Vortex Dynamics in a Compact Kardar-Parisi-Zhang System

A. Zamora, N. Lad, and M. H. Szymanska

Phys. Rev. Lett. 125, 265701 – Published 22 December 2020

Abstract

We study the dynamics of vortices in a two-dimensional, nonequilibrium system, described by the compact Kardar-Parisi-Zhang equation, after a sudden quench across the critical region. Our exact numerical solution of the phase-ordering kinetics shows that the unique interplay between nonequilibrium and the variable degree of spatial anisotropy leads to different critical regimes. We provide an analytical expression for the vortex evolution, based on scaling arguments, which is in agreement with the numerical results, and confirms the form of the interaction potential between vortices in this system.

Received 7 October 2019
Accepted 23 November 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.265701

Physics Subject Headings (PhySH)

BKT transition Critical phenomena Phase diagrams Phase transitions

Nonequilibrium systems Polariton condensate

Kardar–Parisi–Zhang equation Monte Carlo methods

Statistical Physics & Thermodynamics

Authors & Affiliations

A. Zamora, N. Lad , and M. H. Szymanska

Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, United Kingdom

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Issue

Vol. 125, Iss. 26 — 31 December 2020

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Images

Figure 1
Steady state of the cKPZ system. Steady-state density of vortices $ρ$ (in arbitrary units, dashed curves) and magnetization $M$ (solid curves) as a function of the noise parameter $σ$ for different $λ$ . The vertical dashed line indicates the final noise $σ_{f} = 1 / 3$ for the infinite rapid quench shown in Figs. 3 and 4. For the $X Y$ model with $λ = 0$ (green curves) and the SA case with $λ_{x} = - λ_{y}$ (black curves) the system exhibits (quasi)ordered and disordered phases separated by a critical noise $σ_{c} \approx 0.63$ (striped rectangle). However, for the isotropic case with large nonlinearity $λ$ (blue curves), the system shows a vortex-dominated phase with no magnetization even at low noise values.
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Figure 2
$θ (r)$ at late times with marked vortices. 2D maps with position of vortices (black dots) and antivortices (red dots) on top of the $θ (r)$ profile for a single realisation in two different regimes: (i) phase ordering configuration in the SA regime with $λ_{x} = - λ_{y} = 1.7$ (left panel), note that due to periodic boundaries the red V at the bottom is paired with the black AV on the top, and (ii) vortex-dominated phase for $λ_{x} = λ_{y} = 1.5$ in the isotropic regime (right panel). Note that the left panel shows the whole system, whereas the right panel shows a zoomed quarter (see the plot for the whole system in Fig. 8 of the Supplemental Material [35]).
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Figure 3
Vortex dynamics in the SA regime. Number of vortices $n_{v}$ as function of time $t / \log (t)$ after a sudden quench from a high noise disordered initial state to a low noise $σ_{f}$ final state. The dots show the numerical solution of the KPZ equation and the solid lines the theoretical prediction from Eq. (6). We observe a diffusive law with a logarithmic correction: $n_{v} \sim [\log (t) / t]^{α}$ , with $α \approx 1$ for all the different configurations. Specifically $α = 1.016 \pm 0.010, 1.111 \pm 0.011, 1.130 \pm 0.015, 1.142 \pm 0.015$ from top to bottom [35]. The dashed line shows the $α = 1$ case.
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Figure 4
Vortex dynamics in the WA regime. Number of vortices $n_{v}$ as a function of time $t / \log (t)$ after a sudden quench from a disordered initial state to a low noise $σ_{f}$ final state. The dots are from the numerical solution of the cKPZ equation and the solid lines are the analytical estimate from Eq. (6). From top to bottom: $λ = 3.5$ , 3.25, 3.0, 2.75, 2.5, 2.25, 1.9, 0.5, with $λ \equiv λ_{x} = λ_{y}$ . We observe a saturation of the number of vortices at $ρ_{S} (λ)$ for $λ \geq 1.9$ , which is a clear indicator of the vortex-dominated phase [35]. Inset: Blue dots show the characteristic length scale $L_{v} \sim ρ_{S}^{- 1 / 2}$ obtained from numerics and red solid line from the expression (3) for $2.5 \leq λ \leq 4.0$ .
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