Phonon modes of magnetic vortex lattices in finite isospin chiral perturbation theory

We study phonon modes associated with magnetic vortex lattices of finite isospin chiral perturbation theory near the upper critical point by introducing quasimomentum fluctuations to the lattice and calculate dispersion relations associated with the optical and acoustic modes. We find that one of the acoustic modes is massless and that its energy for small transverse quasimomentum is quartic (due the presence of an isospin chemical potential), which is significantly softer than the "supersoft" (quadratic) massless mode of the Abelian Higgs Model (AHM). Due to the presence of derivative interactions, which is absent in the AHM, the speed of the longitudinal mode depends on both the isospin chemical potential and the external magnetic field. Our results suggest that the standard assumption of an ordered lattice in finite isospin QCD should be revisited and the existence of a disordered spaghetti phase of a vortex liquid or gas, should be considered.