Scaling, finite size effects, and crossovers of the resistivity and current-voltage characteristics in two-dimensional superconductors

We revisit the scaling properties of the resistivity and the current-voltage characteristics at and below the Berezinskii-Kosterlitz-Thouless transition, both in zero and nonzero magnetic fields. The scaling properties are derived by integrating the renormalization group flow equations up to a scale where they can be reliably matched to simple analytic expressions. The vortex fugacity turns out to be dangerously irrelevant for these quantities below $T_c$, thereby altering the scaling behavior. We derive the possible crossover effects as the current, magnetic field, or system size is varied, and find a strong multiplicative logarithmic correction near $T_c$, all of which is necessary to account for when interpreting experiments and simulation data. Our analysis clarifies a longstanding discrepancy between the finite size dependence found in many simulations and the current-voltage characteristics of experiments. We further show that the logarithmic correction can be avoided by approaching the transition in a magnetic field, thereby simplifying the scaling analysis. We confirm our results by large-scale numerical simulations, and calculate the dynamic critical exponent $z$, for relaxational Langevin dynamics and for resistively and capacitively shunted Josephson junction dynamics.

Figure 1

Langevin dynamics. (a) Resistivity $\rho$ vs system size $L$ at different temperatures in a magnetic field $B={\Phi }_0/L^2$. The dotted lines are guides for the eyes, and the full green curve at $T=T_c$ is a ${\chi }^2$ fit (using $L=16$–120) to the power law $\rho \sim L^{-z}$, giving $z=2.22$. (b) As in (a), but for zero magnetic field. The full green curve at $T=T_c$ is a ${\chi }^2$ fit (using $L=16$–80) to $\rho \sim L^{-2.22}/{(\ln L-{\ell }_0)}^2$ with fixed $z=2.22$, giving ${\ell }_0=-2.71$. Insets: The effective exponent $z_{\mathrm{eff}}$ vs temperature, obtained from power-law fits. Note how $z_{\mathrm{eff}}$ is almost constant below $T_c$ in (a).

Figure 2

Overdamped RCSJ dynamics (Stewart-McCumber ${\beta }_C=2\pi I_c{R}^{2}C/{\Phi }_0=0.25$). (a) Resistivity $\rho$ vs system size $L$ at different temperatures in a magnetic field $B={\Phi }_0/L^2$. The full green curve at $T=T_c$ is a ${\chi }^2$ fit (using $L=16$–120) to the power law $\rho \sim L^{-z}$, giving $z=1.77$. (b) As in (a), but for zero magnetic field. The full green curve at $T=T_c$ is a ${\chi }^2$ fit (using $L=16$–80) to $\rho \sim L^{-1.77}/{(\ln L-{\ell }_0)}^2$ with fixed $z=1.77$, giving ${\ell }_0=-1.33$. Insets: As in Fig. 1.