Properties of holographic superconductors from Machine Learning

We investigate holographic superconductors using modern optimisation techniques inspired by machine learning. The critical temperature is obtained by minimising the variational functional for the eigenvalue ${\lambda }^2$ with two complementary trial functions: a simple cosine ansatz $F(z)=\cos (az)$ and a flexible exponential polynomial $F(z)=\exp ({\sum }_{n=2}^N{a}_n{z}^n)$, both of which automatically satisfy the standard boundary conditions. For the cosine ansatz, we perform a one-parameter minimisation and obtain ${\lambda }^2(\mathrm{\Delta })$ and $T_c\mathrm{/}\sqrt{\rho }$ over a wide range of $\mathrm{\Delta }$, including the exact values at $\mathrm{\Delta }=1$ and $\mathrm{\Delta }=2$ to high accuracy. The exponential polynomial ansatz, with up to 19 coefficients, is optimised using a multi-start L-BFGS-B algorithm with warm-starting, yielding even better agreement with known exact results. Our numerical data for ${\lambda }^2(\mathrm{\Delta })$ and $T_c\mathrm{/}\sqrt{\rho }$ match the analytical predictions from the literature, confirming the robustness of the variational approach. This work; therefore, demonstrates that a combination of analytic trial functions and modern numerical optimisation provides a powerful, flexible, and efficient tool for exploring holographic superconductors, and can be readily extended to include backreaction or other sectors in this field.