Interlayer exchange interactions, $\mathrm{SU}\mathrm{}\left(4\right)\mathrm{}$ soft waves, and skyrmions in bilayer quantum Hall ferromagnets

The exchange Coulomb interaction is the driving force for quantum coherence in quantum Hall systems. We construct a microscopic Landau-site Hamiltonian for the exchange interaction in bilayer quantum Hall ferromagnets, which is characterized by the SU(4) isospin structure. By taking a continuous limit, the Hamiltonian gives rise to the SU(4) nonlinear sigma model in the von Neumann–lattice formulation. The ground-state energy is evaluated at filling factors $\nu =1,2,3,4.\mathrm{}$ It is shown at $\nu =1\mathrm{}$ that there are three independent soft waves, where only one soft wave is responsible for the coherent tunneling of electrons between the two layers. It is also shown at $\nu =1\mathrm{}$ that there are three independent skyrmion states apart from the translational degree of freedom. They are ${\mathrm{CP}}^3$ skyrmions enjoying the spin-charge entanglement confined within the lowest Landau level.