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 proportional to 1/R-ij(2 sigma), 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) proportional to 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 r << 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) not equal 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.

Automated summary

Understanding the low-temperature pure state structure of spin glasses continues to be a challenge in the field of statistical mechanics. By studying Monte Carlo dynamics in a one-dimensional Ising spin-glass model, it was found that dynamic correlation length grows with time in a power-law manner, and the decay of the correlation function follows a power-law behavior at large times, revealing unique phenomena different from traditional spin glass models.