The structure of the phase diagram for strong interactions becomes richer in the presence of a magnetic background, which enters as a new control parameter for the thermodynamics. Motivated by the relevance of this physical setting for current and future high-energy heavy-ion collision experiments and for the cosmological QCD transitions, we use the linear sigma model coupled to quarks and to Polyakov loops as an effective theory to investigate how the chiral and the deconfining transitions are affected, and present a general picture for the temperature\char21{}magnetic field phase diagram. We compute and discuss each contribution to the effective potential for the approximate order parameters, and uncover new phenomena such as the paramagnetically induced breaking of global Z3 symmetry, and possible splitting of deconfinement and chiral transitions in a strong magnetic field.

We show that in a sufficiently strong magnetic field the QCD vacuum may undergo a transition to a new phase where charged $\rho${}\ifmmode\pm\else\textpm\fi{} mesons are condensed. In this phase the vacuum behaves as an anisotropic inhomogeneous superconductor which supports superconductivity along the axis of the magnetic field. In the directions transverse to the magnetic field the superconductivity is absent. The magnetic-field-induced anisotropic superconductivity\char22{}which is realized in the cold vacuum, i.e. at zero temperature and density\char22{}is a consequence of a nonminimal coupling of the $\rho${} mesons to the electromagnetic field. The onset of the superconductivity of the charged $\rho${}\ifmmode\pm\else\textpm\fi{} mesons should also induce an inhomogeneous superfluidity of the neutral $\rho${}0 mesons. We also argue that due to simple kinematical reasons a strong enough magnetic field makes the lifetime of the $\rho${} mesons longer by closing the main channels of the strong decays of the $\rho${} mesons into charged pions.

Using an extended Nambu\char21{}Jona-Lasinio model as a low-energy effective model of QCD, we show that the vacuum in a strong external magnetic field (stronger than 1016\,\hskip0.056em\hskip0.056emT) experiences a spontaneous phase transition to an electromagnetically superconducting state. The unexpected superconductivity of, basically, empty space is induced by emergence of quark-antiquark vector condensates with quantum numbers of electrically charged rho mesons. The superconducting phase possesses an anisotropic inhomogeneous structure similar to a periodic Abrikosov lattice in a type-II superconductor. The superconducting vacuum is made of a new type of vortices which are topological defects in the charged vector condensates. The superconductivity is realized along the axis of the magnetic field only. We argue that this effect is absent in pure QED.

The chiral magnetic effect is the generation of electric current of quarks along an external magnetic field in the background of topologically nontrivial gluon fields. There is recent evidence that this effect is observed by the STAR Collaboration in heavy-ion collisions at the Relativistic Heavy Ion Collider. In our paper we study qualitative signatures of the chiral magnetic effect using quenched lattice simulations. We find indications that the electric current is indeed enhanced in the direction of the magnetic field both in equilibrium configurations of the quantum gluon fields and in a smooth gluon background with nonzero topological charge. In the confinement phase the magnetic field enhances the local fluctuations of both the electric charge and chiral charge densities. In the deconfinement phase the effects of the magnetic field become smaller, possibly due to thermal screening. Using a simple model of a fireball we obtain a good agreement between our data and experimental results of STAR Collaboration.

We study rotating fermionic matter at finite temperature in the framework of the Nambu-Jona-Lasinio model. In order to respect causality the rigidly rotating system must be bound by a cylindrical boundary with appropriate boundary conditions that confine the fermions inside the cylinder. We show the finite geometry with the MIT boundary conditions affects strongly the phase structure of the model leading to three distinct regions characterized by explicitly broken (gapped), partially restored (nearly gapless) and spontaneously broken (gapped) phases at, respectively, small, moderate and large radius of the cylinder. The presence of the boundary leads to specific steplike irregularities of the chiral condensate as functions of coupling constant, temperature and angular frequency. These steplike features have the same nature as the Shubnikov-de Haas oscillations with the crucial difference that they occur in the absence of both external magnetic field and Fermi surface. At finite temperature the rotation leads to restoration of spontaneously broken chiral symmetry while the vacuum at zero temperature is insensitive to rotation (“cold vacuum cannot rotate”). As the temperature increases the critical angular frequency decreases and the transition becomes softer. A phase diagram in angular frequency-temperature plane is presented. We also show that at fixed temperature the fermion matter in the chirally restored (gapless) phase has a higher moment of inertia compared to the one in the chirally broken (gapped) phase.

The axial magneticeffect, i.e., the generation of an energy current parallel to an axial magnetic field coupling with opposite signs to left- and right-handed fermions, is a nondissipative transport phenomenon intimately related to the gravitational contribution to the axial anomaly. An axial magnetic field emerges naturally in condensed matter in so-called Weyl semimetals. We present a measurable implementation of the axial magnetic effect. We show that the edge states of a Weyl semimetal at finite temperature possess a temperature dependent angular momentum in the direction of the vector potential intrinsic to the system. Such a realization provides a plausible context for the experimental confirmation of the elusive gravitational anomaly.