Reduction of Spin Glasses applied to the Migdal-Kadanoff Hierarchical Lattice

A reduction procedure to obtain ground states of spin glasses on sparse graphs is developed and tested on the hierarchical lattice associated with the Migdal-Kadanoff approximation for low-dimensional lattices. While more generally applicable, these rules here lead to a complete reduction of the lattice. The stiffness exponent governing the scaling of the defect energy $\Delta E$ with system size $L$, $\sigma(\Delta E)\sim L^y$, is obtained as $y_3=0.25546(3)$ by reducing the equivalent of lattices up to $L=2^{100}$ in $d=3$, and as $y_4=0.76382(4)$ for up to $L=2^{35}$ in $d=4$. The reduction rules allow the exact determination of the ground state energy, entropy, and also provide an approximation to the overlap distribution. With these methods, some well-know and some new features of diluted hierarchical lattices are calculated.Comment: 7 pages, RevTex, 6 figures (postscript), added results for d=4, some corrections; final version, as to appear in EPJ

Aging Exponents in Self-Organized Criticality

In a recent Letter [Phys. Rev. Lett. 79, 889 (1997) and cond-mat/9702054] we have demonstrated that the avalanches in the Bak-Sneppen model display aging behavior similar to glassy systems. Numerical results for temporal correlations show a broad distribution with two distinct regimes separated by a time scale which is related to the age of the avalanche. This dynamical breaking of time-translational invariance results in a new critical exponent, $r$. Here we present results for $r$ from extensive numerical simulations of self-organized critical models in $d=1$ and 2. We find $r_{d=1}=0.45\pm 0.05$ and $r_{d=2}=0.23\pm 0.05$ for the Bak-Sneppen model, and our results suggest $r=1/4$ for the analytically tractable multi-trade model in both dimensions.Comment: 8 pages RevTex, 8 ps-figures included. Improved presentation, as to appear in PR

Finite-Size Corrections for Ground States of Edwards-Anderson Spin Glasses

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 becomes a hard combinatorial problem. Finite-size corrections of the form 1/N^w are shown to be consistent throughout with the prediction w=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 $d\to\infty$ appears to match our mean-field results for these corrections. In the glassy phase, w does not approach the value of 2/3 for large d predicted from simulations of the Sherrington-Kirkpatrick spin glass. However, the value of w reached at the upper critical dimension does match certain mean-field spin glass models on sparse random networks of regular degree called Bethe lattices.Comment: 6 pages, RevTex4, all ps figures included, corrected and final version with extended analysis and more data, such as for case d=3. Find additional information at http://www.physics.emory.edu/faculty/boettcher

Mesoscopic real space structures in aging spin-glasses: the Edwards-Anderson model

Isothermal simulational data for the 3D Edwards-Anderson spin glass are collected at several temperatures below $T_{\rm c}$ and, in analogy with a recent model of dense colloidal suspensions,interpreted in terms of clusters of contiguous spins overturned by quakes, non-equilibrium events linked to record sized energy fluctuations. We show numerically that, to a good approximation, these quakes are statistically independent and constitute a Poisson process whose average grows logarithmically in time. The overturned clusters are local projections on one of the two ground states of the model, and grow likewise logarithmically in time. Data collected at different temperatures $T$ can be collapsed by scaling them with $T^{1.75}$, a hitherto unnoticed feature of the E-A model, which we relate on the one hand to the geometry of configuration space and on the other to experimental memory and rejuvenation effects. The rate at which a cluster flips is shown to decrease exponentially with the size of the cluster, as recently assumed in a coarse grained model of dense colloidal dynamics. The evolving structure of clusters in real space is finally sssociated to the decay of the thermo-remanent magnetization. Our analysis provides an unconventional coarse-grained description of spin glass aging as statistically subordinated to a Poisson quaking process and highlights record dynamics as a viable common theoretical framework for aging in different systems.Comment: 13 pages, 6 figs. Revised text and notation, several typos correcte