Vortex Sublattice Melting in a Two-Component Superconductor

We consider the vortices in a superconductor with two individually conserved condensates in a finite magnetic field. The ground state is a lattice of cocentered vortices in both order parameters. We find two phase transitions: (i) a “vortex sublattice melting” transition where vortices in the field with lowest phase stiffness (“light vortices”) lose cocentricity with the vortices with large phase stiffness (“heavy vortices”), entering a liquid state (the structure factor of the light vortices vanishes continuously; this transition is in the 3D$xy$ universality class); (ii) a first-order melting transition of the lattice of heavy vortices, in a liquid of light vortices.

Figure 1

(A) A type-II, $N=2$ superconductor at zero temperature in a magnetic field forms a lattice of composite vortices, i.e., cocentered type-$1$ (red) and type-$2$ (blue) vortices. (B) Low-temperature fluctuations in the composite vortex lattice (VL) generate closed loops of type-$1$ vortices and local splitting of field-induced composite vortices. This phase features superfluidity, as well as longitudinal superconductivity. (C) There is a temperature region in low magnetic field when type-$1$ vortices form a vortex liquid and the corresponding vortex loops are proliferated, while type-$2$ vortices form a VL. Superfluidity is lost, longitudinal superconductivity is retained. The arrow from (B) to (C) illustrates type-$1$ loop proliferation. (D) A vortex liquid of type-$1$ and type-$2$ vortices. Superfluidity and longitudinal superconductivity is lost, i.e., the normal phase [5]. There is no type-$2$ loop proliferation going from (C) to (D), since this is a first-order melting transition of the type-$2$ VL in the background of a liquid of line-tension-less type-$1$ vortices.

Figure 2

Detailed illustration of the low-temperature thermal fluctuations in a VL of composite vortices. A local excursion of a type-$1$ vortex away from the composite VL may be viewed as a type-$1$ bound vortex-loop superposed on the composite VL. The composite vortex line does not interact with a vortex with nontrivial winding in $\Delta \gamma =\Delta ({\theta }^{(1)}–{\theta }^{(2)})$ [6]. A splitting of the composite VL, illustrated in going from (B) to (C) in Fig. 1, may be viewed as a zero-field vortex-loop proliferation of type-$1$ vortices; a 3D$xy$ phase transition universality [12, 13, 14].

Figure 3

MC results for $N=2$ $|{\psi }^{(1)}{|}^2=0.2$, $|{\psi }^{(2)}{|}^2=2$, and $e=1/\sqrt{10}$. (a) $C_V$ (black) and $N_{\mathrm{c}\mathrm{o}}$ (green). The $C_V$ anomaly at $T_{c1}=0.37$, where type-$1$ vortices proliferate, matches the point at which $N_{\mathrm{c}\mathrm{o}}$ drops to zero. Thus, type-$1$ vortices are torn off type-$2$ vortices. The remnant of the zero-field anomaly in $C_V$ is seen as a hump at $T\sim 3.6$. (b) $S^{(1)}(\mathbf{K})$ (red) and $S^{(2)}(\mathbf{K})$ (blue) for the particular Bragg vector $\mathbf{K}=(\pi /4,-\pi /4)$. $S^{(1)}(\mathbf{K})$ vanishes continuously at $T_{\mathrm{c}1}$, while $S^{(2)}(\mathbf{K})$ vanishes discontinuously at $T_{\mathrm{M}}=2.34$. (c) FSS plots of the M3 from which the exponents $\alpha =-0.02\pm 0.05$ and $\nu =0.67\pm 0.03$ are extracted, showing that the sublattice melting is a 3D$xy$ phase transition. (d)–(g) Plots of $S^{(2)}({\mathbf{k}}_{\perp })$ for the temperatures $T_{\mathrm{d}}=0.35$, $T_{\mathrm{e}}=0.4$, $T_{\mathrm{f}}=1.66$, and $T_{\mathrm{g}}=2.85$, respectively. At $T_{\mathrm{d}}$, $T_{\mathrm{e}}$, and $T_{\mathrm{f}}$, the VL remains intact. The VL melts at $T_{\mathrm{M}}$ to give a vortex-liquid ring pattern at $T_{\mathrm{g}}$.