Vortex and droplet engineering in a holographic superconductor

We give a detailed account of the construction of nontrivial localized solutions in a $2+1$ dimensional model of superconductors using a $3+1$ dimensional gravitational dual theory of a black hole coupled to a scalar field. The solutions are found in the presence of a background magnetic field. We use numerical and analytic techniques to solve the full Maxwell-scalar equations of motion in the background geometry, finding condensate droplet solutions, and vortex solutions possessing a conserved winding number. These solutions and their properties, which we uncover, help shed light on key features of the $(B,T)$ phase diagram.

Figure 1

Three solutions with the same $\tilde{R}(1)$ but different ${\partial }_z{\tilde{A}}_t(1)$ that satisfy Dirichlet boundary conditions at $z=0$. The solutions are distinguished by the number of nodes (times they cross the $z$-axis) they have.

Figure 2

Vacuum expectation values for the scalar. Here $T_c$ is defined to be $0.226\alpha \sqrt{\rho }$ and $0.118\alpha \sqrt{\rho }$ for the $\Delta =1$ and $\Delta =2$ operator, respectively.

Figure 3

Droplet solutions for $\gamma =0.5$ on the left and $\gamma =2$ on the right. They correspond to $T/T_c\approx 0.84$ and $T/T_c\approx 0.67$ respectively.

Figure 4

The limiting droplet line in the $g\to \infty$ limit. Below this, droplets disappear.

Figure 5

Limiting droplet line for three values of $g$ with $\tilde{T}=(3-q_c^2)/4\pi$.

Figure 6

Convergence to the $g\to \infty$ critical temperature. The curves asymptote to 0.225492 and 0.118412, respectively. See text for discussion.

Figure 7

The droplet limiting curves for a range of couplings, after rescaling to include the $g\to \infty$ case. $T_c=\alpha \sigma \sqrt{2qg}$.

Figure 8

Vortex Solutions for $\xi =1$ (lhs) and $\xi =2$ (rhs).

Figure 9

Vortex Solutions for $\xi =1$ (lhs) and $\xi =2$ (rhs).