Vortex-antivortex lattices in a holographic superconductor

We employ the Einstein-Abelian-Higgs theory to investigate the structure of vortex-antivortex lattices within a superconductor driven by spatial periodic magnetic fields. By adjusting the parameters of the external magnetic field, including the period (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi mathvariant="script">T</a:mi></a:mrow></a:math>) and the amplitude (<d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msub><d:mi>B</d:mi><d:mn>0</d:mn></d:msub></d:math>), various distinct vortex states emerge. These states encompass the Wigner crystallization state, the vortex cluster state, and the suppressed state. Additionally, we present a comprehensive phase diagram to demarcate the specific regions where these structures emerge, contributing to our understanding of superconductivity in complex magnetic environments.