Holographic Abrikosov lattice: Vortex matter from black hole

Chuan-Yin Xia,Chiang-Mei Chen,Hua-Bi Zeng,Yu Tian,Jan Zaanen

doi:10.1103/physrevd.105.l021901

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.

Highlights

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
A well-known property of the type II superconductors is the quantization of the magnetic flux in the mixed state, where the magnetic field penetrates in the form of vortices that combine with the magnetic field into quantized fluxoids each carrying a quantized magnetic flux Φ0 1⁄4 hc=2e, forming an Abrikosov lattice [1]
The quantitative description of the formation of the Abrikosov lattice in conventional type II superconductors is a classic success of Ginzburg-Landau theory resting on the microscopic by SCOAP3.employing thpe ffiBffi ardeen-Cooper-Schrieffer (BCS) theory

Summary

INTRODUCTION

A well-known property of the type II superconductors is the quantization of the magnetic flux in the mixed state, where the magnetic field penetrates in the form of vortices that combine with the magnetic field into quantized fluxoids each carrying a quantized magnetic flux Φ0 1⁄4 hc=2e, forming an Abrikosov lattice [1]. One adds a charged scalar field to the Einstein-Maxwell theory describing the planar black hole living in the deep interior of the AdS bulk Upon lowering temperature this scalar field may acquire a finite amplitude in the bulk. According to the probe limit the GL parameter κ indicates that it is invariably of the type II kind [24,25,26] Another approach is to involve the time evolution by attempting to solve the time independent equation of motions (EoMs) for the scalar and gauge fields in the bulk spacetime to look for equilibrium solutions [27,28,29,30,31,32,33,34]. We determine the phase diagram at temperatures close to Tc (Fig. 5)

HOLOGRAPHIC MODEL

ABRIKOSOV LATTICES

MAGNETIZATION

PHASE DIAGRAM

SUMMARY