Dynamics of spin systems with randomly asymmetric bonds: Ising spins and Glauber dynamics

The stochastic dynamics of randomly asymmetric fully connected Ising systems is studied. We solve analytically the particularly simple case of fully asymmetric systems. We calculate the relaxation time of the autocorrelation function and show that the system remains paramagnetic even at zero temperature (T=0). The ferromagnetic phase is only slightly affected by the asymmetry, and the paramagnetic-to-ferromagnetic phase transition is characterized by a critical slowing down similar to second-order transition in symmetric (fully connected) systems. Monte Carlo simulations of a fully connected Ising system with random asymmetric interactions, both at finite and zero temperature, are presented. For finite T the autocorrelation function decays completely to zero for all strengths of the asymmetry. The T=0 behavior is more complex. In the fully asymmetric case the system is ergodic, with decaying autocorrelations, in agreement with the theoretical predictions. In the partially asymmetric case all flows terminate at fixed points (i.e., states which are stable to single spin flips). However, the typical time that it takes to converge to a fixed point grows exponentially with the size of the system. This convergence time varies from sample to sample and has a log-normal distribution in large systems. On time scales which are smaller than the convergence time, the system behaves ergodically, and the autocorrelation function decays to zero, much like the finite-temperature case.