Effective gap equation for the inhomogeneous Larkin-Ovchinnikov-Fulde-Ferrel superconductive phase

Abstract

We present an approximate gap equation for different crystalline structures of the Larkin-Ovchinnikov-Fulde-Ferrel phase of high density QCD at $T=0$. This equation is derived by using an effective condensate term obtained by averaging the inhomogeneous condensate over distances of the order of the crystal lattice size. The approximation is expected to work better far off any second-order phase transition. As a function of the difference of the chemical potentials of the up and down quarks, $\ensuremath{\delta}\ensuremath{\mu}$, we get that the octahedron is energetically favored from $\ensuremath{\delta}\ensuremath{\mu}={\ensuremath{\Delta}}_{0}/\sqrt{2}$ to $0.95{\ensuremath{\Delta}}_{0}$, where ${\ensuremath{\Delta}}_{0}$ is the gap for the homogeneous phase, while in the range $0.95{\ensuremath{\Delta}}_{0}--1.32{\ensuremath{\Delta}}_{0}$ the face-centered cube prevails. At $\ensuremath{\delta}\ensuremath{\mu}=1.32{\ensuremath{\Delta}}_{0}$ a first-order phase transition to the normal phase occurs.