Nontrivial maturation metastate-average state in a one-dimensional long-range Ising spin glass: above and below the upper critical range

Understanding the low-temperature pure state structure of spin glasses remains an open problem in the field of statistical mechanics of disordered systems. Here we study Monte Carlo dynamics, performing simulations of the growth of correlations following a quench from infinite temperature to a temperature well below the spin-glass transition temperature $T_c$ for a one-dimensional Ising spin glass model with diluted long-range interactions. In this model, the probability $P_{ij}$ that an edge ${i,j}$ has nonvanishing interaction falls as a power-law with chord distance, $P_{ij}propto1/R_{ij}^{2sigma}$, and we study a range of values of $sigma$ with $1/2<sigma<1$. We consider a correlation function $C_{4}(r,t)$. A dynamic correlation length that shows power-law growth with time $xi(t)propto t^{1/z}$ can be identified in the data and, for large time $t$, $C_{4}(r,t)$ decays as a power law $r^{-alpha_d}$ with distance $r$ when $rll xi(t)$. The calculation can be interpreted in terms of the maturation metastate averaged Gibbs state, or MMAS, and the decay exponent $alpha_d$ differentiates between a trivial MMAS ($alpha_d=0$), as expected in the droplet picture of spin glasses, and a nontrivial MMAS ($alpha_d e 0$), as in the replica-symmetry-breaking (RSB) or chaotic pairs pictures. We find nonzero $alpha_d$ even in the regime $sigma >2/3$ which corresponds to short-range systems below six dimensions. For $sigma < 2/3$, the decay exponent $alpha_d$ follows the RSB prediction for the decay exponent $alpha_s = 3 - 4 sigma$ of the static metastate, consistent with a conjectured statics-dynamics relation, while it approaches $alpha_d=1-sigma$ in the regime $2/3<sigma<1$; however, it deviates from both lines in the vicinity of $sigma=2/3$.