Active-absorbing phase transition and small-world behaviour in Ising model on finite addition type networks in two dimensions

We consider the ordering dynamics of the Ising model on a square lattice where an additional fixed number of bonds connect any two sites chosen randomly from a total of $N$ lattice sites. The total number of shortcuts added is controlled by two parameters $p$ and $\alpha$ for fixed $N$. The structural properties of the network are investigated which show that the small-world behaviour is obtained along the line $\alpha=\frac{\ln (N/2p)}{\ln N}$, which separates regions with ultra-small world like behaviour and short-ranged lattice like behaviour. We obtain a rich phase diagram in the $p-\alpha$ plane showing the existence of different types of active and absorbing states to which the Ising model evolves to and their boundaries.

Critical behavior near the reversible-irreversible transition in periodically driven vortices under random local shear

When many-particle (vortex) assemblies with disordered distribution are subjected to a periodic shear with a small amplitude $${\boldsymbol{d}}$$ d , the particles gradually self-organize to avoid next collisions and transform into an organized configuration. We can detect it from the time-dependent voltage $${\boldsymbol{V}}{\boldsymbol{(}}{\boldsymbol{t}}{\boldsymbol{)}}$$ V ( t ) (average velocity) that increases towards a steady-state value. For small $${\boldsymbol{d}}$$ d , the particles settle into a reversible state where all the particles return to their initial position after each shear cycle, while they reach an irreversible state for $${\boldsymbol{d}}$$ d above a threshold $${{\boldsymbol{d}}}_{{\boldsymbol{c}}}$$ d c . Here, we investigate the general phenomenon of a reversible-irreversible transition (RIT) using periodically driven vortices in a strip-shaped amorphous film with random pinning that causes local shear, as a function of $${\boldsymbol{d}}$$ d . By measuring $${\boldsymbol{V}}{\boldsymbol{(}}{\boldsymbol{t}}{\boldsymbol{)}}$$ V ( t ) , we observe a critical behavior of RIT, not only on the irreversible side, but also on the reversible side of the transition, which is the first under random local shear. The relaxation time $${\boldsymbol{\tau }}{\boldsymbol{(}}{\boldsymbol{d}}{\boldsymbol{)}}$$ τ ( d ) to reach either the reversible or irreversible state shows a power-law divergence at $${{\boldsymbol{d}}}_{{\boldsymbol{c}}}$$ d c . The critical exponent is determined with higher accuracy and is, within errors, in agreement with the value expected for an absorbing phase transition in the two-dimensional directed-percolation universality class. As $${\boldsymbol{d}}$$ d is decreased down to the intervortex spacing in the reversible regime, $${\boldsymbol{\tau }}{\boldsymbol{(}}{\boldsymbol{d}}{\boldsymbol{)}}$$ τ ( d ) deviates downward from the power-law relation, reflecting the suppression of intervortex collisions. We also suggest the possibility of a narrow smectic-flow regime, which is predicted to intervene between fully reversible and irreversible flow.