Pseudospin vortex-antivortex states with interwoven spin textures in double layer quantum Hall systems

Recent experiments on strongly correlated bilayer quantum Hall systems strongly suggest that, contrary to the usual assumption, the electron spin degree of freedom is not completely frozen either in the quantum Hall or in the compressibles states that occur at filling factor $\nu =1.$ These experiments imply that the quasiparticles at $\nu =1$ could have both spin and pseudospin textures i.e. they could be CP$^{3}$ skyrmions. Using a microscopic unrestricted Hartree-Fock approximation, we compute the energy of several crystal states with spin, pseudospin and mixed spin-pseudospin textures around $\nu =1$ as a function of interlayer separation $d$ for different values of tunneling ($\Delta_{SAS}$), Zeeman ($\Delta_{Z}$), and bias ($\Delta_{b}$) energies. We show that in some range of these parameters, crystal states involving a certain amount of spin depolarization have lower energy than the fully spin polarized crystals. We study this depolarization dependence on $d,\Delta_{SAS},\Delta_{Z}$ and $\Delta_{b}$ and discuss how it can lead to the fast NMR relaxation rate observed experimentally.