We propose a test to distinguish, both numerically and theoretically, between the two competing pictures of short-ranged Ising spin glasses at low temperature: ``chaotic'' size dependence. Scaling theories in which at most two pure states (related by a global spin flip) occur require that finite-volume correlations (with, say, periodic boundary conditions) have a well-defined thermodynamic limit. We argue, however, that the picture based on the infinite-ranged Sherrington-Kirkpatrick model, with many noncongruent pure states, leads to a breakdown of the thermodynamic limit. The argument combines rigorous and heuristic elements; one of the fomer is a proof that in the infinite-ranged model itself, non-self-averaging implies chaotic size dependence. Numerical tests, based on chaotic size dependence, could provide a more sensitive measure than the usual overlap distribution P(q) in determining the number of pure states.
We present a large-scale simulation of the three-dimensional and mean-field spin glasses down to a very low but finite temperature. We extrapolate pertinent observables, e.g., the disorder-averaged central weight to zero temperature, finding that many thermodynamic states at a finite temperature and two ground states at zero temperature are fully compatible. While the disorder-averaged central weight monotonically decreases with decreasing temperature, this is far from true for individual samples. This motivates us to link this behaviour with the well-known temperature chaos. At an observing temperature, a sample may or may not have pure state coexistence depending on whether it is undergoing temperature chaos, which is a random process. Therefore, temperature chaos is likely responsible for the emergence of many pure states, providing a natural and intuitive explanation for the coexistence of expensive domain-wall excitations and many pure states at the disorder-averaged level.
We investigate the three-dimensional Edwards-Anderson spin glass model at low temperature on simple cubic lattices of sizes up to
L=12. Our findings show a strong continuity among
T>0 physical features and those found previously at
T=0, leading to a scenario with emerging mean field like characteristics that are enhanced in the large volume limit. For instance, the picture of space filling sponges seems to survive in the large volume limit at
T>0, while entropic effects play a crucial role in determining the free-energy degeneracy of our finite volume states. All of our analysis is applied to equilibrium configurations obtained by a parallel tempering on
512 different disorder realizations. First, we consider the spatial properties of the sites where pairs of independent spin configurations differ and we introduce a modified spin overlap distribution which exhibits a nontrivial limit for large L. Second, after removing the
Z2 $(\ifmmode\pm\else\textpm\fi{}1)$ symmetry, we cluster spin configurations into valleys. On average these valleys have free-energy differences of
O(1), but a difference in the (extensive) internal energy that grows significantly with L; there is thus a large interplay between energy and entropy fluctuations. We also find that valleys typically differ by spongelike space filling clusters, just as found previously for low-energy system-size excitations above the ground state.