Holographic Abrikosov lattice: Vortex matter from black hole
Chuan-Yin Xia, Hua-Bi Zeng, Yu Tian, Chiang-Mei Chen, Jan Zaanen
Abstract
The $\mathrm{AdS}/\mathrm{CFT}$ correspondence provides a unique way to study the vortex matter phases in superconductors. We solve the dynamical evolution of a superconductor in $2+1$ dimensions at a finite temperature subjected to a magnetic field quench in terms of a gravitational ``hairy black hole'' dual living in an asymptotic ${\mathrm{AdS}}_{4}$ space. We exploit this to determine the nature of the equilibrium states realized at long times after the quench of this two dimensional type II superconductor in a perpendicular external uniform magnetic field ${B}_{0}$. This holographic superconductor exhibits the generic lower (${B}_{c1}(T)$) and upper (${B}_{c2}(T)$) critical fields. For ${B}_{0}<{B}_{c1}(T)$ the magnetic field is completely expelled revealing the Meissner phase, while the superconductivity is destroyed when ${B}_{0}>{B}_{c2}(T)$. Abrikosov lattices appear in the range ${B}_{c1}(T)<{B}_{0}<{B}_{c2}(T)$ that realize various configurations in the form of hexagonal, square and slightly irregular square lattices pending the magnetic field strength and the influence of finite size boundaries. We show this to be consistent with the expectations of Ginzburg-Landau theory where the upper and lower critical fields are associated with the inverse squares of the coherence length and magnetic penetration depth, respectively.