Dynamical obstruction in a constrained system and its realization in lattices of superconducting devices

Abstract

Hard constraints imposed in statistical mechanics models can lead to interesting thermodynamical behaviors, but may at the same time raise obstructions in the thoroughfare to thermal equilibration. Here we study a variant of Baxter's three-color model in which local interactions and defects are included, and discuss its connection to triangular arrays of Josephson junctions of superconductors with broken time-reversal symmetry and kagom\'e networks of superconducting wires. The model is equivalent to an Ising model in a hexagonal lattice with the additional constraint that the magnetization of each hexagon is $\ifmmode\pm\else\textpm\fi{}6$ or $0.$ Defects in the superconducting models correspond to violations of this constraint, and include fractional and integer vortices, as well as open strings within two-color loops. In the absence of defects, and for ferromagnetic interactions, we find that the system is critical for a range of temperatures (critical line) that terminates when it undergoes an exotic first-order phase transition with a jump from a zero magnetization state into the fully magnetized state at finite temperature. Dynamically, however, we find that the system becomes frozen into domains. The domain walls are made of perfectly straight segments, and domain growth appears frozen within the time scales studied with Monte Carlo simulations, with the system trapped into a ``polycrystalline'' phase. This dynamical obstruction has its origin in the topology of the allowed reconfigurations in phase space, which consist of updates of closed loops of spins. Only an extreme rare-event dominated proliferation of confined defects may overcome this obstruction, at much longer time scales. Also as a consequence of the dynamical obstruction, there exists a dynamical temperature, lower than the (avoided) static critical temperature, at which the system is seen to jump from a ``supercooled liquid'' to the polycrystalline phase within our Monte Carlo time scale. In contrast, for antiferromagnetic interactions, we argue that the system orders for infinitesimal coupling because of the constraint, and we observe no interesting dynamical effects.