Spin mapping, phase diagram, and collective modes in double layer quantum Hall systems at $\nu =2$

An exact spin mapping is identified to simplify the recently proposed hard-core boson description (Demler and Das Sarma, Phys. Rev. Lett., to be published) of the bilayer quantum Hall system at filling factor 2. The effective spin model describes an easy-plane ferromagnet subject to an external Zeeman field. The phase diagram of this effective model is determined exactly and found to agree with the approximate calculation of Demler and Das Sarma, while the Goldstone-mode spectrum, order parameter stiffness and Kosterlitz-Thouless temperature in the canted antiferromagnetic phase are computed approximately.