We investigate the critical relaxational dynamics of the three-dimensional (3D) lattice ${\mathbb{Z}}_N$ gauge models with $N=6$ and $N=8$, whose equilibrium critical behavior at their topological transitions belongs to the inverted $\mathit{XY}$ (IXY) universality class [this is also the universality class of the continuous transitions of the 3D lattice $U$(1) gauge Higgs models with a one-component complex scalar field], which is connected to the standard $\mathit{XY}$ universality class by a nonlocal duality relation of the partition functions. Specifically, we consider the purely relaxational dynamics realized by a locally reversible Metropolis dynamics, as commonly used in Monte Carlo simulations. To determine the corresponding dynamic exponent $z$, we focus on the out-of-equilibrium critical relaxational flows arising from instantaneous quenches to the critical point, which are analyzed within an out-of-equilibrium finite-size scaling framework. We obtain the estimate $z=2.59\left(3\right)$. A numerical analysis of the equilibrium critical dynamics give consistent, but less accurate, results. This dynamic exponent is expected to characterize the critical slowing down of the purely relaxational dynamics of all topological transitions that belong to the 3D IXY universality class. We note that this result implies that the critical relaxational dynamics of the 3D IXY universality class is slower than that of the standard 3D $\mathit{XY}$ universality class, whose relaxational dynamic exponent $z\approx 2.02$ is significantly smaller, although they share the same length-scale critical exponent $\nu \approx 0.6717$.

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• Critical phenomena
• Phase transitions
• Topological phase transition
• Duality
• Finite-size scaling
• Lattice gauge theory
• Renormalization group