Electromagnetic superconductivity of vacuum induced by strong magnetic field

37 pages, 14 figures, to appear in Lect. Notes Phys. "Strongly interacting matter in magnetic fields" (Springer), edited by D. Kharzeev, K. Landsteiner, A. Schmitt, H.-U. YeeInternational audienceThe quantum vacuum may become an electromagnetic superconductor in the presence of a strong external magnetic field of the order of 10^{16} Tesla. The magnetic field of the required strength (and even stronger) is expected to be generated for a short time in ultraperipheral collisions of heavy ions at the Large Hadron Collider. The superconducting properties of the new phase appear as a result of a magnetic-field-assisted condensation of quark-antiquark pairs with quantum numbers of electrically charged rho mesons. We discuss similarities and differences between the suggested superconducting state of the quantum vacuum, a conventional superconductivity and the Schwinger pair creation. We argue qualitatively and quantitatively why the superconducting state should be a natural ground state of the vacuum at the sufficiently strong magnetic field. We demonstrate the existence of the superconducting phase using both the Nambu-Jona-Lasinio model and an effective bosonic model based on the vector meson dominance (the rho-meson electrodynamics). We discuss various properties of the new phase such as absence of the Meissner effect, anisotropy of superconductivity, spatial inhomogeneity of ground state, emergence of a neutral superfluid component in the ground state and presence of new topological vortices in the quark-antiquark condensates