Wave Function and Emergent SU(2) Symmetry in $\nu_T=1$ Quantum Hall Bilayer*

We propose a trial wave function for the quantum Hall bilayer system of total filling factor $\nu_T=1$ at a layer distance $d$ to magnetic length $\ell$ ratio $d/\ell=\kappa_{c1}\approx1.1$, where the lowest charged excitation is known to have a level crossing. The wave function has two-particle correlations which fit well with those in previous numerical studies, and can be viewed as a Bose-Einstein condensate of free excitons formed by composite bosons and anti-composite bosons in different layers. We show the free nature of these excitons indicating an emergent SU(2) symmetry for the composite bosons at $d/\ell=\kappa_{c1}$, which leads to the level crossing in low-lying charged excitations. We further show the overlap between the trial wave function and the ground state of a small size exact diagonalization is peaked near $d/\ell=\kappa_{c1}$, which supports our theory.