Finite isospin chiral perturbation theory in a magnetic field

The phase diagram of finite-isospin, zero-temperature QCD with the pions coupled to photons in a uniform external magnetic field is explored in the low-field, small-isospin-density regime for which chiral perturbation theory is a valid description. For realistic pion masses, the system behaves as a type-II superconductor: a uniform superconducting state is formed at sufficiently low magnetic fields, a vortex state for intermediate magnetic fields, and finally a normal state for large magnetic fields. In each of these phases (including the vortex phase), ${\pi }^0$ remains uncondensed just as in the zero-external-field problem. The critical magnetic field where the phase transition from the uniform superconducting state to a vortex state occurs was found numerically.

Figure 1

The region where $m_{\mathrm{A}}>m_{\theta }$ is shown in blue, the region where $m_{\theta }>m_{\mathrm{A}}$ is shown in orange, and the green region represents the normal phase, i.e., ${\mu }_{\mathrm{I}}\le m_{\pi }$.

Figure 2

The plot shows the density of the charged pion as a function of radial distance from the center of the vortex, which is at the origin. The result is for ${\tilde{m}}_{\pi }=1.5$ and ${\tilde{\mu }}_I=2.0$.

Figure 3

The plot shows the magnetic field, i.e., $\vec{B}\equiv \vec{\nabla }\times \vec{A}$ as a function of radial distance from the center of the vortex. The result is for ${\tilde{m}}_{\pi }=1.5$ and ${\tilde{\mu }}_I=2.0$.

Figure 4

The plot shows the critical magnetic field for the formation of the vortex state for realistic pion mass ${\tilde{m}}_{\pi }=1.5$ as a function of isospin density.