Critical scaling and aging near the flux-line-depinning transition

Harshwardhan Chaturvedi, Ulrich Dobramysl, Michel Pleimling, and Uwe C. Täuber

Phys. Rev. B 101, 024515 – Published 21 January 2020

Abstract

We utilize Langevin molecular dynamics simulations to study dynamical critical behavior of magnetic flux lines near the depinning transition in type-II superconductors subject to randomly distributed attractive point defects. We employ a coarse-grained elastic line Hamiltonian for the mutually repulsive vortices and purely relaxational kinetics. In order to infer the stationary-state critical exponents for the continuous nonequilibrium depinning transition at zero temperature $T = 0$ and at the critical driving current density $j_{c}$ , we explore two-parameter scaling laws for the flux lines' gyration radius and mean velocity as functions of the two relevant scaling fields $T$ and $j - j_{c}$ . We also investigate critical aging scaling for the two-time height auto-correlation function in the early-time nonequilibrium relaxation regime to independently measure critical exponents. We provide numerical exponent values for the distinct universality classes of noninteracting and repulsive vortices.

Received 15 July 2019
Revised 3 January 2020

DOI:https://doi.org/10.1103/PhysRevB.101.024515

Physics Subject Headings (PhySH)

Dynamical phase transitions Fluctuations & noise Nonequilibrium statistical mechanics Superconductivity

Disordered systems Superconductors

Dissipative particle dynamics Langevin algorithm Scaling methods Stochastic differential equations

Statistical Physics & Thermodynamics

Authors & Affiliations

Harshwardhan Chaturvedi^1,2, Ulrich Dobramysl³, Michel Pleimling ^1,4, and Uwe C. Täuber ¹

¹Department of Physics & Center for Soft Matter and Biological Physics (MC 0435), Virginia Tech, 850 West Campus Drive, Blacksburg, Virginia 24061, USA
²Kaizen Analytix, 2 Ravinia Drive, Atlanta, Georgia 30346, USA
³Wellcome Trust / CRUK Gurdon Institute, University of Cambridge, Tennis Court Rd, Cambridge CB2 1QN, United Kingdom
⁴Academy of Integrated Science (MC 0563), Virginia Tech, 800 West Campus Drive, Blacksburg, Virginia 24061, USA

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Vol. 101, Iss. 2 — 1 January 2020

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Images

Figure 1
Simulation snapshots of the noninteracting flux line system in the (a) pinned ( $F_{d} = 0 ε_{0} b_{0}$ ), (b) critical ( $F_{d} = 0.0095 ε_{0} b_{0}$ ), and (c) moving ( $F_{d} = 0.03 ε_{0} b_{0}$ ) stationary regimes at temperature $T = 0.0009 ε_{0} b_{0} / k_{B}$ . The snapshots represent a side view of the system, i.e., a projection of the three-dimensional system onto the $x z$ plane, with drive $F_{d}$ directed in the positive $x$ direction.
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Figure 2
Velocity-temperature ( $v - T$ ) curves for repulsive vortex lines taken for various values of the drive $F_{d}$ , with the curve at critical drive $F_{c} = (0.013 \pm 0.0005) ε_{0}$ indicated by a solid line.
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Figure 3
Steady-state (a), (b) velocity $v$ (in units of $b_{0} / t_{0}$ ) and (c), (d) radius of gyration $r_{g}$ (in units of $b_{0}$ ) as functions of driving force $F_{d}$ (in units of $ε_{0}$ ) for (a), (c) noninteracting and (b), (d) interacting flux lines. Each quantity is measured at five different temperatures $T$ (values listed in units of $ε_{0} b_{0} / k_{B}$ ).
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Figure 4
Steady-state data from Fig. 3 replotted by means of the two-parameter scaling ansätze (3) and (4) with stationary critical exponents $β, δ$ , and $ν$ . Panels (a), (c) show results for noninteracting vortices, while (b), (d) represent the data for repulsively interacting flux lines. Panels (a), (b) show scaled velocities $v$ for five different temperatures $T$ (in units of $ε_{0} b_{0} / k_{B}$ ) as functions of scaled reduced drive $f$ , and panels (c), (d) display scaled gyration radii $r_{g}$ for the same temperatures, also as functions of $f$ .
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Figure 5
(a),(b) Two-time flux line height autocorrelation functions $C (t, s)$ following critical quenches at $T = 0.0005 ε_{0} b_{0} / k_{B}$ for waiting times $s = 2^{6} t_{0}, 2^{7} t_{0}$ , and $2^{8} t_{0}$ as a function of $t - s$ ; (c),(d) these autocorrelations scaled with $s^{b}$ as functions $t / s$ with $b = 0.56 \pm 0.03$ for (c) and $0.30 \pm 0.03$ for (d). Panels (a), (c), (e), (g) and (b), (d), (f), h) represent the noninteracting and repulsively interacting flux line systems, respectively. The solid black line in (c), (d) shows the power law dependence of the scaled quantities for $T = 0.0005 ε_{0} b_{0} / k_{B}$ on $t / s$ with $λ_{C} / z = 0.61 \pm 0.015$ for (c) and $0.44 \pm 0.03$ for (d). Panels (e), (f) and (g), (h), respectively, show the exponents $b$ and $λ_{C} / z$ estimated for critical quenches at five different temperatures $T = 0.0005$ , 0.0006, 0.0007, 0.0008, and 0.0009 (left to right, in units of $ε_{0} b_{0} / k_{B}$ ); the solid horizontal line in each panel represents the mean value of the data points and the shaded region indicates the error of the mean. The final mean exponent values are stated in Table 2.
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Figure 6
Correlation time $τ$ of (a) noninteracting and (b) interacting vortices as a function of drive $f$ (double-logarithmic scale) for six temperatures $T$ (values listed in units of $ε_{0} b_{0} / k_{B}$ ) along with linear (power law) fits. The negative line slope for a given $T$ yields $ν z$ . In each panel, the inset shows the dynamic exponent $z$ (obtained from the $log τ - log f$ data) as a function of $T$ , along with a linear extrapolation to zero temperature. (Each data point originates from 1000 independent simulation runs.)
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