Numerical results for the Edwards-Anderson spin-glass model at low temperature

Abstract

We have simulated Edwards-Anderson (EA) as well as Sherrington-Kirkpatrick systems of L 3 spins. After averaging over large sets of EA system samples of 3 L 10, we obtain accurate numbers for distributions p(q) of the overlap parameter q at very low-temperature T . We find p(0)/T → 0.233(4) as T → 0. This is in contrast with the droplet scenario of spin glasses. We also study the number of mismatched links-between replica pairs-that come with large scale excitations. Contributions from small scale excitations are discarded. We thus obtain for the fractal dimension of outer surfaces of q ∼ 0 excitations in the EA model d s → 2.59(3) as T → 0. This is in contrast with d s → 3 as T → 0 that is predicted by mean-field theory for the macroscopic limit.

Typical versus averaged overlap distribution in spin glasses: Evidence for droplet scaling theory

“…In the field of classical spin-glasses (see for instance [9][10][11]), there has been an ongoing debate on the nature of the spin-glass phase between the droplet scaling theory [12][13][14], which is based on real space renormalization ideas (explicit real-space renormalization for spin-glasses have been studied in detail within the Migdal-Kadanoff approximation [15]), and the alternative Replica-Symmetry-Breaking scenario [16] based on the mean-field fully connected Sherrington-Kirkpatrick model [17]. The questions under debate include the presence of the number of ground states (two or many) [18][19][20], the properties of the overlap [21][22][23][24][25][26][27][28][29], the statistics of excitations [30,31], the structure of state space [32], the absence or presence of an Almeida-Thouless line in the presence of an magnetic field [33][34][35][36][37][38], etc ... In particular, one of the standard observable to discriminate between the droplet and the replica theories has been the averaged overlap distribution P J (q).…”