Continuous Time Monte Carlo and Spatial Ordering in Driven Lattice Gases: Application to Driven Vortices in Periodic Superconducting Networks

TL;DR

This paper compares different dynamical rules for driven lattice Coulomb gases, showing that continuous time Monte Carlo preserves spatial order better than Metropolis dynamics, and explores the resulting phases and ordering phenomena.

Contribution

It introduces a continuous time Monte Carlo method for driven Coulomb gases, demonstrating its physical accuracy and analyzing spatial ordering and phase behavior.

Findings

Metropolis dynamics destroys equilibrium lattice order under drive.

CTMC preserves spatial order at low temperatures.

Long-range hexatic order appears in driven square lattice liquids.

Abstract

We consider the two dimensional (2D) classical lattice Coulomb gas as a model for magnetic field induced vortices in 2D superconducting networks. Two different dynamical rules are introduced to investigate driven diffusive steady states far from equilibrium as a function of temperature and driving force. The resulting steady states differ dramatically depending on which dynamical rule is used. We show that the commonly used driven diffusive Metropolis Monte Carlo dynamics contains unphysical intrinsic randomness that destroys the spatial ordering present in equilibrium (the vortex lattice) over most of the driven phase diagram. A continuous time Monte Carlo (CTMC) is then developed, which results in spatially ordered driven states at low temperature in finite sized systems. We show that CTMC is the natural discretization of continuum Langevin dynamics, and argue that it gives the correct physical behavior when the discrete grid represents the minima of a periodic potential. We use detailed finite size scaling methods to analyze the spatial structure of the steady states. We find that finite size effects can be subtle and that very long simulation times can be needed to arrive at the correct steady state. For particles moving on a triangular grid, we find that the ordered moving state is a transversely pinned smectic that becomes unstable to an anisotropic liquid on sufficiently large length scales. For particles moving on a square grid, the moving state is a similar smectic at large drives, but we find evidence for a possible moving solid at lower drives. We find that the driven liquid on the square grid has long range hexatic order, and we explain this as a specifically non-equilibrium effect. We show that, in the liquid, fluctuations are diffusive in both the transverse and longitudinal directions.