Nonequilibrium dynamics of random field Ising spin chains: exact results via real space RG

Abstract

The non-equilibrium dynamics of classical random Ising spin chains with non-conserved magne-tization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising spin chains (RFIM) with and without a uniform applied field, as well as on Ising spin glass chains (SG) in an applied field. For the RFIM we consider the universal regime where the random field and the temperature are both much smaller than the exchange coupling. In that regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g. a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the non-equilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly towards a set of system-specific time-dependent postitions that are independent of the initial conditions. Thus the behavior of this non-equilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function S0(t)Sx(t) and the two-time autocorrelations S0(t ′)S0(t). We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the ±J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function S0(t)Sx(t)S0(t ′)Sx(t ′) which can in principle be measured by experiments on spin-glass like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focussing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time tw. The non-equilibrium fluctuation-dissipation ratio X(t, tw) is computed. We find that for (t − tw) ∼ t ˆ α w withˆαwithˆ withˆα < 1, it is equal to its equilibrium value X = 1 although time translational invariance does not hold in this regime. It exhibits for t − tw ∼ tw an aging regime with a non-trivial X = X(t/tw) = 1, but the behaviour is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence towards equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.