Magnetic Vortices in Finite Isospin Chiral Perturbation Theory

We study magnetic vortices in finite isospin chiral perturbation theory, the effective theory of low energy QCD. The relevant degrees of freedom in chiral perturbation theory are three pions (two of which are electromagnetically charged) that arise due to spontaneous breaking of chiral symmetry by the QCD vacuum. Since pions are Goldstone bosons, interactions between them are derivative in nature. At zero external magnetic field, above a critical chemical potential charged pions form a superfluid. In the presence of a uniform, external magnetic field, we show that pions behave as type-II superconductors. We find the magnetic vortex lattice condensation energy and the vortex lattice structure near the upper critical point, and construct the phase diagram by finding the critical magnetic fields as a function of the isospin chemical potential. We also discuss the possibility of neutral pion condensation, which do not couple directly with the external magnetic field but couple with the charged pions via strong interactions.