Arresting dynamics in hardcore spin models

We study the dynamics of hardcore spin models on the square and triangular lattice, constructed by analogy to hard spheres, where the translational degrees of freedom of the spheres are replaced by orientational degrees of freedom of spins on a lattice and the packing fraction as a control parameter is replaced by an exclusion angle. In equilibrium, models on both lattices exhibit a Kosterlitz-Thouless transition at an exclusion angle ${\mathrm{\Delta }}_{\mathrm{KT}}$. We devise compression protocols for hardcore spins and find that any protocol that changes the exclusion angle nonadiabatically, if endowed with only local dynamics, fails to compress random initial states beyond an angle ${\mathrm{\Delta }}_{\mathrm{J}}>{\mathrm{\Delta }}_{\mathrm{KT}}$. This coincides with a doubly algebraic divergence of the relaxation time of compressed states toward equilibrium. We identify a remarkably simple mechanism underpinning this divergent timescale: topological defects involved in the phase ordering kinetics of the system become incompatible with the hardcore spin constraint, leading to a vanishing defect mobility as $\mathrm{\Delta }\to {\mathrm{\Delta }}_{\mathrm{J}}$.

Figure 1

(a) Illustration of the hardcore constraint. For a given spin, red shading indicates the orientations forbidden by its nearest neighbors. (b) An antiferromagnetic vortex on the square lattice. The vortex core is marked in purple. (c) Minimal phase diagram of the model.

Figure 2

(a) State at $\mathrm{\Delta }={\mathrm{\Delta }}_{\mathrm{J}}$, frozen vortex cores are enclosed by dashed lines. (b) Vortex density ${\rho }_{\mathrm{V}}$ of far-from-equilibrium states relaxing toward equilibrium at different exclusion angles $\mathrm{\Delta }$, for $L=60$. (c) Doubly algebraic behavior of relaxation time scale fitted from long-time tail of ${\rho }_{\mathrm{V}}$ as $\mathrm{\Delta }\to {\mathrm{\Delta }}_{\mathrm{J}}$. (d) Vanishing defect mobility as $\mathrm{\Delta }\to {\mathrm{\Delta }}_{\mathrm{J}}$.

Figure 3

(a) Equilibrium phase diagram of the model on the triangular lattice. (b) A vortex on the triangular lattice. (c) Domain walls of chirality both with a kink (left) and without any kinks (right). (d) Distribution of jamming angles ${\mathrm{\Delta }}_J$ as a function of compression speed for $L=42$; lower $N_{\mathrm{rattle}}$ implies faster compression.