Sine-Gordon Description of Beresinskii-Kosterlitz-Thouless Vortices in Superconductors Immersed in an External Magnetic Field

The Beresinskii-Kosterlitz-Thouless (BKT) physics of vortices in two-dimensional superconductors at finite magnetic field is investigated by means of a field-theoretical approach based on the sine-Gordon model. This description leads to a straightforward definition of the field-induced magnetization and shows that the persistence of nonlinear effects at low fields above the transition is a typical signature of the fast divergence of the correlation length within the BKT theory.

Figure 1

(a)–(b) $M(H)$ and $n_F(H)$ below $T_{\mathrm{BKT}}$ from the numerical solution of Eqs. (9, 10). The arrows indicate $H_{c1}$ according to Eq. (14). At $T=0.4T_{\mathrm{BKT}}$ $M(H)$ shows a sharp kink at $H_{c1}$, where $n_F$ drops abruptly to small values. At higher $T$ these features are partly smoothened out by thermal smearing. (c) Temperature dependence of the mass term ${\Delta }_{H=0}$. Notice that ${\Delta }_{H=0}\sim {\Delta }_{\Lambda }$ at $T_{\mathrm{BKT}}$, where $\Lambda =6.9\times {10}^5\AA$, ${\Delta }_R\sim 4\times {10}^{-6}$, and ${\Delta }_{\Lambda }=1.9\times {10}^{-5}$.

Figure 2

(a) $M(H)$ above and below $T_{\mathrm{BKT}}$ from the numerical solution of Eqs. (9, 10). (b) Solid line: the threshold field $H_l^{a,b}$ as a function of $T$. The points show the analytical estimates (14, 15), which agree with the numerical result except in a small range near $T_{\mathrm{BKT}}$. (c) $M(H)$ in logarithmic scale (curves are spaced by 1 K).