the correct linear behaviour of P (h) for the SK model requires the full limit of infinite replicasymmetry breaking order; any finite-order truncation yielding a fictitious (if decreasing with RSB-order) hole in P (h) near h = 0.…”

“…Since then there have been several further studies of P (h) for the SK model (see, for example, Refs. [7], [8], [9], and [10]) and the nature of the local field distributions is well understood. On the other hand, there has been little work on the study of P (h) for other spin-glass models.…”

Construction and purpose of effective field theories for frustrated magnetic order

“…The simpler error-function model is contained for α = 2 ) = √ πerf (x)/(2x). The best parameter set for the full range 0 < a < ∞, matching also the extrema of q (a), was found to be {α ≈ 0.53, γ ≈ 1.71, w ≈ 0.02, ξ ≈ 1.16} [9]. We show a comparison with 40−step RSB (80-dimensional extremization of the SK-energy) in Fig.…”

“…The glassy order however by itself is a highly interesting physical object: being on one hand an equilibrium solution and on the other as a fluctuation solution in a new pseudo-time space [9] as we shall explain below.…”