Relaxation time for competing short- and long-range interactions in the model A dynamic universality class

We study the relaxation dynamics at criticality in the one-dimensional spin-$1/2$ Nagle-Kardar model, where short- and long-range interactions can compete. The phase diagram of this model shows lines of first and second-order phase transitions, separated by a tricritical point. We consider Glauber dynamics, focusing on the slowing-down of the magnetization $m$ both along the critical line and at the tricritical point. Starting from the master equation and performing a coarse-graining procedure, we obtain a Fokker-Planck equation for $m$ and the fraction of defects. Using central manifold theory, we analytically show that $m$ decays asymptotically as $t^{-1/2}$ along the critical line, and as $t^{-1/4}$ at the tricritical point. This result implies that the dynamical critical exponent is $z=2$, proving that the macroscopic critical dynamics of the Nagle-Kardar model falls within the dynamic universality class of purely relaxational dynamics with a non-conserved order parameter (model A). Large deviation techniques enable us to show that the average first passage time between local equilibrium states follows an Arrhenius law.