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We propose a trial wave function for the quantum Hall bilayer system of totalfilling 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 isknown to have a level crossing. The wave function has two-particle correlationswhich fit well with those in previous numerical studies, and can be viewed as aBose-Einstein condensate of free excitons formed by composite bosons andanti-composite bosons in different layers. We show the free nature of theseexcitons indicating an emergent SU(2) symmetry for the composite bosons at$d/\ell=\kappa_{c1}$, which leads to the level crossing in low-lying chargedexcitations. We further show the overlap between the trial wave function andthe ground state of a small size exact diagonalization is peaked near$d/\ell=\kappa_{c1}$, which supports our theory.

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